- - + - 1. For Loops + 1. for Loop @@ -718,7 +718,7 @@
-
- 2. While Loop
+ 2. while Loop
@@ -749,9 +749,9 @@
- - + - 1. Call The Stack + 1. Call Stack @@ -781,9 +781,9 @@
-
-
+
- 2.2.3 Compare The Two
+ 2.2.3 Comparison
@@ -905,9 +905,9 @@
- - + - 1. For Loops + 1. for Loop @@ -916,7 +916,7 @@
-
- 2. While Loop
+ 2. while Loop
@@ -947,9 +947,9 @@
- - + - 1. Call The Stack + 1. Call Stack @@ -979,9 +979,9 @@
-
-
+
- 2.2.3 Compare The Two
+ 2.2.3 Comparison
@@ -1023,13 +1023,13 @@
-
2.2 Iteration vs. Recursion¶
-In data structures and algorithms, it is common to repeat a task, which is closely related to the complexity of the algorithm. There are two basic program structures that we usually use to repeat a task: iteration and recursion.
+2.2 Iteration and Recursion¶
+In algorithms, repeatedly performing a task is common and closely related to complexity analysis. Therefore, before introducing time complexity and space complexity, let's first understand how to implement task repetition in programs, focusing on two basic programming control structures: iteration and recursion.
2.2.1 Iteration¶
-An "iteration iteration" is a control structure that repeats a task. In iteration, a program repeats the execution of a piece of code until the condition is no longer satisfied.
-1. For Loops¶
-
-forloops are one of the most common forms of iteration, suitable when the number of iterations is known in advance.The following function implements the summation \(1 + 2 + \dots + n\) based on a
+forloop, and the result is recorded using the variableres. Note thatrange(a, b)in Python corresponds to a "left-closed-right-open" interval, which is traversed in the range \(a, a + 1, \dots, b-1\)."Iteration" is a control structure for repeatedly performing a task. In iteration, a program repeats a block of code as long as a certain condition is met, until this condition is no longer satisfied.
+1. for Loop¶
+The
+forloop is one of the most common forms of iteration, suitable for use when the number of iterations is known in advance.The following function implements the sum \(1 + 2 + \dots + n\) using a
forloop, with the sum result recorded in the variableres. Note that in Python,range(a, b)corresponds to a "left-closed, right-open" interval, covering \(a, a + 1, \dots, b-1\):-@@ -1176,14 +1176,14 @@The Figure 2-1 shows the flow block diagram of this summation function.
-
-
Figure 2-1 Flow block diagram of the summation function
+The flowchart below represents this sum function.
+
+Figure 2-1 Flowchart of the Sum Function
-The number of operations in this summation function is proportional to the size of the input data \(n\), or a "linear relationship". In fact, time complexity describes this "linear relationship". This is described in more detail in the next section.
-2. While Loop¶
-Similar to a
-forloop, awhileloop is a way to implement iteration. In awhileloop, the program first checks the condition at each turn, and if the condition is true, it continues, otherwise it ends the loop.Below, we use a
+whileloop to realize the summation \(1 + 2 + \dots + n\) .The number of operations in this sum function is proportional to the input data size \(n\), or in other words, it has a "linear relationship". This is actually what time complexity describes. This topic will be detailed in the next section.
+2. while Loop¶
+Similar to the
+forloop, thewhileloop is another method to implement iteration. In awhileloop, the program checks the condition in each round; if the condition is true, it continues, otherwise, the loop ends.Below we use a
whileloop to implement the sum \(1 + 2 + \dots + n\):-@@ -1356,8 +1356,8 @@In
-whileloops, since the steps of initializing and updating condition variables are independent of the loop structure, it has more degrees of freedom thanforloops.For example, in the following code, the condition variable \(i\) is updated twice per round, which is not convenient to implement with a
+forloop.The
+whileloop is more flexible than theforloop. In awhileloop, we can freely design the initialization and update steps of the condition variable.For example, in the following code, the condition variable \(i\) is updated twice in each round, which would be inconvenient to implement with a
forloop:-@@ -1553,9 +1553,9 @@Overall,
+forloops have more compact code andwhileloops are more flexible, and both can implement iteration structures. The choice of which one to use should be based on the needs of the particular problem.Overall,
forloops are more concise, whilewhileloops are more flexible. Both can implement iterative structures. Which one to use should be determined based on the specific requirements of the problem.3. Nested Loops¶
-We can nest one loop structure inside another, using the
+forloop as an example:We can nest one loop structure within another. Below is an example using
forloops:-@@ -1744,25 +1744,25 @@The Figure 2-2 gives the block diagram of the flow of this nested loop.
-
-
Figure 2-2 Block diagram of the flow of nested loops
+The flowchart below represents this nested loop.
+
+Figure 2-2 Flowchart of the Nested Loop
-In this case, the number of operations of the function is proportional to \(n^2\), or the algorithm's running time is "squared" to the size of the input data \(n\).
-We can continue to add nested loops, and each nest is a "dimension up", which will increase the time complexity to "cubic relations", "quadratic relations", and so on.
+In this case, the number of operations in the function is proportional to \(n^2\), or the algorithm's running time and the input data size \(n\) have a "quadratic relationship".
+We can continue adding nested loops, each nesting is a "dimensional escalation," which will increase the time complexity to "cubic," "quartic," and so on.
2.2.2 Recursion¶
-"Recursion recursion is an algorithmic strategy to solve a problem by calling the function itself. It consists of two main phases.
+"Recursion" is an algorithmic strategy that solves problems by having a function call itself. It mainly consists of two phases.
-
-
- recursive: the program calls itself deeper and deeper, usually passing smaller or simpler arguments, until a "termination condition" is reached. -
- Recursion: After the "termination condition" is triggered, the program returns from the deepest level of the recursion function, level by level, aggregating the results of each level. +
- Recursion: The program continuously calls itself, usually with smaller or more simplified parameters, until reaching a "termination condition." +
- Return: Upon triggering the "termination condition," the program begins to return from the deepest recursive function, aggregating the results of each layer.
And from an implementation point of view, recursion code contains three main elements.
+From an implementation perspective, recursive code mainly includes three elements.
-
-
- Termination condition: used to decide when to switch from "recursive" to "inductive". -
- Recursion call: corresponds to "recursion", where the function calls itself, usually with smaller or more simplified input parameters. -
- return result: corresponds to "return", returning the result of the current recursion level to the previous one. +
- Termination Condition: Determines when to switch from "recursion" to "return." +
- Recursive Call: Corresponds to "recursion," where the function calls itself, usually with smaller or more simplified parameters. +
- Return Result: Corresponds to "return," where the result of the current recursion level is returned to the previous layer.
Observe the following code, we only need to call the function
+recur(n), and the calculation of \(1 + 2 + \dots + n\) is done:Observe the following code, where calling the function
recur(n)completes the computation of \(1 + 2 + \dots + n\):-@@ -1923,38 +1923,38 @@The Figure 2-3 shows the recursion of the function.
-
-
Figure 2-3 Recursion process for the summation function
+The Figure 2-3 shows the recursive process of this function.
+
+Figure 2-3 Recursive Process of the Sum Function
-Although iteration and recursion can yield the same results from a computational point of view, they represent two completely different paradigms for thinking about and solving problems.
+Although iteration and recursion can achieve the same results from a computational standpoint, they represent two entirely different paradigms of thinking and solving problems.
-
-
- Iteration: solving problems "from the bottom up". Start with the most basic steps and repeat or add to them until the task is completed. -
- Recursion: solving problems "from the top down". The original problem is broken down into smaller subproblems that have the same form as the original problem. Next, the subproblem continues to be broken down into smaller subproblems until it stops at the base case (the solution to the base case is known). +
- Iteration: Solves problems "from the bottom up." It starts with the most basic steps, then repeatedly adds or accumulates these steps until the task is complete. +
- Recursion: Solves problems "from the top down." It breaks down the original problem into smaller sub-problems, each of which has the same form as the original problem. These sub-problems are then further decomposed into even smaller sub-problems, stopping at the base case (whose solution is known).
As an example of the above summation function, set the problem \(f(n) = 1 + 2 + \dots + n\) .
+Taking the sum function as an example, let's define the problem as \(f(n) = 1 + 2 + \dots + n\).
-
-
- Iteration: the summation process is simulated in a loop, iterating from \(1\) to \(n\) and executing the summation operation in each round to find \(f(n)\). -
- Recursion: decompose the problem into subproblems \(f(n) = n + f(n-1)\) and keep (recursively) decomposing until the base case \(f(1) = 1\) terminates. +
- Iteration: In a loop, simulate the summing process, iterating from \(1\) to \(n\), performing the sum operation in each round, to obtain \(f(n)\). +
- Recursion: Break down the problem into sub-problems \(f(n) = n + f(n-1)\), continuously (recursively) decomposing until reaching the base case \(f(1) = 1\) and then stopping.
1. Call The Stack¶
-Each time a recursion function calls itself, the system allocates memory for the newly opened function to store local variables, call addresses, other information, and so on. This results in two things.
+1. Call Stack¶
+Each time a recursive function calls itself, the system allocates memory for the newly initiated function to store local variables, call addresses, and other information. This leads to two main consequences.
-
-
- The context data for a function is stored in an area of memory called "stack frame space" and is not freed until the function returns. As a result, recursion is usually more memory-intensive than iteration. -
- Recursion calls to functions incur additional overhead. Therefore recursion is usually less time efficient than loops. +
- The function's context data is stored in a memory area called "stack frame space" and is only released after the function returns. Therefore, recursion generally consumes more memory space than iteration. +
- Recursive calls introduce additional overhead. Hence, recursion is usually less time-efficient than loops.
As shown in the Figure 2-4 , before the termination condition is triggered, there are \(n\) unreturned recursion functions at the same time, with a recursion depth of \(n\) .
-
-
Figure 2-4 Recursion call depth
+As shown in the Figure 2-4 , there are \(n\) unreturned recursive functions before triggering the termination condition, indicating a recursion depth of \(n\).
+
+Figure 2-4 Recursion Call Depth
-In practice, the depth of recursion allowed by a programming language is usually limited, and too deep a recursion may result in a stack overflow error.
+In practice, the depth of recursion allowed by programming languages is usually limited, and excessively deep recursion can lead to stack overflow errors.
2. Tail Recursion¶
-Interestingly, if a function makes a recursion call only at the last step before returning, the function can be optimized by the compiler or interpreter to be comparable to iteration in terms of space efficiency. This situation is called "tail recursion tail recursion".
+Interestingly, if a function makes its recursive call as the last step before returning, it can be optimized by compilers or interpreters to be as space-efficient as iteration. This scenario is known as "tail recursion".
-
-
- Ordinary recursion: when a function returns to a function at a higher level, it needs to continue executing the code, so the system needs to save the context of the previous call. -
- tail recursion: the recursion call is the last operation before the function returns, which means that the function does not need to continue with other operations after returning to the previous level, so the system does not need to save the context of the previous function. +
- Regular Recursion: The function needs to perform more code after returning to the previous level, so the system needs to save the context of the previous call. +
- Tail Recursion: The recursive call is the last operation before the function returns, meaning no further actions are required upon returning to the previous level, so the system doesn't need to save the context of the previous level's function.
In the case of calculating \(1 + 2 + \dots + n\), for example, we can implement tail recursion by setting the result variable
+resas a function parameter.For example, in calculating \(1 + 2 + \dots + n\), we can make the result variable
resa parameter of the function, thereby achieving tail recursion:-@@ -2091,30 +2091,30 @@The execution of tail recursion is shown in the Figure 2-5 . Comparing normal recursion and tail recursion, the execution point of the summation operation is different.
+The execution process of tail recursion is shown in the following figure. Comparing regular recursion and tail recursion, the point of the summation operation is different.
-
-
- Ordinary recursion: the summing operation is performed during the "return" process, and the summing operation is performed again after returning from each level. -
- Tail recursion: the summing operation is performed in a "recursion" process, the "recursion" process simply returns in levels. +
- Regular Recursion: The summation operation occurs during the "return" phase, requiring another summation after each layer returns. +
- Tail Recursion: The summation operation occurs during the "recursion" phase, and the "return" phase only involves returning through each layer.
-
Figure 2-5 tail recursion process
+
+Figure 2-5 Tail Recursion Process
Tip
-Note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if a function is tail recursive, you may still encounter stack overflow problems.
+Note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if the function is in the form of tail recursion, it may still encounter stack overflow issues.
3. Recursion Tree¶
-When dealing with algorithmic problems related to divide and conquer, recursion is often more intuitive and easier to read than iteration. Take the Fibonacci sequence as an example.
+When dealing with algorithms related to "divide and conquer", recursion often offers a more intuitive approach and more readable code than iteration. Take the "Fibonacci sequence" as an example.
-Question
-Given a Fibonacci series \(0, 1, 1, 2, 3, 5, 8, 13, \dots\) , find the \(n\)th number of the series.
+Given a Fibonacci sequence \(0, 1, 1, 2, 3, 5, 8, 13, \dots\), find the \(n\)th number in the sequence.
Let the \(n\)th number of the Fibonacci series be \(f(n)\) , which leads to two easy conclusions.
+Let the \(n\)th number of the Fibonacci sequence be \(f(n)\), it's easy to deduce two conclusions:
-
-
- The first two numbers of the series are \(f(1) = 0\) and \(f(2) = 1\). -
- Each number in the series is the sum of the previous two numbers, i.e. \(f(n) = f(n - 1) + f(n - 2)\) . +
- The first two numbers of the sequence are \(f(1) = 0\) and \(f(2) = 1\). +
- Each number in the sequence is the sum of the two preceding ones, that is, \(f(n) = f(n - 1) + f(n - 2)\).
Recursion code can be written by making recursion calls according to the recursion relationship, using the first two numbers as termination conditions. Call
+fib(n)to get the \(n\)th number of the Fibonacci series.Using the recursive relation, and considering the first two numbers as termination conditions, we can write the recursive code. Calling
fib(n)will yield the \(n\)th number of the Fibonacci sequence:-@@ -2275,63 +2275,63 @@Looking at the above code, we have recursively called two functions within a function, this means that from one call, two call branches are created. As shown in the Figure 2-6 , this recursion will result in a recursion tree with the number of levels \(n\).
-
-
Figure 2-6 Recursion tree for Fibonacci series
+Observing the above code, we see that it recursively calls two functions within itself, meaning that one call generates two branching calls. As illustrated below, this continuous recursive calling eventually creates a "recursion tree" with a depth of \(n\).
+
+Figure 2-6 Fibonacci Sequence Recursion Tree
-Essentially, recursion embodies the paradigm of "breaking down a problem into smaller sub-problems", and this divide and conquer strategy is essential.
+Fundamentally, recursion embodies the paradigm of "breaking down a problem into smaller sub-problems." This divide-and-conquer strategy is crucial.
-
-
- From an algorithmic point of view, many important algorithmic strategies such as searching, sorting algorithm, backtracking, divide and conquer, dynamic programming, etc. directly or indirectly apply this way of thinking. -
- From a data structure point of view, recursion is naturally suited to problems related to linked lists, trees and graphs because they are well suited to be analyzed with the idea of partitioning. +
- From an algorithmic perspective, many important strategies like searching, sorting, backtracking, divide-and-conquer, and dynamic programming directly or indirectly use this way of thinking. +
- From a data structure perspective, recursion is naturally suited for dealing with linked lists, trees, and graphs, as they are well suited for analysis using the divide-and-conquer approach.
2.2.3 Compare The Two¶
-To summarize the above, as shown in the Table 2-1 , iteration and recursion differ in implementation, performance and applicability.
-Table 2-1 Comparison of iteration and recursion features
+2.2.3 Comparison¶
+Summarizing the above content, the following table shows the differences between iteration and recursion in terms of implementation, performance, and applicability.
+Table: Comparison of Iteration and Recursion Characteristics
- iteration -recursion +Iteration +Recursion - implementation -circular structure -function call itself +Approach +Loop structure +Function calls itself - time-efficient -typically efficient, no function call overhead -overhead on every function call +Time Efficiency +Generally higher efficiency, no function call overhead +Each function call generates overhead Memory Usage -Usually uses a fixed size of memory space -Cumulative function calls may use a lot of stack frame space +Typically uses a fixed size of memory space +Accumulative function calls can use a substantial amount of stack frame space - Applicable Problems -For simple cyclic tasks, code is intuitive and readable -For sub-problem decomposition, such as trees, graphs, divide and conquer, backtracking, etc., the code structure is concise and clear +Suitable Problems +Suitable for simple loop tasks, intuitive and readable code +Suitable for problem decomposition, like trees, graphs, divide-and-conquer, backtracking, etc., concise and clear code structure -Tip
-If you find the following solutions difficult to understand, you can review them after reading the "Stack" chapter.
+If you find the following content difficult to understand, consider revisiting it after reading the "Stack" chapter.
So what is the intrinsic connection between iteration and recursion? In the case of the recursive function described above, the summing operation takes place in the "return" phase of the recursion. This means that the function that is initially called is actually the last to complete its summing operation, This mechanism works in the same way as the stack's "first in, last out" principle.
-In fact, recursion terms like "call stack" and "stack frame space" already imply a close relationship between recursion and the stack.
+So, what is the intrinsic connection between iteration and recursion? Taking the above recursive function as an example, the summation operation occurs during the recursion's "return" phase. This means that the initially called function is actually the last to complete its summation operation, mirroring the "last in, first out" principle of a stack.
+In fact, recursive terms like "call stack" and "stack frame space" hint at the close relationship between recursion and stacks.
-
-
- Recursive: When a function is called, the system allocates a new stack frame on the "call stack" for the function, which is used to store the function's local variables, parameters, return address, and other data. -
- Return to: When a function completes execution and returns, the corresponding stack frame is removed from the "call stack", restoring the function's previous execution environment. +
- Recursion: When a function is called, the system allocates a new stack frame on the "call stack" for that function, storing local variables, parameters, return addresses, and other data. +
- Return: When a function completes execution and returns, the corresponding stack frame is removed from the "call stack," restoring the execution environment of the previous function.
Thus, we can use an explicit stack to model the behavior of the call stack, thus transforming recursion into an iteration form:
+Therefore, we can use an explicit stack to simulate the behavior of the call stack, thus transforming recursion into an iterative form:
-@@ -2589,12 +2589,12 @@Observing the code above, it becomes more complex when recursion is converted to iteration. Although iteration and recursion can be converted to each other in many cases, it is not always worth doing so for two reasons.
+Observing the above code, when recursion is transformed into iteration, the code becomes more complex. Although iteration and recursion can often be transformed into each other, it's not always advisable to do so for two reasons:
-
-
- The transformed code may be more difficult to understand and less readable. -
- For some complex problems, simulating the behavior of the system call stack can be very difficult. +
- The transformed code may become harder to understand and less readable. +
- For some complex problems, simulating the behavior of the system's call stack can be quite challenging.
In short, the choice of iteration or recursion depends on the nature of the particular problem. In programming practice, it is crucial to weigh the advantages and disadvantages of both and choose the appropriate method based on the context.
+In summary, choosing between iteration and recursion depends on the nature of the specific problem. In programming practice, weighing the pros and cons of each and choosing the appropriate method for the situation is essential.
diff --git a/en/chapter_computational_complexity/performance_evaluation/index.html b/en/chapter_computational_complexity/performance_evaluation/index.html index 0ae003018..2a035ef45 100644 --- a/en/chapter_computational_complexity/performance_evaluation/index.html +++ b/en/chapter_computational_complexity/performance_evaluation/index.html @@ -91,7 +91,7 @@- + Skip to content @@ -676,9 +676,9 @@-
-
+
- 2.1.1 Practical Testing
+ 2.1.1 Actual Testing
@@ -820,9 +820,9 @@
-
-
+
- 2.1.1 Practical Testing
+ 2.1.1 Actual Testing
@@ -873,43 +873,43 @@
-
2.1 Evaluation Of Algorithm Efficiency¶
-In algorithm design, we aim to achieve two goals in succession:
+2.1 Algorithm Efficiency Assessment¶
+In algorithm design, we pursue the following two objectives in sequence.
-
-
- Finding a Solution to the Problem: The algorithm needs to reliably find the correct solution within the specified input range. -
- Seeking the Optimal Solution: There may be multiple ways to solve the same problem, and our goal is to find the most efficient algorithm possible. +
- Finding a Solution to the Problem: The algorithm should reliably find the correct solution within the stipulated range of inputs. +
- Seeking the Optimal Solution: For the same problem, multiple solutions might exist, and we aim to find the most efficient algorithm possible.
In other words, once the ability to solve the problem is established, the efficiency of the algorithm emerges as the main benchmark for assessing its quality, which includes the following two aspects.
+In other words, under the premise of being able to solve the problem, algorithm efficiency has become the main criterion for evaluating the merits of an algorithm, which includes the following two dimensions.
- Time Efficiency: The speed at which an algorithm runs. -
- Space Efficiency: The amount of memory space the algorithm consumes. +
- Space Efficiency: The size of the memory space occupied by an algorithm.
In short, our goal is to design data structures and algorithms that are both "fast and economical". Effectively evaluating algorithm efficiency is crucial, as it allows for the comparison of different algorithms and guides the design and optimization process.
-There are mainly two approaches for assessing efficiency: practical testing and theoretical estimation.
-2.1.1 Practical Testing¶
-Let's consider a scenario where we have two algorithms,
-AandB, both capable of solving the same problem. To compare their efficiency, the most direct method is to use a computer to run both algorithms while monitoring and recording their execution time and memory usage. This approach provides a realistic assessment of their performance, but it also has significant limitations.On one hand, it's challenging to eliminate the interference of the test environment. Hardware configurations can significantly affect the performance of algorithms. For instance, on one computer, Algorithm
-Amight run faster than AlgorithmB, but the results could be the opposite on another computer with different specifications. This means we would need to conduct tests on a variety of machines and calculate an average efficiency, which is impractical.Furthermore, conducting comprehensive tests is resource-intensive. The efficiency of algorithms can vary with different volumes of input data. For example, with smaller data sets, Algorithm A might run faster than Algorithm B; however, this could change with larger data sets. Therefore, to reach a convincing conclusion, it's necessary to test a range of data sizes, which requires excessive computational resources.
+In short, our goal is to design data structures and algorithms that are both fast and memory-efficient. Effectively assessing algorithm efficiency is crucial because only then can we compare various algorithms and guide the process of algorithm design and optimization.
+There are mainly two methods of efficiency assessment: actual testing and theoretical estimation.
+2.1.1 Actual Testing¶
+Suppose we have algorithms
+AandB, both capable of solving the same problem, and we need to compare their efficiencies. The most direct method is to use a computer to run these two algorithms and monitor and record their runtime and memory usage. This assessment method reflects the actual situation but has significant limitations.On one hand, it's difficult to eliminate interference from the testing environment. Hardware configurations can affect algorithm performance. For example, algorithm
+Amight run faster thanBon one computer, but the opposite result may occur on another computer with different configurations. This means we would need to test on a variety of machines to calculate average efficiency, which is impractical.On the other hand, conducting a full test is very resource-intensive. As the volume of input data changes, the efficiency of the algorithms may vary. For example, with smaller data volumes, algorithm
Amight run faster thanB, but the opposite might be true with larger data volumes. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires significant computational resources.2.1.2 Theoretical Estimation¶
-Given the significant limitations of practical testing, we can consider assessing algorithm efficiency solely through calculations. This method of estimation is known as 'asymptotic complexity analysis,' often simply referred to as 'complexity analysis.
-Complexity analysis illustrates the relationship between the time (and space) resources required by an algorithm and the size of its input data. It describes the growing trend in the time and space required for the execution of an algorithm as the size of the input data increases. This definition might sound a bit complex, so let's break it down into three key points for easier understanding.
+Due to the significant limitations of actual testing, we can consider evaluating algorithm efficiency solely through calculations. This estimation method is known as "asymptotic complexity analysis," or simply "complexity analysis."
+Complexity analysis reflects the relationship between the time and space resources required for algorithm execution and the size of the input data. It describes the trend of growth in the time and space required by the algorithm as the size of the input data increases. This definition might sound complex, but we can break it down into three key points to understand it better.
-
-
- In complexity analysis, 'time and space' directly relate to 'time complexity' and 'space complexity,' respectively. -
- The statement "as the size of the input data increases" highlights that complexity analysis examines the interplay between the size of the input data and the algorithm's efficiency. -
- Lastly, the phrase "the growing trend in time and space required" emphasizes that the focus of complexity analysis is not on the specific values of running time or space occupied, but on the rate at which these requirements increase with larger input sizes. +
- "Time and space resources" correspond to "time complexity" and "space complexity," respectively. +
- "As the size of input data increases" means that complexity reflects the relationship between algorithm efficiency and the volume of input data. +
- "The trend of growth in time and space" indicates that complexity analysis focuses not on the specific values of runtime or space occupied but on the "rate" at which time or space grows.
Complexity analysis overcomes the drawbacks of practical testing methods in two key ways:.
+Complexity analysis overcomes the disadvantages of actual testing methods, reflected in the following aspects:
-
-
- It is independent of the testing environment, meaning the analysis results are applicable across all operating platforms. -
- It effectively demonstrates the efficiency of algorithms with varying data volumes, particularly highlighting performance in large-scale data scenarios. +
- It is independent of the testing environment and applicable to all operating platforms. +
- It can reflect algorithm efficiency under different data volumes, especially in the performance of algorithms with large data volumes.
-Tip
-If you're still finding the concept of complexity confusing, don't worry. We will cover it in more detail in the subsequent chapters.
+If you're still confused about the concept of complexity, don't worry. We will introduce it in detail in subsequent chapters.
Complexity analysis provides us with a 'ruler' for evaluating the efficiency of algorithms, enabling us to measure the time and space resources required to execute a given algorithm and to compare the efficiency of different algorithms.
-Complexity is a mathematical concept that might seem abstract and somewhat challenging for beginners. From this perspective, introducing complexity analysis at the very beginning may not be the most suitable approach. However, when discussing the characteristics of a particular data structure or algorithm, analyzing its operational speed and space usage is often inevitable.
-Therefore, it is recommended that before diving deeply into data structures and algorithms, one should first gain a basic understanding of complexity analysis. This foundational knowledge will facilitate the complexity analysis of simple algorithms.
+Complexity analysis provides us with a "ruler" to measure the time and space resources needed to execute an algorithm and compare the efficiency between different algorithms.
+Complexity is a mathematical concept and may be abstract and challenging for beginners. From this perspective, complexity analysis might not be the best content to introduce first. However, when discussing the characteristics of a particular data structure or algorithm, it's hard to avoid analyzing its speed and space usage.
+In summary, it's recommended that you establish a preliminary understanding of complexity analysis before diving deep into data structures and algorithms, so that you can carry out simple complexity analyses of algorithms.
diff --git a/en/chapter_computational_complexity/space_complexity/index.html b/en/chapter_computational_complexity/space_complexity/index.html index 2380128ff..de15c2e05 100644 --- a/en/chapter_computational_complexity/space_complexity/index.html +++ b/en/chapter_computational_complexity/space_complexity/index.html @@ -739,9 +739,9 @@- - + - 2.4.1 Algorithmic Correlation Space + 2.4.1 Space Related to Algorithms @@ -778,7 +778,7 @@
- - 2. Linear Order \(O(N)\) + 2. Linear Order \(O(n)\) @@ -787,7 +787,7 @@
- - 3. Quadratic Order \(O(N^2)\) + 3. Quadratic Order \(O(n^2)\) @@ -796,7 +796,7 @@
- - 4. Exponential Order \(O(2^N)\) + 4. Exponential Order \(O(2^n)\) @@ -805,7 +805,7 @@
- - 5. Logarithmic Order \(O(\Log N)\) + 5. Logarithmic Order \(O(\log n)\) @@ -817,9 +817,9 @@
-
-
+
- 2.4.4 Weighing Time And Space
+ 2.4.4 Balancing Time and Space
@@ -889,9 +889,9 @@
- - + - 2.4.1 Algorithmic Correlation Space + 2.4.1 Space Related to Algorithms @@ -928,7 +928,7 @@
- - 2. Linear Order \(O(N)\) + 2. Linear Order \(O(n)\) @@ -937,7 +937,7 @@
- - 3. Quadratic Order \(O(N^2)\) + 3. Quadratic Order \(O(n^2)\) @@ -946,7 +946,7 @@
- - 4. Exponential Order \(O(2^N)\) + 4. Exponential Order \(O(2^n)\) @@ -955,7 +955,7 @@
- - 5. Logarithmic Order \(O(\Log N)\) + 5. Logarithmic Order \(O(\log n)\) @@ -967,9 +967,9 @@
-
-
+
- 2.4.4 Weighing Time And Space
+ 2.4.4 Balancing Time and Space
@@ -1012,25 +1012,26 @@
2.4 Space Complexity¶
-The space complexity is used to measure the growth trend of memory consumption as the scale of data increases for an algorithm solution. This concept is analogous to time complexity by replacing "runtime" with "memory space".
-2.4.1 Algorithmic Correlation Space¶
-The memory space used by algorithms during its execution include the following types.
+"Space complexity" is used to measure the growth trend of the memory space occupied by an algorithm as the amount of data increases. This concept is very similar to time complexity, except that "running time" is replaced with "occupied memory space".
+2.4.1 Space Related to Algorithms¶
+The memory space used by an algorithm during its execution mainly includes the following types.
-
-
- Input Space: Used to store the input data for the algorithm. -
- Temporary Space: Used to store variables, objects, function contexts, and other data of the algorithm during runtime. +
- Input Space: Used to store the input data of the algorithm. +
- Temporary Space: Used to store variables, objects, function contexts, and other data during the algorithm's execution.
- Output Space: Used to store the output data of the algorithm.
In general, the "Input Space" is excluded from the statistics of space complexity.
-The Temporary Space can be further divided into three parts.
+Generally, the scope of space complexity statistics includes both "Temporary Space" and "Output Space".
+Temporary space can be further divided into three parts.
-
-
- Temporary Data: Used to store various constants, variables, objects, etc., during the the algorithm's execution. -
- Stack Frame Space: Used to hold the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is freed when the function returns. -
- Instruction Space: Used to hold compiled program instructions, usually ignored in practical statistics. +
- Temporary Data: Used to save various constants, variables, objects, etc., during the algorithm's execution. +
- Stack Frame Space: Used to save the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns. +
- Instruction Space: Used to store compiled program instructions, which are usually negligible in actual statistics.
When analyzing the space complexity of a piece of program, three parts are usually taken into account: Temporary Data, Stack Frame Space and Output Data.
-
-
Figure 2-15 Associated spaces used by the algorithm
+When analyzing the space complexity of a program, we typically count the Temporary Data, Stack Frame Space, and Output Data, as shown in the Figure 2-15 .
+
+Figure 2-15 Space Types Used in Algorithms
+The relevant code is as follows:
-@@ -1304,12 +1305,12 @@2.4.2 Calculation Method¶
-The calculation method for space complexity is pretty similar to time complexity, with the only difference being that the focus shifts from "operation count" to "space usage size".
-On top of that, unlike time complexity, we usually only focus on the worst-case space complexity. This is because memory space is a hard requirement, and we have to make sure that there is enough memory space reserved for all possibilities incurred by input data.
-Looking at the following code, the "worst" in worst-case space complexity has two layers of meaning.
+The method for calculating space complexity is roughly similar to that of time complexity, with the only change being the shift of the statistical object from "number of operations" to "size of used space".
+However, unlike time complexity, we usually only focus on the worst-case space complexity. This is because memory space is a hard requirement, and we must ensure that there is enough memory space reserved under all input data.
+Consider the following code, the term "worst-case" in worst-case space complexity has two meanings.
-
-
- Based on the worst-case input data: when \(n < 10\), the space complexity is \(O(1)\); however, when \(n > 10\), the initialized array
numsoccupies \(O(n)\) space; thus the worst-case space complexity is \(O(n)\).
- - Based on the peak memory during algorithm execution: for example, the program occupies \(O(1)\) space until the last line is executed; when the array
numsis initialized, the program occupies \(O(n)\) space; thus the worst-case space complexity is \(O(n)\).
+ - Based on the worst input data: When \(n < 10\), the space complexity is \(O(1)\); but when \(n > 10\), the initialized array
numsoccupies \(O(n)\) space, thus the worst-case space complexity is \(O(n)\).
+ - Based on the peak memory used during the algorithm's execution: For example, before executing the last line, the program occupies \(O(1)\) space; when initializing the array
nums, the program occupies \(O(n)\) space, hence the worst-case space complexity is \(O(n)\).
@@ -1426,11 +1427,7 @@In recursion functions, it is important to take into count the measurement of stack frame space. For example in the following code:
--
-
- The function
loop()calls \(n\) timesfunction()in a loop, and each round offunction()returns and frees stack frame space, so the space complexity is still \(O(1)\).
- - The recursion function
recur()will have \(n\) unreturnedrecur()during runtime, thus occupying \(O(n)\) of stack frame space.
-
In recursive functions, stack frame space must be taken into count. Consider the following code:
+@@ -1646,20 +1643,25 @@The time complexity of both
+loop()andrecur()functions is \(O(n)\), but their space complexities differ.-
+
- The
loop()function callsfunction()\(n\) times in a loop, where each iteration'sfunction()returns and releases its stack frame space, so the space complexity remains \(O(1)\).
+ - The recursive function
recur()will have \(n\) instances of unreturnedrecur()existing simultaneously during its execution, thus occupying \(O(n)\) stack frame space.
+
2.4.3 Common Types¶
-Assuming the input data size is \(n\), the figure illustrates common types of space complexity (ordered from low to high).
+Let the size of the input data be \(n\), the following chart displays common types of space complexities (arranged from low to high).
\[ \begin{aligned} O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline -\text{constant order} < \text{logarithmic order} < \text{linear order} < \text{square order} < \text{exponential order} +\text{Constant Order} < \text{Logarithmic Order} < \text{Linear Order} < \text{Quadratic Order} < \text{Exponential Order} \end{aligned} \]-
-
Figure 2-16 Common space complexity types
+
+Figure 2-16 Common Types of Space Complexity
1. Constant Order \(O(1)\)¶
-Constant order is common for constants, variables, and objects whose quantity is unrelated to the size of the input data \(n\).
-It is important to note that memory occupied by initializing a variable or calling a function in a loop is released once the next iteration begins. Therefore, there is no accumulation of occupied space and the space complexity remains \(O(1)\) :
+Constant order is common in constants, variables, objects that are independent of the size of input data \(n\).
+Note that memory occupied by initializing variables or calling functions in a loop, which is released upon entering the next cycle, does not accumulate over space, thus the space complexity remains \(O(1)\):
-@@ -1971,8 +1973,8 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline2. Linear Order \(O(N)\)¶
-Linear order is commonly found in arrays, linked lists, stacks, queues, and similar structures where the number of elements is proportional to \(n\):
+2. Linear Order \(O(n)\)¶
+Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to \(n\):
-@@ -2213,7 +2215,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newlineAs shown in the Figure 2-17 , the depth of recursion for this function is \(n\), which means that there are \(n\) unreturned
+linear_recur()functions at the same time, using \(O(n)\) size stack frame space:As shown below, this function's recursive depth is \(n\), meaning there are \(n\) instances of unreturned
linear_recur()function, using \(O(n)\) size of stack frame space:-@@ -2333,11 +2335,11 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
-
Figure 2-17 Linear order space complexity generated by recursion function
+
+Figure 2-17 Recursive Function Generating Linear Order Space Complexity
-3. Quadratic Order \(O(N^2)\)¶
-Quadratic order is common in matrices and graphs, where the number of elements is in a square relationship with \(n\):
+3. Quadratic Order \(O(n^2)\)¶
+Quadratic order is common in matrices and graphs, where the number of elements is quadratic to \(n\):
-@@ -2530,7 +2532,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newlineAs shown in the Figure 2-18 , the recursion depth of this function is \(n\), and an array is initialized in each recursion function with lengths \(n\), \(n-1\), \(\dots\), \(2\), \(1\), and an average length of \(n / 2\), thus occupying \(O(n^2)\) space overall:
+As shown below, the recursive depth of this function is \(n\), and in each recursive call, an array is initialized with lengths \(n\), \(n-1\), \(\dots\), \(2\), \(1\), averaging \(n/2\), thus overall occupying \(O(n^2)\) space:
-@@ -2668,11 +2670,11 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
-
Figure 2-18 Square-order space complexity generated by the recursion function
+
+Figure 2-18 Recursive Function Generating Quadratic Order Space Complexity
-4. Exponential Order \(O(2^N)\)¶
-Exponential order is common in binary trees. Looking at the Figure 2-19 , a "full binary tree" of degree \(n\) has \(2^n - 1\) nodes, occupying \(O(2^n)\) space:
+4. Exponential Order \(O(2^n)\)¶
+Exponential order is common in binary trees. Observe the below image, a "full binary tree" with \(n\) levels has \(2^n - 1\) nodes, occupying \(O(2^n)\) space:
-@@ -2818,16 +2820,16 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
-
Figure 2-19 Exponential order space complexity generated by a full binary tree
+
+Figure 2-19 Full Binary Tree Generating Exponential Order Space Complexity
-5. Logarithmic Order \(O(\Log N)\)¶
-Logarithmic order is commonly used in divide and conquer algorithms. For example, in a merge sort, given an array of length \(n\) as the input, each round of recursion divides the array in half from its midpoint to form a recursion tree of height \(\log n\), using \(O(\log n)\) stack frame space.
-Another example is to convert a number into a string. Given a positive integer \(n\) with a digit count of \(\log_{10} n + 1\), the corresponding string length is \(\log_{10} n + 1\). Therefore, the space complexity is \(O(\log_{10} n + 1) = O(\log n)\).
-2.4.4 Weighing Time And Space¶
-Ideally, we would like to optimize both the time complexity and the space complexity of an algorithm. However, in reality, simultaneously optimizing time and space complexity is often challenging.
-Reducing time complexity usually comes at the expense of increasing space complexity, and vice versa. The approach of sacrificing memory space to improve algorithm speed is known as "trading space for time", while the opposite is called "trading time for space".
-The choice between these approaches depends on which aspect we prioritize. In most cases, time is more valuable than space, so "trading space for time" is usually the more common strategy. Of course, in situations with large data volumes, controlling space complexity is also crucial.
+5. Logarithmic Order \(O(\log n)\)¶
+Logarithmic order is common in divide-and-conquer algorithms. For example, in merge sort, an array of length \(n\) is recursively divided in half each round, forming a recursion tree of height \(\log n\), using \(O(\log n)\) stack frame space.
+Another example is converting a number to a string. Given a positive integer \(n\), its number of digits is \(\log_{10} n + 1\), corresponding to the length of the string, thus the space complexity is \(O(\log_{10} n + 1) = O(\log n)\).
+2.4.4 Balancing Time and Space¶
+Ideally, we aim for both time complexity and space complexity to be optimal. However, in practice, optimizing both simultaneously is often difficult.
+Lowering time complexity usually comes at the cost of increased space complexity, and vice versa. The approach of sacrificing memory space to improve algorithm speed is known as "space-time tradeoff"; the reverse is known as "time-space tradeoff".
+The choice depends on which aspect we value more. In most cases, time is more precious than space, so "space-time tradeoff" is often the more common strategy. Of course, controlling space complexity is also very important when dealing with large volumes of data.
diff --git a/en/chapter_computational_complexity/summary/index.html b/en/chapter_computational_complexity/summary/index.html index 489baa996..375153f16 100644 --- a/en/chapter_computational_complexity/summary/index.html +++ b/en/chapter_computational_complexity/summary/index.html @@ -758,9 +758,9 @@-
-
+
- 1. Highlights
+ 1. Key Review
@@ -818,9 +818,9 @@
-
-
+
- 1. Highlights
+ 1. Key Review
@@ -872,49 +872,49 @@
2.5 Summary¶
-1. Highlights¶
-Evaluation of Algorithm Efficiency
+1. Key Review¶
+Algorithm Efficiency Assessment
-
-
- Time and space efficiency are the two leading evaluation indicators to measure an algorithm. -
- We can evaluate the efficiency of an algorithm through real-world testing. Still, it isn't easy to eliminate the side effects from the testing environment, and it consumes a lot of computational resources. -
- Complexity analysis overcomes the drawbacks of real-world testing. The analysis results can apply to all operating platforms and reveal the algorithm's efficiency under variant data scales. +
- Time efficiency and space efficiency are the two main criteria for assessing the merits of an algorithm. +
- We can assess algorithm efficiency through actual testing, but it's challenging to eliminate the influence of the test environment, and it consumes substantial computational resources. +
- Complexity analysis can overcome the disadvantages of actual testing. Its results are applicable across all operating platforms and can reveal the efficiency of algorithms at different data scales.
Time Complexity
-
-
- Time complexity is used to measure the trend of algorithm running time as the data size grows., which can effectively evaluate the algorithm's efficiency. However, it may fail in some cases, such as when the input volume is small or the time complexities are similar, making it difficult to precisely compare the efficiency of algorithms. -
- The worst time complexity is denoted by big \(O\) notation, which corresponds to the asymptotic upper bound of the function, reflecting the growth rate in the number of operations \(T(n)\) as \(n\) tends to positive infinity. -
- The estimation of time complexity involves two steps: first, counting the number of operations, and then determining the asymptotic upper bound. -
- Common time complexities, from lowest to highest, are \(O(1)\), \(O(\log n)\), \(O(n)\), \(O(n \log n)\), \(O(n^2)\), \(O(2^n)\), and \(O(n!)\). -
- The time complexity of certain algorithms is not fixed and depends on the distribution of the input data. The time complexity can be categorized into worst-case, best-case, and average. The best-case time complexity is rarely used because the input data must meet strict conditions to achieve the best-case. -
- The average time complexity reflects the efficiency of an algorithm with random data inputs, which is closest to the performance of algorithms in real-world scenarios. Calculating the average time complexity requires statistical analysis of input data and a synthesized mathematical expectation. +
- Time complexity measures the trend of an algorithm's running time with the increase in data volume, effectively assessing algorithm efficiency. However, it can fail in certain cases, such as with small input data volumes or when time complexities are the same, making it challenging to precisely compare the efficiency of algorithms. +
- Worst-case time complexity is denoted using big O notation, representing the asymptotic upper bound, reflecting the growth level of the number of operations \(T(n)\) as \(n\) approaches infinity. +
- Calculating time complexity involves two steps: first counting the number of operations, then determining the asymptotic upper bound. +
- Common time complexities, arranged from low to high, include \(O(1)\), \(O(\log n)\), \(O(n)\), \(O(n \log n)\), \(O(n^2)\), \(O(2^n)\), and \(O(n!)\), among others. +
- The time complexity of some algorithms is not fixed and depends on the distribution of input data. Time complexities are divided into worst, best, and average cases. The best case is rarely used because input data generally needs to meet strict conditions to achieve the best case. +
- Average time complexity reflects the efficiency of an algorithm under random data inputs, closely resembling the algorithm's performance in actual applications. Calculating average time complexity requires accounting for the distribution of input data and the subsequent mathematical expectation.
Space Complexity
-
-
- Space complexity serves a similar purpose to time complexity and is used to measure the trend of space occupied by an algorithm as the data volume increases. -
- The memory space associated with the operation of an algorithm can be categorized into input space, temporary space, and output space. Normally, the input space is not considered when determining space complexity. The temporary space can be classified into instruction space, data space, and stack frame space, and the stack frame space usually only affects the space complexity for recursion functions. -
- We mainly focus on the worst-case space complexity, which refers to the measurement of an algorithm's space usage when given the worst-case input data and during the worst-case execution scenario. -
- Common space complexities are \(O(1)\), \(O(\log n)\), \(O(n)\), \(O(n^2)\) and \(O(2^n)\) from lowest to highest. +
- Space complexity, similar to time complexity, measures the trend of memory space occupied by an algorithm with the increase in data volume. +
- The relevant memory space used during the algorithm's execution can be divided into input space, temporary space, and output space. Generally, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions. +
- We usually focus only on the worst-case space complexity, which means calculating the space complexity of the algorithm under the worst input data and at the worst moment of operation. +
- Common space complexities, arranged from low to high, include \(O(1)\), \(O(\log n)\), \(O(n)\), \(O(n^2)\), and \(O(2^n)\), among others.
2. Q & A¶
Is the space complexity of tail recursion \(O(1)\)?
-Theoretically, the space complexity of a tail recursion function can be optimized to \(O(1)\). However, most programming languages (e.g., Java, Python, C++, Go, C#, etc.) do not support auto-optimization for tail recursion, so the space complexity is usually considered as \(O(n)\).
+Theoretically, the space complexity of a tail-recursive function can be optimized to \(O(1)\). However, most programming languages (such as Java, Python, C++, Go, C#) do not support automatic optimization of tail recursion, so it's generally considered to have a space complexity of \(O(n)\).
-What is the difference between the terms function and method?
-A function can be executed independently, and all arguments are passed explicitly. A method is associated with an object and is implicitly passed to the object that calls it, allowing it to operate on the data contained within an instance of a class.
-Let's illustrate with a few common programming languages.
+What is the difference between the terms 'function' and 'method'?
+A "function" can be executed independently, with all parameters passed explicitly. A "method" is associated with an object and is implicitly passed to the object calling it, able to operate on the data contained within an instance of a class.
+Here are some examples from common programming languages:
-
-
- C is a procedural programming language without object-oriented concepts, so it has only functions. However, we can simulate object-oriented programming by creating structures (struct), and the functions associated with structures are equivalent to methods in other languages. -
- Java and C# are object-oriented programming languages, and blocks of code (methods) are typically part of a class. Static methods behave like a function because it is bound to the class and cannot access specific instance variables. -
- Both C++ and Python support both procedural programming (functions) and object-oriented programming (methods). +
- C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with these structures are equivalent to methods in other programming languages. +
- Java and C# are object-oriented programming languages where code blocks (methods) are typically part of a class. Static methods behave like functions because they are bound to the class and cannot access specific instance variables. +
- C++ and Python support both procedural programming (functions) and object-oriented programming (methods).
-diff --git a/en/chapter_computational_complexity/time_complexity/index.html b/en/chapter_computational_complexity/time_complexity/index.html index fffad9315..3abb9a633 100644 --- a/en/chapter_computational_complexity/time_complexity/index.html +++ b/en/chapter_computational_complexity/time_complexity/index.html @@ -718,46 +718,46 @@Does the figure "Common Types of Space Complexity" reflect the absolute size of the occupied space?
-No, that figure shows the space complexity, which reflects the growth trend, not the absolute size of the space occupied.
-For example, if you take \(n = 8\) , the values of each curve do not align with the function because each curve contains a constant term used to compress the range of values to a visually comfortable range.
-In practice, since we usually don't know each method's "constant term" complexity, it is generally impossible to choose the optimal solution for \(n = 8\) based on complexity alone. But it's easier to choose for \(n = 8^5\) as the growth trend is already dominant.
+Does the 'Common Types of Space Complexity' figure reflect the absolute size of occupied space?
+No, the figure shows space complexities, which reflect growth trends, not the absolute size of the occupied space.
+If you take \(n = 8\), you might find that the values of each curve don't correspond to their functions. This is because each curve includes a constant term, intended to compress the value range into a visually comfortable range.
+In practice, since we usually don't know the "constant term" complexity of each method, it's generally not possible to choose the best solution for \(n = 8\) based solely on complexity. However, for \(n = 8^5\), it's much easier to choose, as the growth trend becomes dominant.
-
-
+
- 1. Highlights
+ 1. Key Review
@@ -872,49 +872,49 @@
-
-
+
- 2.1.1 Practical Testing
+ 2.1.1 Actual Testing
@@ -873,43 +873,43 @@
-
diff --git a/en/chapter_introduction/algorithms_are_everywhere/index.html b/en/chapter_introduction/algorithms_are_everywhere/index.html index 57fb556e7..145db1fd4 100644 --- a/en/chapter_introduction/algorithms_are_everywhere/index.html +++ b/en/chapter_introduction/algorithms_are_everywhere/index.html @@ -799,47 +799,47 @@ -Abstract
-Data structures are like a solid and varied framework.
-It provides a blueprint for the orderly organization of data upon which algorithms can come alive.
+Data structures resemble a stable and diverse framework.
+They serve as a blueprint for organizing data orderly, enabling algorithms to come to life upon this foundation.
1.1 Algorithms Are Everywhere¶
-When we hear the word "algorithm", we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which is ubiquitous in our daily lives.
-Before we formally discuss algorithms, an interesting fact is worth sharing: you have already learned many algorithms unconsciously and have become accustomed to applying them in your daily life. Below, I will give a few specific examples to prove this point.
-Example 1: Looking Up a Dictionary. In a standard dictionary, each word corresponds to a phonetic transcription and the dictionary is organized alphabetically based on these transcriptions. Let's say we're looking for a word that begins with the letter \(r\). This is typically done in the following way:
+1.1 Algorithms are Everywhere¶
+When we hear the word "algorithm," we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.
+Before formally discussing algorithms, there's an interesting fact worth sharing: you have already unconsciously learned many algorithms and have become accustomed to applying them in your daily life. Here, I will give a few specific examples to prove this point.
+Example 1: Looking Up a Dictionary. In an English dictionary, words are listed alphabetically. Suppose we're searching for a word that starts with the letter \(r\). This is typically done in the following way:
-
-
- Open the dictionary around its midpoint and note the first letter on that page, assuming it to be \(m\). -
- Given the sequence of words following the initial letter \(m\), estimate where words starting with the letter \(r\) might be located within the alphabetical order. -
- Iterate steps
1.and2.until you find the page where the word begins with the letter \(r\).
+ - Open the dictionary to about halfway and check the first letter on the page, let's say the letter is \(m\). +
- Since \(r\) comes after \(m\) in the alphabet, we can ignore the first half of the dictionary and focus on the latter half. +
- Repeat steps
1.and2.until you find the page where the word starts with \(r\).
--
+
-
+
-
+
-
+
-
+
Figure 1-1 Dictionary search step
+Figure 1-1 Process of Looking Up a Dictionary
-The skill of looking up a dictionary, essential for elementary school students, is actually the renowned binary search algorithm. Through the lens of data structures, we can view the dictionary as a sorted "array"; while from an algorithmic perspective, the series of operations in looking up a dictionary can be seen as "binary search".
-Example 2: Organizing Playing Cards. When playing cards, we need to arrange the cards in ascending order each game, as shown in the following process.
+This essential skill for elementary students, looking up a dictionary, is actually the famous "Binary Search" algorithm. From a data structure perspective, we can consider the dictionary as a sorted "array"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as "Binary Search."
+Example 2: Organizing Playing Cards. When playing cards, we need to arrange the cards in our hand in ascending order, as shown in the following process.
-
-
- Divide the playing cards into "ordered" and "unordered" parts, assuming initially that the leftmost card is already ordered. -
- Take out a card from the unordered part and insert it into the correct position in the ordered part; once completed, the leftmost two cards will be in an ordered sequence. -
- Continue the loop described in step
2., each iteration involving insertion of one card from the unordered segment into the ordered portion, until all cards are appropriately ordered.
+ - Divide the playing cards into "ordered" and "unordered" sections, assuming initially the leftmost card is already in order. +
- Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order. +
- Continue to repeat step
2.until all cards are in order.
-
Figure 1-2 Playing cards sorting process
+
+Figure 1-2 Playing Cards Sorting Process
-The above method of organizing playing cards is essentially the "insertion sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting library functions include insertion sort.
+The above method of organizing playing cards is essentially the "Insertion Sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.
Example 3: Making Change. Suppose we buy goods worth \(69\) yuan at a supermarket and give the cashier \(100\) yuan, then the cashier needs to give us \(31\) yuan in change. They would naturally complete the thought process as shown below.
- The options are currencies smaller than \(31\), including \(1\), \(5\), \(10\), and \(20\). @@ -851,8 +851,8 @@
- Graphs, compared to linked lists, provide richer logical information but require more memory space.
- Data structures are the foundation of algorithms. They provide structured data storage and methods for manipulating data for algorithms.
- Algorithms are the stage where data structures come into play. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved. diff --git a/en/chapter_preface/about_the_book/index.html b/en/chapter_preface/about_the_book/index.html index 6c8f6d5f4..15d0f7db8 100644 --- a/en/chapter_preface/about_the_book/index.html +++ b/en/chapter_preface/about_the_book/index.html @@ -91,7 +91,7 @@
-
-
+
- 0.1.1 Target Readers
+ 0.1.1 Target Audience
@@ -829,9 +829,9 @@
-
-
+
- 0.1.1 Target Readers
+ 0.1.1 Target Audience
@@ -891,43 +891,44 @@
-
0.1 The Book¶
-The aim of this project is to create an open source, free, novice-friendly introductory tutorial on data structures and algorithms.
+0.1 About This Book¶
+This open-source project aims to create a free, and beginner-friendly crash course on data structures and algorithms.
-
-
- Animated graphs are used throughout the book to structure the knowledge of data structures and algorithms in a way that is clear and easy to understand with a smooth learning curve. -
- The source code of the algorithms can be run with a single click, supporting Java, C++, Python, Go, JS, TS, C#, Swift, Rust, Dart, Zig and other languages. -
- Readers are encouraged to help each other and make progress in the chapter discussion forums, and questions and comments can usually be answered within two days. +
- Using animated illustrations, it delivers structured insights into data structures and algorithmic concepts, ensuring comprehensibility and a smooth learning curve. +
- Run code with just one click, supporting Java, C++, Python, Go, JS, TS, C#, Swift, Rust, Dart, Zig and other languages. +
- Readers are encouraged to engage with each other in the discussion area for each section, questions and comments are usually answered within two days.
0.1.1 Target Readers¶
-If you are a beginner to algorithms, have never touched an algorithm before, or already have some experience brushing up on data structures and algorithms, and have a vague understanding of data structures and algorithms, repeatedly jumping sideways between what you can and can't do, then this book is just for you!
-If you have already accumulated a certain amount of questions and are familiar with most of the question types, then this book can help you review and organize the algorithm knowledge system, and the repository source code can be used as a "brushing tool library" or "algorithm dictionary".
-If you are an algorithm expert, we look forward to receiving your valuable suggestions or participate in the creation together.
+0.1.1 Target Audience¶
+If you are new to algorithms with limited exposure, or you have accumulated some experience in algorithms, but you only have a vague understanding of data structures and algorithms, and you are constantly jumping between "yep" and "hmm", then this book is for you!
+If you have already accumulated a certain amount of problem-solving experience, and are familiar with most types of problems, then this book can help you review and organize your algorithm knowledge system. The repository's source code can be used as a "problem-solving toolkit" or an "algorithm cheat sheet".
+If you are an algorithm expert, we look forward to receiving your valuable suggestions, or join us and collaborate.
-precondition
-You will need to have at least a basic knowledge of programming in any language and be able to read and write simple code.
+Prerequisites
+You should know how to write and read simple code in at least one programming language.
0.1.2 Content Structure¶
-The main contents of the book are shown in the Figure 0-1 .
+The main content of the book is shown in the following figure.
-
-
- Complexity Analysis: dimensions and methods of evaluation of data structures and algorithms. Methods of deriving time complexity, space complexity, common types, examples, etc. -
- Data Structures: basic data types, classification methods of data structures. Definition, advantages and disadvantages, common operations, common types, typical applications, implementation methods of data structures such as arrays, linked lists, stacks, queues, hash tables, trees, heaps, graphs, etc. -
- Algorithms: definitions, advantages and disadvantages, efficiency, application scenarios, solution steps, sample topics of search, sorting algorithms, divide and conquer, backtracking algorithms, dynamic programming, greedy algorithms, and more. +
- Complexity Analysis: explores aspects and methods for evaluating data structures and algorithms. Covers methods of deriving time complexity and space complexity, along with common types and examples. +
- Data Structures: focuses on fundamental data types, classification methods, definitions, pros and cons, common operations, types, applications, and implementation methods of data structures such as array, linked list, stack, queue, hash table, tree, heap, graph, etc. +
- Algorithms: defines algorithms, discusses their pros and cons, efficiency, application scenarios, problem-solving steps, and includes sample questions for various algorithms such as search, sorting, divide and conquer, backtracking, dynamic programming, greedy algorithms, and more.
-
Figure 0-1 Hello Algo content structure
+
+Figure 0-1 Main Content of the Book
0.1.3 Acknowledgements¶
-During the creation of this book, I received help from many people, including but not limited to:
+Throughout the creation of this book, numerous individuals provided invaluable assistance, including but not limited to:
-
-
- Thank you to my mentor at the company, Dr. Shih Lee, for encouraging me to "get moving" during one of our conversations, which strengthened my resolve to write this book. -
- I would like to thank my girlfriend Bubbles for being the first reader of this book, and for making many valuable suggestions from the perspective of an algorithm whiz, making this book more suitable for newbies. -
- Thanks to Tengbao, Qibao, and Feibao for coming up with a creative name for this book that evokes fond memories of writing the first line of code "Hello World!". -
- Thanks to Sutong for designing the beautiful cover and logo for this book and patiently revising it many times under my OCD. -
- Thanks to @squidfunk for writing layout suggestions and for developing the open source documentation theme Material-for-MkDocs. +
- Thanks to my mentor at the company, Dr. Xi Li, who encouraged me in a conversation to "get moving fast," which solidified my determination to write this book; +
- Thanks to my girlfriend Paopao, as the first reader of this book, for offering many valuable suggestions from the perspective of a beginner in algorithms, making this book more suitable for newbies; +
- Thanks to Tengbao, Qibao, and Feibao for coming up with a creative name for this book, evoking everyone's fond memories of writing their first line of code "Hello World!"; +
- Thanks to Xiaoquan for providing professional help in intellectual property, which has played a significant role in the development of this open-source book; +
- Thanks to Sutong for designing a beautiful cover and logo for this book, and for patiently making multiple revisions under my insistence; +
- Thanks to @squidfunk for providing writing and typesetting suggestions, as well as his developed open-source documentation theme Material-for-MkDocs.
During the writing process, I read many textbooks and articles on data structures and algorithms. These works provide excellent models for this book and ensure the accuracy and quality of its contents. I would like to thank all my teachers and predecessors for their outstanding contributions!
-This book promotes a hands-on approach to learning, and in this respect is heavily inspired by "Dive into Deep Learning". I highly recommend this excellent book to you.
-A heartfelt thank you to my parents, it is your constant support and encouragement that gives me the opportunity to do this fun-filled thing.
+Throughout the writing journey, I delved into numerous textbooks and articles on data structures and algorithms. These works served as exemplary models, ensuring the accuracy and quality of this book's content. I extend my gratitude to all who preceded me for their invaluable contributions!
+This book advocates a combination of hands-on and minds-on learning, inspired in this regard by "Dive into Deep Learning". I highly recommend this excellent book to all readers.
+Heartfelt thanks to my parents, whose ongoing support and encouragement have allowed me to do this interesting work.
diff --git a/en/chapter_preface/index.html b/en/chapter_preface/index.html index 710aa37e8..f2b6f4262 100644 --- a/en/chapter_preface/index.html +++ b/en/chapter_preface/index.html @@ -811,7 +811,7 @@Abstract
Algorithms are like a beautiful symphony, with each line of code flowing like a rhythm.
-May this book ring softly in your head, leaving a unique and profound melody.
+May this book ring softly in your mind, leaving a unique and profound melody.
本章内容¶
-
diff --git a/en/chapter_preface/summary/index.html b/en/chapter_preface/summary/index.html
index 4d7cfdf5e..2049ac5de 100644
--- a/en/chapter_preface/summary/index.html
+++ b/en/chapter_preface/summary/index.html
@@ -804,9 +804,9 @@
- The main audience of this book is beginners in algorithm. If you already have some basic knowledge, this book can help you systematically review your algorithm knowledge, and the source code in this book can also be used as a "Coding Toolkit".
- The book consists of three main sections, Complexity Analysis, Data Structures, and Algorithms, covering most of the topics in the field.
- For newcomers to algorithms, it is crucial to read an introductory book in the beginning stages to avoid many detours or common pitfalls. -
- Animations and graphs within the book are usually used to introduce key points and difficult knowledge. These should be given more attention when reading the book. +
- Animations and figures within the book are usually used to introduce key points and difficult knowledge. These should be given more attention when reading the book.
- Practice is the best way to learn programming. It is highly recommended that you run the source code and type in the code yourself. -
- Each chapter in the web version of this book features a discussion forum, and you are welcome to share your questions and insights at any time. +
- Each chapter in the web version of this book features a discussion section, and you are welcome to share your questions and insights at any time.
Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code
Dive In Clone Repo Get PDF
The English edition is brewing...
Feel free to engage in Chinese-to-English translation and pull request review! For guidelines, please see #914.
EndorsementsQuote
\"An easy-to-understand book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!\"
\u2014\u2014 Junhui Deng, Professor of Computer Science, Tsinghua University
Quote
\"If I had 'Hello Algo' when I was learning data structures and algorithms, it would have been 10 times easier!\"
\u2014\u2014 Mu Li, Senior Principal Scientist, Amazon
Animated illustrationsEasy to understandSmooth learning curve
\"A picture is worth a thousand words.\"
Off-the-Shelf CodeMulti programming languagesRun with one click
\"Talk is cheap. Show me the code.\"
Learning TogetherDiscussion and questions welcomeReaders progress together
\"Chase the wind and moon, never stopping\"
\"Beyond the plains, there are spring mountains\"
PrefaceTwo years ago, I shared the \"Sword Offer\" series of problem solutions on LeetCode, which received much love and support from many students. During my interactions with readers, the most common question I encountered was \"How to get started with algorithms.\" Gradually, I developed a deep interest in this question.
Blindly solving problems seems to be the most popular method, being simple, direct, and effective. However, problem-solving is like playing a \"Minesweeper\" game, where students with strong self-learning abilities can successfully clear the mines one by one, but those with insufficient foundations may end up bruised from explosions, retreating step by step in frustration. Thoroughly reading textbooks is also common, but for students aiming for job applications, the energy consumed by graduation, resume submissions, and preparing for written tests and interviews makes tackling thick books a daunting challenge.
If you are facing similar troubles, then you are lucky to have found this book. This book is my answer to this question, not necessarily the best solution, but at least an active attempt. Although this book won't directly land you an Offer, it will guide you through the \"knowledge map\" of data structures and algorithms, help you understand the shape, size, and distribution of different \"mines,\" and equip you with various \"demining methods.\" With these skills, I believe you can more comfortably solve problems and read literature, gradually building a complete knowledge system.
I deeply agree with Professor Feynman's saying: \"Knowledge isn't free. You have to pay attention.\" In this sense, this book is not entirely \"free.\" To not disappoint the precious \"attention\" you pay to this book, I will do my utmost, investing the greatest \"attention\" to complete the creation of this book.
AuthorYudong Jin(Krahets), Senior Algorithm Engineer in a top tech company, Master's degree from Shanghai Jiao Tong University. The highest-read blogger across the entire LeetCode, his published \"Illustration of Algorithm Data Structures\" has been subscribed to by over 300k.
ContributionThis book is continuously improved with the joint efforts of many contributors from the open-source community. Thanks to each writer who invested their time and energy, listed in the order generated by GitHub:
The code review work for this book was completed by Gonglja, gvenusleo, hpstory, justin\u2010tse, krahets, night-cruise, nuomi1, Reanon, and sjinzh (listed in alphabetical order). Thanks to them for their time and effort, ensuring the standardization and uniformity of the code in various languages.
GongljaC, C++ gvenusleoDart hpstoryC# justin-tseJS, TS krahetsJava, Python night-cruiseRust nuomi1Swift ReanonGo, C sjinzhRust, Zig"},{"location":"chapter_computational_complexity/","title":"Chapter 2. \u00a0 Complexity Analysis","text":"Abstract
Complexity analysis is like a space-time guide in the vast universe of algorithms.
It leads us to explore deeply in the dimensions of time and space, in search of more elegant solutions.
"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 2.1 \u00a0 Evaluating Algorithm Efficiency
- 2.2 \u00a0 Iteration and Recursion
- 2.3 \u00a0 Time Complexity
- 2.4 \u00a0 Space Complexity
- 2.5 \u00a0 Summary
In data structures and algorithms, it is common to repeat a task, which is closely related to the complexity of the algorithm. There are two basic program structures that we usually use to repeat a task: iteration and recursion.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221-iteration","title":"2.2.1 \u00a0 Iteration","text":"An \"iteration iteration\" is a control structure that repeats a task. In iteration, a program repeats the execution of a piece of code until the condition is no longer satisfied.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for-loops","title":"1. \u00a0 For Loops","text":"forloops are one of the most common forms of iteration, suitable when the number of iterations is known in advance.The following function implements the summation \\(1 + 2 + \\dots + n\\) based on a
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyforloop, and the result is recorded using the variableres. Note thatrange(a, b)in Python corresponds to a \"left-closed-right-open\" interval, which is traversed in the range \\(a, a + 1, \\dots, b-1\\).
iteration.cppdef for_loop(n: int) -> int:\n \"\"\"for \u5faa\u73af\"\"\"\n res = 0\n # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in range(1, n + 1):\n res += i\n return res\n
iteration.java/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; ++i) {\n res += i;\n }\n return res;\n}\n
iteration.cs/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.go/* for \u5faa\u73af */\nint ForLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.swift/* for \u5faa\u73af */\nfunc forLoop(n int) int {\n res := 0\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i := 1; i <= n; i++ {\n res += i\n }\n return res\n}\n
iteration.js/* for \u5faa\u73af */\nfunc forLoop(n: Int) -> Int {\n var res = 0\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in 1 ... n {\n res += i\n }\n return res\n}\n
iteration.ts/* for \u5faa\u73af */\nfunction forLoop(n) {\n let res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.dart/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\n let res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.rs/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.c/* for \u5faa\u73af */\nfn for_loop(n: i32) -> i32 {\n let mut res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in 1..=n {\n res += i;\n }\n res\n} \n
iteration.zig/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\n var res: i32 = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (1..n+1) |i| {\n res = res + @as(i32, @intCast(i));\n }\n return res;\n} \nThe Figure 2-1 shows the flow block diagram of this summation function.
Figure 2-1 \u00a0 Flow block diagram of the summation function
The number of operations in this summation function is proportional to the size of the input data \\(n\\), or a \"linear relationship\". In fact, time complexity describes this \"linear relationship\". This is described in more detail in the next section.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while-loop","title":"2. \u00a0 While Loop","text":"Similar to a
forloop, awhileloop is a way to implement iteration. In awhileloop, the program first checks the condition at each turn, and if the condition is true, it continues, otherwise it ends the loop.Below, we use a
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pywhileloop to realize the summation \\(1 + 2 + \\dots + n\\) .
iteration.cppdef while_loop(n: int) -> int:\n \"\"\"while \u5faa\u73af\"\"\"\n res = 0\n i = 1 # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n:\n res += i\n i += 1 # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n return res\n
iteration.java/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.cs/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.go/* while \u5faa\u73af */\nint WhileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.swift/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\n res := 0\n // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n i := 1\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++\n }\n return res\n}\n
iteration.js/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -> Int {\n var res = 0\n var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n {\n res += i\n i += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res\n}\n
iteration.ts/* while \u5faa\u73af */\nfunction whileLoop(n) {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.dart/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.rs/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.c/* while \u5faa\u73af */\nfn while_loop(n: i32) -> i32 {\n let mut res = 0;\n let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n {\n res += i;\n i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n res\n}\n
iteration.zig/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\n var res: i32 = 0;\n var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += @intCast(i);\n i += 1;\n }\n return res;\n}\nIn
whileloops, since the steps of initializing and updating condition variables are independent of the loop structure, it has more degrees of freedom thanforloops.For example, in the following code, the condition variable \\(i\\) is updated twice per round, which is not convenient to implement with a
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyforloop.
iteration.cppdef while_loop_ii(n: int) -> int:\n \"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\n res = 0\n i = 1 # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n # \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n:\n res += i\n # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1\n i *= 2\n return res\n
iteration.java/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.cs/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.go/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, 4, 5...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1; \n i *= 2;\n }\n return res;\n}\n
iteration.swift/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n res := 0\n // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n i := 1\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n for i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++\n i *= 2\n }\n return res\n}\n
iteration.js/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -> Int {\n var res = 0\n var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1\n i *= 2\n }\n return res\n}\n
iteration.ts/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.dart/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.rs/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.c/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -> i32 {\n let mut res = 0;\n let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1;\n i *= 2;\n }\n res\n}\n
iteration.zig/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n// while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n var res: i32 = 0;\n var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += @intCast(i);\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1;\n i *= 2;\n }\n return res;\n}\nOverall,
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-nested-loops","title":"3. \u00a0 Nested Loops","text":"forloops have more compact code andwhileloops are more flexible, and both can implement iteration structures. The choice of which one to use should be based on the needs of the particular problem.We can nest one loop structure inside another, using the
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyforloop as an example:
iteration.cppdef nested_for_loop(n: int) -> str:\n \"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\n res = \"\"\n # \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in range(1, n + 1):\n # \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in range(1, n + 1):\n res += f\"({i}, {j}), \"\n return res\n
iteration.java/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\n ostringstream res;\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; ++i) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; ++j) {\n res << \"(\" << i << \", \" << j << \"), \";\n }\n }\n return res.str();\n}\n
iteration.cs/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n StringBuilder res = new StringBuilder();\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res.append(\"(\" + i + \", \" + j + \"), \");\n }\n }\n return res.toString();\n}\n
iteration.go/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\n StringBuilder res = new();\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res.Append($\"({i}, {j}), \");\n }\n }\n return res.ToString();\n}\n
iteration.swift/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\n res := \"\"\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i := 1; i <= n; i++ {\n for j := 1; j <= n; j++ {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n res += fmt.Sprintf(\"(%d, %d), \", i, j)\n }\n }\n return res\n}\n
iteration.js/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -> String {\n var res = \"\"\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in 1 ... n {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in 1 ... n {\n res.append(\"(\\(i), \\(j)), \")\n }\n }\n return res\n}\n
iteration.ts/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\n let res = '';\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (let j = 1; j <= n; j++) {\n res += `(${i}, ${j}), `;\n }\n }\n return res;\n}\n
iteration.dart/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\n let res = '';\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (let j = 1; j <= n; j++) {\n res += `(${i}, ${j}), `;\n }\n }\n return res;\n}\n
iteration.rs/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n String res = \"\";\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res += \"($i, $j), \";\n }\n }\n return res;\n}\n
iteration.c/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -> String {\n let mut res = vec![];\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in 1..=n {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in 1..=n {\n res.push(format!(\"({}, {}), \", i, j));\n }\n }\n res.join(\"\")\n}\n
iteration.zig/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n // n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\n int size = n * n * 26 + 1;\n char *res = malloc(size * sizeof(char));\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n char tmp[26];\n snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n strncat(res, tmp, size - strlen(res) - 1);\n }\n }\n return res;\n}\n// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n var res = std.ArrayList(u8).init(allocator);\n defer res.deinit();\n var buffer: [20]u8 = undefined;\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (1..n+1) |i| {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (1..n+1) |j| {\n var _str = try std.fmt.bufPrint(&buffer, \"({d}, {d}), \", .{i, j});\n try res.appendSlice(_str);\n }\n }\n return res.toOwnedSlice();\n}\nThe Figure 2-2 gives the block diagram of the flow of this nested loop.
Figure 2-2 \u00a0 Block diagram of the flow of nested loops
In this case, the number of operations of the function is proportional to \\(n^2\\), or the algorithm's running time is \"squared\" to the size of the input data \\(n\\).
We can continue to add nested loops, and each nest is a \"dimension up\", which will increase the time complexity to \"cubic relations\", \"quadratic relations\", and so on.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222-recursion","title":"2.2.2 \u00a0 Recursion","text":"\"Recursion recursion is an algorithmic strategy to solve a problem by calling the function itself. It consists of two main phases.
- recursive: the program calls itself deeper and deeper, usually passing smaller or simpler arguments, until a \"termination condition\" is reached.
- Recursion: After the \"termination condition\" is triggered, the program returns from the deepest level of the recursion function, level by level, aggregating the results of each level.
And from an implementation point of view, recursion code contains three main elements.
- Termination condition: used to decide when to switch from \"recursive\" to \"inductive\".
- Recursion call: corresponds to \"recursion\", where the function calls itself, usually with smaller or more simplified input parameters.
- return result: corresponds to \"return\", returning the result of the current recursion level to the previous one.
Observe the following code, we only need to call the function
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyrecur(n), and the calculation of \\(1 + 2 + \\dots + n\\) is done:
recursion.cppdef recur(n: int) -> int:\n \"\"\"\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if n == 1:\n return 1\n # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n res = recur(n - 1)\n # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n
recursion.java/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.cs/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.go/* \u9012\u5f52 */\nint Recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = Recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.swift/* \u9012\u5f52 */\nfunc recur(n int) int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n res := recur(n - 1)\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n}\n
recursion.js/* \u9012\u5f52 */\nfunc recur(n: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n let res = recur(n: n - 1)\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n}\n
recursion.ts/* \u9012\u5f52 */\nfunction recur(n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n const res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.dart/* \u9012\u5f52 */\nfunction recur(n: number): number {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n const res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.rs/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.c/* \u9012\u5f52 */\nfn recur(n: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1;\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n let res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n n + res\n}\n
recursion.zig/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1) {\n return 1;\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n var res: i32 = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\nThe Figure 2-3 shows the recursion of the function.
Figure 2-3 \u00a0 Recursion process for the summation function
Although iteration and recursion can yield the same results from a computational point of view, they represent two completely different paradigms for thinking about and solving problems.
- Iteration: solving problems \"from the bottom up\". Start with the most basic steps and repeat or add to them until the task is completed.
- Recursion: solving problems \"from the top down\". The original problem is broken down into smaller subproblems that have the same form as the original problem. Next, the subproblem continues to be broken down into smaller subproblems until it stops at the base case (the solution to the base case is known).
As an example of the above summation function, set the problem \\(f(n) = 1 + 2 + \\dots + n\\) .
- Iteration: the summation process is simulated in a loop, iterating from \\(1\\) to \\(n\\) and executing the summation operation in each round to find \\(f(n)\\).
- Recursion: decompose the problem into subproblems \\(f(n) = n + f(n-1)\\) and keep (recursively) decomposing until the base case \\(f(1) = 1\\) terminates.
Each time a recursion function calls itself, the system allocates memory for the newly opened function to store local variables, call addresses, other information, and so on. This results in two things.
- The context data for a function is stored in an area of memory called \"stack frame space\" and is not freed until the function returns. As a result, recursion is usually more memory-intensive than iteration.
- Recursion calls to functions incur additional overhead. Therefore recursion is usually less time efficient than loops.
As shown in the Figure 2-4 , before the termination condition is triggered, there are \\(n\\) unreturned recursion functions at the same time, with a recursion depth of \\(n\\) .
Figure 2-4 \u00a0 Recursion call depth
In practice, the depth of recursion allowed by a programming language is usually limited, and too deep a recursion may result in a stack overflow error.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-tail-recursion","title":"2. \u00a0 Tail Recursion","text":"Interestingly, if a function makes a recursion call only at the last step before returning, the function can be optimized by the compiler or interpreter to be comparable to iteration in terms of space efficiency. This situation is called \"tail recursion tail recursion\".
- Ordinary recursion: when a function returns to a function at a higher level, it needs to continue executing the code, so the system needs to save the context of the previous call.
- tail recursion: the recursion call is the last operation before the function returns, which means that the function does not need to continue with other operations after returning to the previous level, so the system does not need to save the context of the previous function.
In the case of calculating \\(1 + 2 + \\dots + n\\), for example, we can implement tail recursion by setting the result variable
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyresas a function parameter.
recursion.cppdef tail_recur(n, res):\n \"\"\"\u5c3e\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if n == 0:\n return res\n # \u5c3e\u9012\u5f52\u8c03\u7528\n return tail_recur(n - 1, res + n)\n
recursion.java/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.cs/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.go/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return TailRecur(n - 1, res + n);\n}\n
recursion.swift/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n-1, res+n)\n}\n
recursion.js/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n: n - 1, res: res + n)\n}\n
recursion.ts/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.dart/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.rs/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.c/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res;\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n tail_recur(n - 1, res + n)\n}\n
recursion.zig/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0) {\n return res;\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\nThe execution of tail recursion is shown in the Figure 2-5 . Comparing normal recursion and tail recursion, the execution point of the summation operation is different.
- Ordinary recursion: the summing operation is performed during the \"return\" process, and the summing operation is performed again after returning from each level.
- Tail recursion: the summing operation is performed in a \"recursion\" process, the \"recursion\" process simply returns in levels.
Figure 2-5 \u00a0 tail recursion process
Tip
Note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if a function is tail recursive, you may still encounter stack overflow problems.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-recursion-tree","title":"3. \u00a0 Recursion Tree","text":"When dealing with algorithmic problems related to divide and conquer, recursion is often more intuitive and easier to read than iteration. Take the Fibonacci sequence as an example.
Question
Given a Fibonacci series \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\) , find the \\(n\\)th number of the series.
Let the \\(n\\)th number of the Fibonacci series be \\(f(n)\\) , which leads to two easy conclusions.
- The first two numbers of the series are \\(f(1) = 0\\) and \\(f(2) = 1\\).
- Each number in the series is the sum of the previous two numbers, i.e. \\(f(n) = f(n - 1) + f(n - 2)\\) .
Recursion code can be written by making recursion calls according to the recursion relationship, using the first two numbers as termination conditions. Call
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyfib(n)to get the \\(n\\)th number of the Fibonacci series.
recursion.cppdef fib(n: int) -> int:\n \"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 or n == 2:\n return n - 1\n # \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n res = fib(n - 1) + fib(n - 2)\n # \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n
recursion.java/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.cs/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.go/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = Fib(n - 1) + Fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.swift/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n res := fib(n-1) + fib(n-2)\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n}\n
recursion.js/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n let res = fib(n: n - 1) + fib(n: n - 2)\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n}\n
recursion.ts/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n === 1 || n === 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n const res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.dart/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n === 1 || n === 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n const res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.rs/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.c/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1;\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n let res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c\n res\n}\n
recursion.zig/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 or n == 2) {\n return n - 1;\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n var res: i32 = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\nLooking at the above code, we have recursively called two functions within a function, this means that from one call, two call branches are created. As shown in the Figure 2-6 , this recursion will result in a recursion tree with the number of levels \\(n\\).
Figure 2-6 \u00a0 Recursion tree for Fibonacci series
Essentially, recursion embodies the paradigm of \"breaking down a problem into smaller sub-problems\", and this divide and conquer strategy is essential.
- From an algorithmic point of view, many important algorithmic strategies such as searching, sorting algorithm, backtracking, divide and conquer, dynamic programming, etc. directly or indirectly apply this way of thinking.
- From a data structure point of view, recursion is naturally suited to problems related to linked lists, trees and graphs because they are well suited to be analyzed with the idea of partitioning.
To summarize the above, as shown in the Table 2-1 , iteration and recursion differ in implementation, performance and applicability.
Table 2-1 \u00a0 Comparison of iteration and recursion features
iteration recursion implementation circular structure function call itself time-efficient typically efficient, no function call overhead overhead on every function call Memory Usage Usually uses a fixed size of memory space Cumulative function calls may use a lot of stack frame space Applicable Problems For simple cyclic tasks, code is intuitive and readable For sub-problem decomposition, such as trees, graphs, divide and conquer, backtracking, etc., the code structure is concise and clearTip
If you find the following solutions difficult to understand, you can review them after reading the \"Stack\" chapter.
So what is the intrinsic connection between iteration and recursion? In the case of the recursive function described above, the summing operation takes place in the \"return\" phase of the recursion. This means that the function that is initially called is actually the last to complete its summing operation, This mechanism works in the same way as the stack's \"first in, last out\" principle.
In fact, recursion terms like \"call stack\" and \"stack frame space\" already imply a close relationship between recursion and the stack.
- Recursive: When a function is called, the system allocates a new stack frame on the \"call stack\" for the function, which is used to store the function's local variables, parameters, return address, and other data.
- Return to: When a function completes execution and returns, the corresponding stack frame is removed from the \"call stack\", restoring the function's previous execution environment.
Thus, we can use an explicit stack to model the behavior of the call stack, thus transforming recursion into an iteration form:
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py
recursion.cppdef for_loop_recur(n: int) -> int:\n \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n # \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack = []\n res = 0\n # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in range(n, 0, -1):\n # \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.append(i)\n # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while stack:\n # \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop()\n # res = 1+2+3+...+n\n return res\n
recursion.java/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack<int> stack;\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.empty()) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.top();\n stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.cs/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n Stack<Integer> stack = new Stack<>();\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.isEmpty()) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.go/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n Stack<int> stack = new();\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.Push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.Count > 0) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.Pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.swift/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack := list.New()\n res := 0\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i := n; i > 0; i-- {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.PushBack(i)\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n for stack.Len() != 0 {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.Back().Value.(int)\n stack.Remove(stack.Back())\n }\n // res = 1+2+3+...+n\n return res\n}\n
recursion.js/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -> Int {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n var stack: [Int] = []\n var res = 0\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in stride(from: n, to: 0, by: -1) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.append(i)\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while !stack.isEmpty {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.removeLast()\n }\n // res = 1+2+3+...+n\n return res\n}\n
recursion.ts/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n const stack = [];\n let res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (let i = 1; i <= n; i++) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.length) { \n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.dart/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \n const stack: number[] = [];\n let res: number = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (let i = 1; i <= n; i++) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.length) { \n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.rs/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n List<int> stack = [];\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.add(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.isEmpty) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.removeLast();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.c/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -> i32 {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n let mut stack = Vec::new();\n let mut res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in (1..=n).rev() {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while !stack.is_empty() {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop().unwrap();\n }\n // res = 1+2+3+...+n\n res\n}\n
recursion.zig/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n int stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\n int top = -1; // \u6808\u9876\u7d22\u5f15\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack[1 + top++] = i;\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (top >= 0) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack[top--];\n }\n // res = 1+2+3+...+n\n return res;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n var stack: [n]i32 = undefined;\n var res: i32 = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n var i: usize = n;\n while (i > 0) {\n stack[i - 1] = @intCast(i);\n i -= 1;\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n var index: usize = n;\n while (index > 0) {\n index -= 1;\n res += stack[index];\n }\n // res = 1+2+3+...+n\n return res;\n}\nObserving the code above, it becomes more complex when recursion is converted to iteration. Although iteration and recursion can be converted to each other in many cases, it is not always worth doing so for two reasons.
- The transformed code may be more difficult to understand and less readable.
- For some complex problems, simulating the behavior of the system call stack can be very difficult.
In short, the choice of iteration or recursion depends on the nature of the particular problem. In programming practice, it is crucial to weigh the advantages and disadvantages of both and choose the appropriate method based on the context.
"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 Evaluation Of Algorithm Efficiency","text":"In algorithm design, we aim to achieve two goals in succession:
- Finding a Solution to the Problem: The algorithm needs to reliably find the correct solution within the specified input range.
- Seeking the Optimal Solution: There may be multiple ways to solve the same problem, and our goal is to find the most efficient algorithm possible.
In other words, once the ability to solve the problem is established, the efficiency of the algorithm emerges as the main benchmark for assessing its quality, which includes the following two aspects.
- Time Efficiency: The speed at which an algorithm runs.
- Space Efficiency: The amount of memory space the algorithm consumes.
In short, our goal is to design data structures and algorithms that are both \"fast and economical\". Effectively evaluating algorithm efficiency is crucial, as it allows for the comparison of different algorithms and guides the design and optimization process.
There are mainly two approaches for assessing efficiency: practical testing and theoretical estimation.
"},{"location":"chapter_computational_complexity/performance_evaluation/#211-practical-testing","title":"2.1.1 \u00a0 Practical Testing","text":"Let's consider a scenario where we have two algorithms,
AandB, both capable of solving the same problem. To compare their efficiency, the most direct method is to use a computer to run both algorithms while monitoring and recording their execution time and memory usage. This approach provides a realistic assessment of their performance, but it also has significant limitations.On one hand, it's challenging to eliminate the interference of the test environment. Hardware configurations can significantly affect the performance of algorithms. For instance, on one computer, Algorithm
Amight run faster than AlgorithmB, but the results could be the opposite on another computer with different specifications. This means we would need to conduct tests on a variety of machines and calculate an average efficiency, which is impractical.Furthermore, conducting comprehensive tests is resource-intensive. The efficiency of algorithms can vary with different volumes of input data. For example, with smaller data sets, Algorithm A might run faster than Algorithm B; however, this could change with larger data sets. Therefore, to reach a convincing conclusion, it's necessary to test a range of data sizes, which requires excessive computational resources.
"},{"location":"chapter_computational_complexity/performance_evaluation/#212-theoretical-estimation","title":"2.1.2 \u00a0 Theoretical Estimation","text":"Given the significant limitations of practical testing, we can consider assessing algorithm efficiency solely through calculations. This method of estimation is known as 'asymptotic complexity analysis,' often simply referred to as 'complexity analysis.
Complexity analysis illustrates the relationship between the time (and space) resources required by an algorithm and the size of its input data. It describes the growing trend in the time and space required for the execution of an algorithm as the size of the input data increases. This definition might sound a bit complex, so let's break it down into three key points for easier understanding.
- In complexity analysis, 'time and space' directly relate to 'time complexity' and 'space complexity,' respectively.
- The statement \"as the size of the input data increases\" highlights that complexity analysis examines the interplay between the size of the input data and the algorithm's efficiency.
- Lastly, the phrase \"the growing trend in time and space required\" emphasizes that the focus of complexity analysis is not on the specific values of running time or space occupied, but on the rate at which these requirements increase with larger input sizes.
Complexity analysis overcomes the drawbacks of practical testing methods in two key ways:.
- It is independent of the testing environment, meaning the analysis results are applicable across all operating platforms.
- It effectively demonstrates the efficiency of algorithms with varying data volumes, particularly highlighting performance in large-scale data scenarios.
Tip
If you're still finding the concept of complexity confusing, don't worry. We will cover it in more detail in the subsequent chapters.
Complexity analysis provides us with a 'ruler' for evaluating the efficiency of algorithms, enabling us to measure the time and space resources required to execute a given algorithm and to compare the efficiency of different algorithms.
Complexity is a mathematical concept that might seem abstract and somewhat challenging for beginners. From this perspective, introducing complexity analysis at the very beginning may not be the most suitable approach. However, when discussing the characteristics of a particular data structure or algorithm, analyzing its operational speed and space usage is often inevitable.
Therefore, it is recommended that before diving deeply into data structures and algorithms, one should first gain a basic understanding of complexity analysis. This foundational knowledge will facilitate the complexity analysis of simple algorithms.
"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 Space Complexity","text":"The space complexity is used to measure the growth trend of memory consumption as the scale of data increases for an algorithm solution. This concept is analogous to time complexity by replacing \"runtime\" with \"memory space\".
"},{"location":"chapter_computational_complexity/space_complexity/#241-algorithmic-correlation-space","title":"2.4.1 \u00a0 Algorithmic Correlation Space","text":"The memory space used by algorithms during its execution include the following types.
- Input Space: Used to store the input data for the algorithm.
- Temporary Space: Used to store variables, objects, function contexts, and other data of the algorithm during runtime.
- Output Space: Used to store the output data of the algorithm.
In general, the \"Input Space\" is excluded from the statistics of space complexity.
The Temporary Space can be further divided into three parts.
- Temporary Data: Used to store various constants, variables, objects, etc., during the the algorithm's execution.
- Stack Frame Space: Used to hold the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is freed when the function returns.
- Instruction Space: Used to hold compiled program instructions, usually ignored in practical statistics.
When analyzing the space complexity of a piece of program, three parts are usually taken into account: Temporary Data, Stack Frame Space and Output Data.
Figure 2-15 \u00a0 Associated spaces used by the algorithm
PythonC++JavaC#GoSwiftJSTSDartRustCZigclass Node:\n \"\"\"Classes\"\"\"\"\n def __init__(self, x: int):\n self.val: int = x # node value\n self.next: Node | None = None # reference to the next node\n\ndef function() -> int:\n \"\"\"\"Functions\"\"\"\"\"\n # Perform certain operations...\n return 0\n\ndef algorithm(n) -> int: # input data\n A = 0 # temporary data (constant, usually in uppercase)\n b = 0 # temporary data (variable)\n node = Node(0) # temporary data (object)\n c = function() # Stack frame space (call function)\n return A + b + c # output data\n/* Structures */\nstruct Node {\n int val;\n Node *next;\n Node(int x) : val(x), next(nullptr) {}\n};\n\n/* Functions */\nint func() {\n // Perform certain operations...\n return 0;\n}\n\nint algorithm(int n) { // input data\n const int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n Node* node = new Node(0); // temporary data (object)\n int c = func(); // stack frame space (call function)\n return a + b + c; // output data\n}\n/* Classes */\nclass Node {\n int val;\n Node next;\n Node(int x) { val = x; }\n}\n\n/* Functions */\nint function() {\n // Perform certain operations...\n return 0;\n}\n\nint algorithm(int n) { // input data\n final int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n Node node = new Node(0); // temporary data (object)\n int c = function(); // stack frame space (call function)\n return a + b + c; // output data\n}\n/* Classes */\nclass Node {\n int val;\n Node next;\n Node(int x) { val = x; }\n}\n\n/* Functions */\nint Function() {\n // Perform certain operations...\n return 0;\n}\n\nint Algorithm(int n) { // input data\n const int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n Node node = new(0); // temporary data (object)\n int c = Function(); // stack frame space (call function)\n return a + b + c; // output data\n}\n/* Structures */\ntype node struct {\n val int\n next *node\n}\n\n/* Create node structure */\nfunc newNode(val int) *node {\n return &node{val: val}\n}\n\n/* Functions */\nfunc function() int {\n // Perform certain operations...\n return 0\n}\n\nfunc algorithm(n int) int { // input data\n const a = 0 // temporary data (constant)\n b := 0 // temporary storage of data (variable)\n newNode(0) // temporary data (object)\n c := function() // stack frame space (call function)\n return a + b + c // output data\n}\n/* Classes */\nclass Node {\n var val: Int\n var next: Node?\n\n init(x: Int) {\n val = x\n }\n}\n\n/* Functions */\nfunc function() -> Int {\n // Perform certain operations...\n return 0\n}\n\nfunc algorithm(n: Int) -> Int { // input data\n let a = 0 // temporary data (constant)\n var b = 0 // temporary data (variable)\n let node = Node(x: 0) // temporary data (object)\n let c = function() // stack frame space (call function)\n return a + b + c // output data\n}\n/* Classes */\nclass Node {\n val;\n next;\n constructor(val) {\n this.val = val === undefined ? 0 : val; // node value\n this.next = null; // reference to the next node\n }\n}\n\n/* Functions */\nfunction constFunc() {\n // Perform certain operations\n return 0;\n}\n\nfunction algorithm(n) { // input data\n const a = 0; // temporary data (constant)\n let b = 0; // temporary data (variable)\n const node = new Node(0); // temporary data (object)\n const c = constFunc(); // Stack frame space (calling function)\n return a + b + c; // output data\n}\n/* Classes */\nclass Node {\n val: number;\n next: Node | null;\n constructor(val?: number) {\n this.val = val === undefined ? 0 : val; // node value\n this.next = null; // reference to the next node\n }\n}\n\n/* Functions */\nfunction constFunc(): number {\n // Perform certain operations\n return 0;\n}\n\nfunction algorithm(n: number): number { // input data\n const a = 0; // temporary data (constant)\n let b = 0; // temporary data (variable)\n const node = new Node(0); // temporary data (object)\n const c = constFunc(); // Stack frame space (calling function)\n return a + b + c; // output data\n}\n/* Classes */\nclass Node {\n int val;\n Node next;\n Node(this.val, [this.next]);\n}\n\n/* Functions */\nint function() {\n // Perform certain operations...\n return 0;\n}\n\nint algorithm(int n) { // input data\n const int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n Node node = Node(0); // temporary data (object)\n int c = function(); // stack frame space (call function)\n return a + b + c; // output data\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Structures */\nstruct Node {\n val: i32,\n next: Option<Rc<RefCell<Node>>>,\n}\n\n/* Creating a Node structure */\nimpl Node {\n fn new(val: i32) -> Self {\n Self { val: val, next: None }\n }\n}\n\n/* Functions */\nfn function() -> i32 { \n // Perform certain operations...\n return 0;\n}\n\nfn algorithm(n: i32) -> i32 { // input data\n const a: i32 = 0; // temporary data (constant)\n let mut b = 0; // temporary data (variable)\n let node = Node::new(0); // temporary data (object)\n let c = function(); // stack frame space (call function)\n return a + b + c; // output data\n}\n/* Functions */\nint func() {\n // Perform certain operations...\n return 0;\n}\n\nint algorithm(int n) { // input data\n const int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n int c = func(); // stack frame space (call function)\n return a + b + c; // output data\n}\n
"},{"location":"chapter_computational_complexity/space_complexity/#242-calculation-method","title":"2.4.2 \u00a0 Calculation Method","text":"\nThe calculation method for space complexity is pretty similar to time complexity, with the only difference being that the focus shifts from \"operation count\" to \"space usage size\".
On top of that, unlike time complexity, we usually only focus on the worst-case space complexity. This is because memory space is a hard requirement, and we have to make sure that there is enough memory space reserved for all possibilities incurred by input data.
Looking at the following code, the \"worst\" in worst-case space complexity has two layers of meaning.
- Based on the worst-case input data: when \\(n < 10\\), the space complexity is \\(O(1)\\); however, when \\(n > 10\\), the initialized array
numsoccupies \\(O(n)\\) space; thus the worst-case space complexity is \\(O(n)\\). - Based on the peak memory during algorithm execution: for example, the program occupies \\(O(1)\\) space until the last line is executed; when the array
numsis initialized, the program occupies \\(O(n)\\) space; thus the worst-case space complexity is \\(O(n)\\).
def algorithm(n: int):\n a = 0 # O(1)\n b = [0] * 10000 # O(1)\n if n > 10:\n nums = [0] * n # O(n)\nvoid algorithm(int n) {\n int a = 0; // O(1)\n vector<int> b(10000); // O(1)\n if (n > 10)\n vector<int> nums(n); // O(n)\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n int[] b = new int[10000]; // O(1)\n if (n > 10)\n int[] nums = new int[n]; // O(n)\n}\nvoid Algorithm(int n) {\n int a = 0; // O(1)\n int[] b = new int[10000]; // O(1)\n if (n > 10) {\n int[] nums = new int[n]; // O(n)\n }\n}\nfunc algorithm(n int) {\n a := 0 // O(1)\n b := make([]int, 10000) // O(1)\n var nums []int\n if n > 10 {\n nums := make([]int, n) // O(n)\n }\n fmt.Println(a, b, nums)\n}\nfunc algorithm(n: Int) {\n let a = 0 // O(1)\n let b = Array(repeating: 0, count: 10000) // O(1)\n if n > 10 {\n let nums = Array(repeating: 0, count: n) // O(n)\n }\n}\nfunction algorithm(n) {\n const a = 0; // O(1)\n const b = new Array(10000); // O(1)\n if (n > 10) {\n const nums = new Array(n); // O(n)\n }\n}\nfunction algorithm(n: number): void {\n const a = 0; // O(1)\n const b = new Array(10000); // O(1)\n if (n > 10) {\n const nums = new Array(n); // O(n)\n }\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n List<int> b = List.filled(10000, 0); // O(1)\n if (n > 10) {\n List<int> nums = List.filled(n, 0); // O(n)\n }\n}\nfn algorithm(n: i32) {\n let a = 0; // O(1)\n let b = [0; 10000]; // O(1)\n if n > 10 {\n let nums = vec![0; n as usize]; // O(n)\n }\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n int b[10000]; // O(1)\n if (n > 10)\n int nums[n] = {0}; // O(n)\n}\n\nIn recursion functions, it is important to take into count the measurement of stack frame space. For example in the following code:
- The function
loop()calls \\(n\\) timesfunction()in a loop, and each round offunction()returns and frees stack frame space, so the space complexity is still \\(O(1)\\). - The recursion function
recur()will have \\(n\\) unreturnedrecur()during runtime, thus occupying \\(O(n)\\) of stack frame space.
def function() -> int:\n # Perform certain operations\n return 0\n\ndef loop(n: int):\n \"\"\"Loop O(1)\"\"\"\"\"\n for _ in range(n):\n function()\n\ndef recur(n: int) -> int:\n \"\"\"Recursion O(n)\"\"\"\"\"\n if n == 1: return\n return recur(n - 1)\nint func() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nint function() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nint Function() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid Loop(int n) {\n for (int i = 0; i < n; i++) {\n Function();\n }\n}\n/* Recursion O(n) */\nint Recur(int n) {\n if (n == 1) return 1;\n return Recur(n - 1);\n}\nfunc function() int {\n // Perform certain operations\n return 0\n}\n\n/* Cycle O(1) */\nfunc loop(n int) {\n for i := 0; i < n; i++ {\n function()\n }\n}\n\n/* Recursion O(n) */\nfunc recur(n int) {\n if n == 1 {\n return\n }\n recur(n - 1)\n}\n@discardableResult\nfunc function() -> Int {\n // Perform certain operations\n return 0\n}\n\n/* Cycle O(1) */\nfunc loop(n: Int) {\n for _ in 0 ..< n {\n function()\n }\n}\n\n/* Recursion O(n) */\nfunc recur(n: Int) {\n if n == 1 {\n return\n }\n recur(n: n - 1)\n}\nfunction constFunc() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nfunction loop(n) {\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n/* Recursion O(n) */\nfunction recur(n) {\n if (n === 1) return;\n return recur(n - 1);\n}\nfunction constFunc(): number {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nfunction loop(n: number): void {\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n/* Recursion O(n) */\nfunction recur(n: number): void {\n if (n === 1) return;\n return recur(n - 1);\n}\nint function() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nfn function() -> i32 {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nfn loop(n: i32) {\n for i in 0..n {\n function();\n }\n}\n/* Recursion O(n) */\nvoid recur(n: i32) {\n if n == 1 {\n return;\n }\n recur(n - 1);\n}\nint func() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\n
"},{"location":"chapter_computational_complexity/space_complexity/#243-common-types","title":"2.4.3 \u00a0 Common Types","text":"\nAssuming the input data size is \\(n\\), the figure illustrates common types of space complexity (ordered from low to high).
\\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n^2) < O(2^n) \\newline \\text{constant order} < \\text{logarithmic order} < \\text{linear order} < \\text{square order} < \\text{exponential order} \\end{aligned} \\]Figure 2-16 \u00a0 Common space complexity types
"},{"location":"chapter_computational_complexity/space_complexity/#1-constant-order-o1","title":"1. \u00a0 Constant Order \\(O(1)\\)","text":"Constant order is common for constants, variables, and objects whose quantity is unrelated to the size of the input data \\(n\\).
It is important to note that memory occupied by initializing a variable or calling a function in a loop is released once the next iteration begins. Therefore, there is no accumulation of occupied space and the space complexity remains \\(O(1)\\) :
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef function() -> int:\n \"\"\"\u51fd\u6570\"\"\"\n # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n\ndef constant(n: int):\n \"\"\"\u5e38\u6570\u9636\"\"\"\n # \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n a = 0\n nums = [0] * 10000\n node = ListNode(0)\n # \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in range(n):\n c = 0\n # \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in range(n):\n function()\n
space_complexity.java/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const int a = 0;\n int b = 0;\n vector<int> nums(10000);\n ListNode node(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n
space_complexity.cs/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n final int a = 0;\n int b = 0;\n int[] nums = new int[10000];\n ListNode node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n
space_complexity.go/* \u51fd\u6570 */\nint Function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n int a = 0;\n int b = 0;\n int[] nums = new int[10000];\n ListNode node = new(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n Function();\n }\n}\n
space_complexity.swift/* \u51fd\u6570 */\nfunc function() int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0\n b := 0\n nums := make([]int, 10000)\n ListNode := newNode(0)\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n var c int\n for i := 0; i < n; i++ {\n c = 0\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for i := 0; i < n; i++ {\n function()\n }\n fmt.Println(a, b, nums, c, ListNode)\n}\n
space_complexity.js/* \u51fd\u6570 */\n@discardableResult\nfunc function() -> Int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n let a = 0\n var b = 0\n let nums = Array(repeating: 0, count: 10000)\n let node = ListNode(x: 0)\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in 0 ..< n {\n let c = 0\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in 0 ..< n {\n function()\n }\n}\n
space_complexity.ts/* \u51fd\u6570 */\nfunction constFunc() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0;\n const b = 0;\n const nums = new Array(10000);\n const node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n const c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n
space_complexity.dart/* \u51fd\u6570 */\nfunction constFunc(): number {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0;\n const b = 0;\n const nums = new Array(10000);\n const node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n const c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n
space_complexity.rs/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n final int a = 0;\n int b = 0;\n List<int> nums = List.filled(10000, 0);\n ListNode node = ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (var i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (var i = 0; i < n; i++) {\n function();\n }\n}\n
space_complexity.c/* \u51fd\u6570 */\nfn function() ->i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const A: i32 = 0;\n let b = 0;\n let nums = vec![0; 10000];\n let node = ListNode::new(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for i in 0..n {\n let c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for i in 0..n {\n function();\n }\n}\n
space_complexity.zig/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const int a = 0;\n int b = 0;\n int nums[1000];\n ListNode *node = newListNode(0);\n free(node);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n
"},{"location":"chapter_computational_complexity/space_complexity/#2-linear-order-on","title":"2. \u00a0 Linear Order \\(O(N)\\)","text":"// \u51fd\u6570\nfn function() i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a: i32 = 0;\n var b: i32 = 0;\n var nums = [_]i32{0}**10000;\n var node = inc.ListNode(i32){.val = 0};\n var i: i32 = 0;\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n while (i < n) : (i += 1) {\n var c: i32 = 0;\n _ = c;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n i = 0;\n while (i < n) : (i += 1) {\n _ = function();\n }\n _ = a;\n _ = b;\n _ = nums;\n _ = node;\n}\nLinear order is commonly found in arrays, linked lists, stacks, queues, and similar structures where the number of elements is proportional to \\(n\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef linear(n: int):\n \"\"\"\u7ebf\u6027\u9636\"\"\"\n # \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n nums = [0] * n\n # \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n hmap = dict[int, str]()\n for i in range(n):\n hmap[i] = str(i)\n
space_complexity.java/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n vector<int> nums(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n vector<ListNode> nodes;\n for (int i = 0; i < n; i++) {\n nodes.push_back(ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n unordered_map<int, string> map;\n for (int i = 0; i < n; i++) {\n map[i] = to_string(i);\n }\n}\n
space_complexity.cs/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int[] nums = new int[n];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n nodes.add(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Map<Integer, String> map = new HashMap<>();\n for (int i = 0; i < n; i++) {\n map.put(i, String.valueOf(i));\n }\n}\n
space_complexity.go/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int[] nums = new int[n];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = [];\n for (int i = 0; i < n; i++) {\n nodes.Add(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Dictionary<int, string> map = [];\n for (int i = 0; i < n; i++) {\n map.Add(i, i.ToString());\n }\n}\n
space_complexity.swift/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n _ = make([]int, n)\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var nodes []*node\n for i := 0; i < n; i++ {\n nodes = append(nodes, newNode(i))\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n m := make(map[int]string, n)\n for i := 0; i < n; i++ {\n m[i] = strconv.Itoa(i)\n }\n}\n
space_complexity.js/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n let nums = Array(repeating: 0, count: n)\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let nodes = (0 ..< n).map { ListNode(x: $0) }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, \"\\($0)\") })\n}\n
space_complexity.ts/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n const nums = new Array(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const nodes = [];\n for (let i = 0; i < n; i++) {\n nodes.push(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const map = new Map();\n for (let i = 0; i < n; i++) {\n map.set(i, i.toString());\n }\n}\n
space_complexity.dart/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n const nums = new Array(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const nodes: ListNode[] = [];\n for (let i = 0; i < n; i++) {\n nodes.push(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const map = new Map();\n for (let i = 0; i < n; i++) {\n map.set(i, i.toString());\n }\n}\n
space_complexity.rs/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n List<int> nums = List.filled(n, 0);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = [];\n for (var i = 0; i < n; i++) {\n nodes.add(ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Map<int, String> map = HashMap();\n for (var i = 0; i < n; i++) {\n map.putIfAbsent(i, () => i.toString());\n }\n}\n
space_complexity.c/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n let mut nums = vec![0; n as usize];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let mut nodes = Vec::new();\n for i in 0..n {\n nodes.push(ListNode::new(i))\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let mut map = HashMap::new();\n for i in 0..n {\n map.insert(i, i.to_string());\n }\n}\n
space_complexity.zig/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n int key;\n int val;\n UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int *nums = malloc(sizeof(int) * n);\n free(nums);\n\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n ListNode **nodes = malloc(sizeof(ListNode *) * n);\n for (int i = 0; i < n; i++) {\n nodes[i] = newListNode(i);\n }\n // \u5185\u5b58\u91ca\u653e\n for (int i = 0; i < n; i++) {\n free(nodes[i]);\n }\n free(nodes);\n\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n HashTable *h = NULL;\n for (int i = 0; i < n; i++) {\n HashTable *tmp = malloc(sizeof(HashTable));\n tmp->key = i;\n tmp->val = i;\n HASH_ADD_INT(h, key, tmp);\n }\n\n // \u5185\u5b58\u91ca\u653e\n HashTable *curr, *tmp;\n HASH_ITER(hh, h, curr, tmp) {\n HASH_DEL(h, curr);\n free(curr);\n }\n}\n// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n var nums = [_]i32{0}**n;\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n defer nodes.deinit();\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n try nodes.append(i);\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n defer map.deinit();\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n defer std.heap.page_allocator.free(string);\n try map.put(i, string);\n }\n _ = nums;\n}\nAs shown in the Figure 2-17 , the depth of recursion for this function is \\(n\\), which means that there are \\(n\\) unreturned
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.pylinear_recur()functions at the same time, using \\(O(n)\\) size stack frame space:
space_complexity.cppdef linear_recur(n: int):\n \"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n print(\"\u9012\u5f52 n =\", n)\n if n == 1:\n return\n linear_recur(n - 1)\n
space_complexity.java/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n cout << \"\u9012\u5f52 n = \" << n << endl;\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n
space_complexity.cs/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n System.out.println(\"\u9012\u5f52 n = \" + n);\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n
space_complexity.go/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\n Console.WriteLine(\"\u9012\u5f52 n = \" + n);\n if (n == 1) return;\n LinearRecur(n - 1);\n}\n
space_complexity.swift/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\n fmt.Println(\"\u9012\u5f52 n =\", n)\n if n == 1 {\n return\n }\n spaceLinearRecur(n - 1)\n}\n
space_complexity.js/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\n print(\"\u9012\u5f52 n = \\(n)\")\n if n == 1 {\n return\n }\n linearRecur(n: n - 1)\n}\n
space_complexity.ts/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\n console.log(`\u9012\u5f52 n = ${n}`);\n if (n === 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.dart/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\n console.log(`\u9012\u5f52 n = ${n}`);\n if (n === 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.rs/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n print('\u9012\u5f52 n = $n');\n if (n == 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.c/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\n println!(\"\u9012\u5f52 n = {}\", n);\n if n == 1 {return};\n linear_recur(n - 1);\n}\n
space_complexity.zig/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n printf(\"\u9012\u5f52 n = %d\\r\\n\", n);\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\n std.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\n if (n == 1) return;\n linearRecur(n - 1);\n}\nFigure 2-17 \u00a0 Linear order space complexity generated by recursion function
"},{"location":"chapter_computational_complexity/space_complexity/#3-quadratic-order-on2","title":"3. \u00a0 Quadratic Order \\(O(N^2)\\)","text":"Quadratic order is common in matrices and graphs, where the number of elements is in a square relationship with \\(n\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef quadratic(n: int):\n \"\"\"\u5e73\u65b9\u9636\"\"\"\n # \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n num_matrix = [[0] * n for _ in range(n)]\n
space_complexity.java/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n vector<vector<int>> numMatrix;\n for (int i = 0; i < n; i++) {\n vector<int> tmp;\n for (int j = 0; j < n; j++) {\n tmp.push_back(0);\n }\n numMatrix.push_back(tmp);\n }\n}\n
space_complexity.cs/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n int[][] numMatrix = new int[n][n];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<Integer>> numList = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n List<Integer> tmp = new ArrayList<>();\n for (int j = 0; j < n; j++) {\n tmp.add(0);\n }\n numList.add(tmp);\n }\n}\n
space_complexity.go/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n int[,] numMatrix = new int[n, n];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numList = [];\n for (int i = 0; i < n; i++) {\n List<int> tmp = [];\n for (int j = 0; j < n; j++) {\n tmp.Add(0);\n }\n numList.Add(tmp);\n }\n}\n
space_complexity.swift/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n numMatrix := make([][]int, n)\n for i := 0; i < n; i++ {\n numMatrix[i] = make([]int, n)\n }\n}\n
space_complexity.js/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n
space_complexity.ts/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numMatrix = Array(n)\n .fill(null)\n .map(() => Array(n).fill(null));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numList = [];\n for (let i = 0; i < n; i++) {\n const tmp = [];\n for (let j = 0; j < n; j++) {\n tmp.push(0);\n }\n numList.push(tmp);\n }\n}\n
space_complexity.dart/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numMatrix = Array(n)\n .fill(null)\n .map(() => Array(n).fill(null));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numList = [];\n for (let i = 0; i < n; i++) {\n const tmp = [];\n for (let j = 0; j < n; j++) {\n tmp.push(0);\n }\n numList.push(tmp);\n }\n}\n
space_complexity.rs/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numMatrix = List.generate(n, (_) => List.filled(n, 0));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numList = [];\n for (var i = 0; i < n; i++) {\n List<int> tmp = [];\n for (int j = 0; j < n; j++) {\n tmp.add(0);\n }\n numList.add(tmp);\n }\n}\n
space_complexity.c/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n let num_matrix = vec![vec![0; n as usize]; n as usize];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n let mut num_list = Vec::new();\n for i in 0..n {\n let mut tmp = Vec::new();\n for j in 0..n {\n tmp.push(0);\n }\n num_list.push(tmp);\n }\n}\n
space_complexity.zig/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n int **numMatrix = malloc(sizeof(int *) * n);\n for (int i = 0; i < n; i++) {\n int *tmp = malloc(sizeof(int) * n);\n for (int j = 0; j < n; j++) {\n tmp[j] = 0;\n }\n numMatrix[i] = tmp;\n }\n\n // \u5185\u5b58\u91ca\u653e\n for (int i = 0; i < n; i++) {\n free(numMatrix[i]);\n }\n free(numMatrix);\n}\n// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n defer nodes.deinit();\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n defer tmp.deinit();\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n try tmp.append(0);\n }\n try nodes.append(tmp);\n }\n}\nAs shown in the Figure 2-18 , the recursion depth of this function is \\(n\\), and an array is initialized in each recursion function with lengths \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\), and an average length of \\(n / 2\\), thus occupying \\(O(n^2)\\) space overall:
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef quadratic_recur(n: int) -> int:\n \"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n <= 0:\n return 0\n # \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n nums = [0] * n\n return quadratic_recur(n - 1)\n
space_complexity.java/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n vector<int> nums(n);\n cout << \"\u9012\u5f52 n = \" << n << \" \u4e2d\u7684 nums \u957f\u5ea6 = \" << nums.size() << endl;\n return quadraticRecur(n - 1);\n}\n
space_complexity.cs/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n int[] nums = new int[n];\n System.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\n return quadraticRecur(n - 1);\n}\n
space_complexity.go/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\n if (n <= 0) return 0;\n int[] nums = new int[n];\n Console.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\n return QuadraticRecur(n - 1);\n}\n
space_complexity.swift/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n if n <= 0 {\n return 0\n }\n nums := make([]int, n)\n fmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\n return spaceQuadraticRecur(n - 1)\n}\n
space_complexity.js/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -> Int {\n if n <= 0 {\n return 0\n }\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n let nums = Array(repeating: 0, count: n)\n print(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\n return quadraticRecur(n: n - 1)\n}\n
space_complexity.ts/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\n if (n <= 0) return 0;\n const nums = new Array(n);\n console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n return quadraticRecur(n - 1);\n}\n
space_complexity.dart/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\n if (n <= 0) return 0;\n const nums = new Array(n);\n console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n return quadraticRecur(n - 1);\n}\n
space_complexity.rs/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0) return 0;\n List<int> nums = List.filled(n, 0);\n print('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\n return quadraticRecur(n - 1);\n}\n
space_complexity.c/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -> i32 {\n if n <= 0 {return 0};\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n let nums = vec![0; n as usize];\n println!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\n return quadratic_recur(n - 1);\n}\n
space_complexity.zig/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n int *nums = malloc(sizeof(int) * n);\n printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\n int res = quadraticRecur(n - 1);\n free(nums);\n return res;\n}\n// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n if (n <= 0) return 0;\n var nums = [_]i32{0}**n;\n std.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\n return quadraticRecur(n - 1);\n}\nFigure 2-18 \u00a0 Square-order space complexity generated by the recursion function
"},{"location":"chapter_computational_complexity/space_complexity/#4-exponential-order-o2n","title":"4. \u00a0 Exponential Order \\(O(2^N)\\)","text":"Exponential order is common in binary trees. Looking at the Figure 2-19 , a \"full binary tree\" of degree \\(n\\) has \\(2^n - 1\\) nodes, occupying \\(O(2^n)\\) space:
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef build_tree(n: int) -> TreeNode | None:\n \"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\n if n == 0:\n return None\n root = TreeNode(0)\n root.left = build_tree(n - 1)\n root.right = build_tree(n - 1)\n return root\n
space_complexity.java/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n if (n == 0)\n return nullptr;\n TreeNode *root = new TreeNode(0);\n root->left = buildTree(n - 1);\n root->right = buildTree(n - 1);\n return root;\n}\n
space_complexity.cs/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\n if (n == 0)\n return null;\n TreeNode root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.go/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\n if (n == 0) return null;\n TreeNode root = new(0) {\n left = BuildTree(n - 1),\n right = BuildTree(n - 1)\n };\n return root;\n}\n
space_complexity.swift/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\n if n == 0 {\n return nil\n }\n root := newTreeNode(0)\n root.left = buildTree(n - 1)\n root.right = buildTree(n - 1)\n return root\n}\n
space_complexity.js/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -> TreeNode? {\n if n == 0 {\n return nil\n }\n let root = TreeNode(x: 0)\n root.left = buildTree(n: n - 1)\n root.right = buildTree(n: n - 1)\n return root\n}\n
space_complexity.ts/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\n if (n === 0) return null;\n const root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.dart/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n if (n === 0) return null;\n const root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.rs/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\n if (n == 0) return null;\n TreeNode root = TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.c/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -> Option<Rc<RefCell<TreeNode>>> {\n if n == 0 {return None};\n let root = TreeNode::new(0);\n root.borrow_mut().left = build_tree(n - 1);\n root.borrow_mut().right = build_tree(n - 1);\n return Some(root);\n}\n
space_complexity.zig/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n if (n == 0)\n return NULL;\n TreeNode *root = newTreeNode(0);\n root->left = buildTree(n - 1);\n root->right = buildTree(n - 1);\n return root;\n}\n// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n if (n == 0) return null;\n const root = try mem_allocator.create(inc.TreeNode(i32));\n root.init(0);\n root.left = try buildTree(mem_allocator, n - 1);\n root.right = try buildTree(mem_allocator, n - 1);\n return root;\n}\nFigure 2-19 \u00a0 Exponential order space complexity generated by a full binary tree
"},{"location":"chapter_computational_complexity/space_complexity/#5-logarithmic-order-olog-n","title":"5. \u00a0 Logarithmic Order \\(O(\\Log N)\\)","text":"Logarithmic order is commonly used in divide and conquer algorithms. For example, in a merge sort, given an array of length \\(n\\) as the input, each round of recursion divides the array in half from its midpoint to form a recursion tree of height \\(\\log n\\), using \\(O(\\log n)\\) stack frame space.
Another example is to convert a number into a string. Given a positive integer \\(n\\) with a digit count of \\(\\log_{10} n + 1\\), the corresponding string length is \\(\\log_{10} n + 1\\). Therefore, the space complexity is \\(O(\\log_{10} n + 1) = O(\\log n)\\).
"},{"location":"chapter_computational_complexity/space_complexity/#244-weighing-time-and-space","title":"2.4.4 \u00a0 Weighing Time And Space","text":"Ideally, we would like to optimize both the time complexity and the space complexity of an algorithm. However, in reality, simultaneously optimizing time and space complexity is often challenging.
Reducing time complexity usually comes at the expense of increasing space complexity, and vice versa. The approach of sacrificing memory space to improve algorithm speed is known as \"trading space for time\", while the opposite is called \"trading time for space\".
The choice between these approaches depends on which aspect we prioritize. In most cases, time is more valuable than space, so \"trading space for time\" is usually the more common strategy. Of course, in situations with large data volumes, controlling space complexity is also crucial.
"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 Summary","text":""},{"location":"chapter_computational_complexity/summary/#1-highlights","title":"1. \u00a0 Highlights","text":"Evaluation of Algorithm Efficiency
- Time and space efficiency are the two leading evaluation indicators to measure an algorithm.
- We can evaluate the efficiency of an algorithm through real-world testing. Still, it isn't easy to eliminate the side effects from the testing environment, and it consumes a lot of computational resources.
- Complexity analysis overcomes the drawbacks of real-world testing. The analysis results can apply to all operating platforms and reveal the algorithm's efficiency under variant data scales.
Time Complexity
- Time complexity is used to measure the trend of algorithm running time as the data size grows., which can effectively evaluate the algorithm's efficiency. However, it may fail in some cases, such as when the input volume is small or the time complexities are similar, making it difficult to precisely compare the efficiency of algorithms.
- The worst time complexity is denoted by big \\(O\\) notation, which corresponds to the asymptotic upper bound of the function, reflecting the growth rate in the number of operations \\(T(n)\\) as \\(n\\) tends to positive infinity.
- The estimation of time complexity involves two steps: first, counting the number of operations, and then determining the asymptotic upper bound.
- Common time complexities, from lowest to highest, are \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n \\log n)\\), \\(O(n^2)\\), \\(O(2^n)\\), and \\(O(n!)\\).
- The time complexity of certain algorithms is not fixed and depends on the distribution of the input data. The time complexity can be categorized into worst-case, best-case, and average. The best-case time complexity is rarely used because the input data must meet strict conditions to achieve the best-case.
- The average time complexity reflects the efficiency of an algorithm with random data inputs, which is closest to the performance of algorithms in real-world scenarios. Calculating the average time complexity requires statistical analysis of input data and a synthesized mathematical expectation.
Space Complexity
- Space complexity serves a similar purpose to time complexity and is used to measure the trend of space occupied by an algorithm as the data volume increases.
- The memory space associated with the operation of an algorithm can be categorized into input space, temporary space, and output space. Normally, the input space is not considered when determining space complexity. The temporary space can be classified into instruction space, data space, and stack frame space, and the stack frame space usually only affects the space complexity for recursion functions.
- We mainly focus on the worst-case space complexity, which refers to the measurement of an algorithm's space usage when given the worst-case input data and during the worst-case execution scenario.
- Common space complexities are \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n^2)\\) and \\(O(2^n)\\) from lowest to highest.
Is the space complexity of tail recursion \\(O(1)\\)?
Theoretically, the space complexity of a tail recursion function can be optimized to \\(O(1)\\). However, most programming languages (e.g., Java, Python, C++, Go, C#, etc.) do not support auto-optimization for tail recursion, so the space complexity is usually considered as \\(O(n)\\).
What is the difference between the terms function and method?
A function can be executed independently, and all arguments are passed explicitly. A method is associated with an object and is implicitly passed to the object that calls it, allowing it to operate on the data contained within an instance of a class.
Let's illustrate with a few common programming languages.
- C is a procedural programming language without object-oriented concepts, so it has only functions. However, we can simulate object-oriented programming by creating structures (struct), and the functions associated with structures are equivalent to methods in other languages.
- Java and C# are object-oriented programming languages, and blocks of code (methods) are typically part of a class. Static methods behave like a function because it is bound to the class and cannot access specific instance variables.
- Both C++ and Python support both procedural programming (functions) and object-oriented programming (methods).
Does the figure \"Common Types of Space Complexity\" reflect the absolute size of the occupied space?
No, that figure shows the space complexity, which reflects the growth trend, not the absolute size of the space occupied.
For example, if you take \\(n = 8\\) , the values of each curve do not align with the function because each curve contains a constant term used to compress the range of values to a visually comfortable range.
In practice, since we usually don't know each method's \"constant term\" complexity, it is generally impossible to choose the optimal solution for \\(n = 8\\) based on complexity alone. But it's easier to choose for \\(n = 8^5\\) as the growth trend is already dominant.
"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 Time Complexity","text":"Runtime can be a visual and accurate reflection of the efficiency of an algorithm. What should we do if we want to accurately predict the runtime of a piece of code?
- Determine the running platform, including hardware configuration, programming language, system environment, etc., all of which affect the efficiency of the code.
- Evaluates the running time required for various computational operations, e.g., the addition operation
+takes 1 ns, the multiplication operation*takes 10 ns, the print operationprint()takes 5 ns, and so on. - Counts all the computational operations in the code and sums the execution times of all the operations to get the runtime.
For example, in the following code, the input data size is \\(n\\) :
PythonC++JavaC#GoSwiftJSTSDartRustCZig# Under an operating platform\ndef algorithm(n: int):\n a = 2 # 1 ns\n a = a + 1 # 1 ns\n a = a * 2 # 10 ns\n # Cycle n times\n for _ in range(n): # 1 ns\n print(0) # 5 ns\n// Under a particular operating platform\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n cout << 0 << endl; // 5 ns\n }\n}\n// Under a particular operating platform\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n System.out.println(0); // 5 ns\n }\n}\n// Under a particular operating platform\nvoid Algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n Console.WriteLine(0); // 5 ns\n }\n}\n// Under a particular operating platform\nfunc algorithm(n int) {\n a := 2 // 1 ns\n a = a + 1 // 1 ns\n a = a * 2 // 10 ns\n // Loop n times\n for i := 0; i < n; i++ { // 1 ns\n fmt.Println(a) // 5 ns\n }\n}\n// Under a particular operating platform\nfunc algorithm(n: Int) {\n var a = 2 // 1 ns\n a = a + 1 // 1 ns\n a = a * 2 // 10 ns\n // Loop n times\n for _ in 0 ..< n { // 1 ns\n print(0) // 5 ns\n }\n}\n// Under a particular operating platform\nfunction algorithm(n) {\n var a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for(let i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n console.log(0); // 5 ns\n }\n}\n// Under a particular operating platform\nfunction algorithm(n: number): void {\n var a: number = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for(let i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n console.log(0); // 5 ns\n }\n}\n// Under a particular operating platform\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n print(0); // 5 ns\n }\n}\n// Under a particular operating platform\nfn algorithm(n: i32) {\n let mut a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for _ in 0..n { // 1 ns for each round i++\n println!(\"{}\", 0); // 5 ns\n }\n}\n// Under a particular operating platform\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n printf(\"%d\", 0); // 5 ns\n }\n}\n// Under a particular operating platform\nfn algorithm(n: usize) void {\n var a: i32 = 2; // 1 ns\n a += 1; // 1 ns\n a *= 2; // 10 ns\n // Loop n times\n for (0..n) |_| { // 1 ns\n std.debug.print(\"{}\\n\", .{0}); // 5 ns\n }\n}\nBased on the above method, the algorithm running time can be obtained as \\(6n + 12\\) ns :
\\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\]In practice, however, statistical algorithm runtimes are neither reasonable nor realistic. First, we do not want to tie the estimation time to the operation platform, because the algorithm needs to run on a variety of different platforms. Second, it is difficult for us to be informed of the runtime of each operation, which makes the prediction process extremely difficult.
"},{"location":"chapter_computational_complexity/time_complexity/#231-trends-in-statistical-time-growth","title":"2.3.1 \u00a0 Trends In Statistical Time Growth","text":"The time complexity analysis counts not the algorithm running time, but the tendency of the algorithm running time to increase as the amount of data gets larger.
The concept of \"time-growing trend\" is rather abstract, so let's try to understand it through an example. Suppose the size of the input data is \\(n\\), and given three algorithmic functions
PythonC++JavaC#GoSwiftJSTSDartRustCZigA,BandC:# Time complexity of algorithm A: constant order\ndef algorithm_A(n: int):\n print(0)\n# Time complexity of algorithm B: linear order\ndef algorithm_B(n: int):\n for _ in range(n):\n print(0)\n# Time complexity of algorithm C: constant order\ndef algorithm_C(n: int):\n for _ in range(1000000):\n print(0)\n// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n cout << 0 << endl;\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n cout << 0 << endl;\n }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n cout << 0 << endl;\n }\n}\n// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n System.out.println(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n System.out.println(0);\n }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n System.out.println(0);\n }\n}\n// Time complexity of algorithm A: constant order\nvoid AlgorithmA(int n) {\n Console.WriteLine(0);\n}\n// Time complexity of algorithm B: linear order\nvoid AlgorithmB(int n) {\n for (int i = 0; i < n; i++) {\n Console.WriteLine(0);\n }\n}\n// Time complexity of algorithm C: constant order\nvoid AlgorithmC(int n) {\n for (int i = 0; i < 1000000; i++) {\n Console.WriteLine(0);\n }\n}\n// Time complexity of algorithm A: constant order\nfunc algorithm_A(n int) {\n fmt.Println(0)\n}\n// Time complexity of algorithm B: linear order\nfunc algorithm_B(n int) {\n for i := 0; i < n; i++ {\n fmt.Println(0)\n }\n}\n// Time complexity of algorithm C: constant order\nfunc algorithm_C(n int) {\n for i := 0; i < 1000000; i++ {\n fmt.Println(0)\n }\n}\n// Time complexity of algorithm A: constant order\nfunc algorithmA(n: Int) {\n print(0)\n}\n\n// Time complexity of algorithm B: linear order\nfunc algorithmB(n: Int) {\n for _ in 0 ..< n {\n print(0)\n }\n}\n\n// Time complexity of algorithm C: constant order\nfunc algorithmC(n: Int) {\n for _ in 0 ..< 1000000 {\n print(0)\n }\n}\n// Time complexity of algorithm A: constant order\nfunction algorithm_A(n) {\n console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n) {\n for (let i = 0; i < n; i++) {\n console.log(0);\n }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n) {\n for (let i = 0; i < 1000000; i++) {\n console.log(0);\n }\n}\n// Time complexity of algorithm A: constant order\nfunction algorithm_A(n: number): void {\n console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n: number): void {\n for (let i = 0; i < n; i++) {\n console.log(0);\n }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n: number): void {\n for (let i = 0; i < 1000000; i++) {\n console.log(0);\n }\n}\n// Time complexity of algorithm A: constant order\nvoid algorithmA(int n) {\n print(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithmB(int n) {\n for (int i = 0; i < n; i++) {\n print(0);\n }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithmC(int n) {\n for (int i = 0; i < 1000000; i++) {\n print(0);\n }\n}\n// Time complexity of algorithm A: constant order\nfn algorithm_A(n: i32) {\n println!(\"{}\", 0);\n}\n// Time complexity of algorithm B: linear order\nfn algorithm_B(n: i32) {\n for _ in 0..n {\n println!(\"{}\", 0);\n }\n}\n// Time complexity of algorithm C: constant order\nfn algorithm_C(n: i32) {\n for _ in 0..1000000 {\n println!(\"{}\", 0);\n }\n}\n// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n printf(\"%d\", 0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n printf(\"%d\", 0);\n }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n printf(\"%d\", 0);\n }\n}\n// Time complexity of algorithm A: constant order\nfn algorithm_A(n: usize) void {\n _ = n;\n std.debug.print(\"{}\\n\", .{0});\n}\n// Time complexity of algorithm B: linear order\nfn algorithm_B(n: i32) void {\n for (0..n) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n}\n// Time complexity of algorithm C: constant order\nfn algorithm_C(n: i32) void {\n _ = n;\n for (0..1000000) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n}\nThe Figure 2-7 shows the time complexity of the above three algorithmic functions.
- Algorithm
Ahas only \\(1\\) print operations, and the running time of the algorithm does not increase with \\(n\\). We call the time complexity of this algorithm \"constant order\". - The print operation in algorithm
Brequires \\(n\\) cycles, and the running time of the algorithm increases linearly with \\(n\\). The time complexity of this algorithm is called \"linear order\". - The print operation in algorithm
Crequires \\(1000000\\) loops, which is a long runtime, but it is independent of the size of the input data \\(n\\). Therefore, the time complexity ofCis the same as that ofA, which is still of \"constant order\".
Figure 2-7 \u00a0 Time growth trends for algorithms A, B and C
What are the characteristics of time complexity analysis compared to direct statistical algorithmic running time?
- The time complexity can effectively evaluate the efficiency of an algorithm. For example, the running time of algorithm
Bincreases linearly and is slower than algorithmAfor \\(n > 1\\) and slower than algorithmCfor \\(n > 1,000,000\\). In fact, as long as the input data size \\(n\\) is large enough, algorithms with \"constant order\" of complexity will always outperform algorithms with \"linear order\", which is exactly what the time complexity trend means. - The time complexity of the projection method is simpler. Obviously, neither the running platform nor the type of computational operation is related to the growth trend of the running time of the algorithm. Therefore, in the time complexity analysis, we can simply consider the execution time of all computation operations as the same \"unit time\", and thus simplify the \"statistics of the running time of computation operations\" to the \"statistics of the number of computation operations\", which is the same as the \"statistics of the number of computation operations\". The difficulty of the estimation is greatly reduced by considering the execution time of all operations as the same \"unit time\".
- There are also some limitations of time complexity. For example, although algorithms
AandChave the same time complexity, the actual running time varies greatly. Similarly, although the time complexity of algorithmBis higher than that ofC, algorithmBsignificantly outperforms algorithmCwhen the size of the input data \\(n\\) is small. In these cases, it is difficult to judge the efficiency of an algorithm based on time complexity alone. Of course, despite the above problems, complexity analysis is still the most effective and commonly used method to judge the efficiency of algorithms.
Given a function with input size \\(n\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef algorithm(n: int):\n a = 1 # +1\n a = a + 1 # +1\n a = a * 2 # +1\n # Cycle n times\n for i in range(n): # +1\n print(0) # +1\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n cout << 0 << endl; // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n System.out.println(0); // +1\n }\n}\nvoid Algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n Console.WriteLine(0); // +1\n }\n}\nfunc algorithm(n int) {\n a := 1 // +1\n a = a + 1 // +1\n a = a * 2 // +1\n // Loop n times\n for i := 0; i < n; i++ { // +1\n fmt.Println(a) // +1\n }\n}\nfunc algorithm(n: Int) {\n var a = 1 // +1\n a = a + 1 // +1\n a = a * 2 // +1\n // Loop n times\n for _ in 0 ..< n { // +1\n print(0) // +1\n }\n}\nfunction algorithm(n) {\n var a = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // Loop n times\n for(let i = 0; i < n; i++){ // +1 (execute i ++ every round)\n console.log(0); // +1\n }\n}\nfunction algorithm(n: number): void{\n var a: number = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // Loop n times\n for(let i = 0; i < n; i++){ // +1 (execute i ++ every round)\n console.log(0); // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n print(0); // +1\n }\n}\nfn algorithm(n: i32) {\n let mut a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n\n // Loop n times\n for _ in 0..n { // +1 (execute i ++ every round)\n println!(\"{}\", 0); // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n printf(\"%d\", 0); // +1\n }\n} \nfn algorithm(n: usize) void {\n var a: i32 = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // Loop n times\n for (0..n) |_| { // +1 (execute i ++ every round)\n std.debug.print(\"{}\\n\", .{0}); // +1\n }\n}\nLet the number of operations of the algorithm be a function of the size of the input data \\(n\\), denoted as \\(T(n)\\) , then the number of operations of the above function is:
\\[ T(n) = 3 + 2n \\]\\(T(n)\\) is a primary function, which indicates that the trend of its running time growth is linear, and thus its time complexity is of linear order.
We denote the time complexity of the linear order as \\(O(n)\\) , and this mathematical notation is called the \"big \\(O\\) notation big-\\(O\\) notation\", which denotes the \"asymptotic upper bound\" of the function \\(T(n)\\).
Time complexity analysis is essentially the computation of asymptotic upper bounds on the \"number of operations function \\(T(n)\\)\", which has a clear mathematical definition.
Function asymptotic upper bound
If there exists a positive real number \\(c\\) and a real number \\(n_0\\) such that \\(T(n) \\leq c \\cdot f(n)\\) for all \\(n > n_0\\) , then it can be argued that \\(f(n)\\) gives an asymptotic upper bound on \\(T(n)\\) , denoted as \\(T(n) = O(f(n))\\) .
As shown in the Figure 2-8 , computing the asymptotic upper bound is a matter of finding a function \\(f(n)\\) such that \\(T(n)\\) and \\(f(n)\\) are at the same growth level as \\(n\\) tends to infinity, differing only by a multiple of the constant term \\(c\\).
Figure 2-8 \u00a0 asymptotic upper bound of function
"},{"location":"chapter_computational_complexity/time_complexity/#233-method-of-projection","title":"2.3.3 \u00a0 Method Of Projection","text":"Asymptotic upper bounds are a bit heavy on math, so don't worry if you feel you don't have a full solution. Because in practice, we only need to master the projection method, and the mathematical meaning can be gradually comprehended.
By definition, after determining \\(f(n)\\), we can get the time complexity \\(O(f(n))\\). So how to determine the asymptotic upper bound \\(f(n)\\)? The overall is divided into two steps: first count the number of operations, and then determine the asymptotic upper bound.
"},{"location":"chapter_computational_complexity/time_complexity/#1-the-first-step-counting-the-number-of-operations","title":"1. \u00a0 The First Step: Counting The Number Of Operations","text":"For the code, it is sufficient to calculate from top to bottom line by line. However, since the constant term \\(c \\cdot f(n)\\) in the above \\(c \\cdot f(n)\\) can take any size, the various coefficients and constant terms in the number of operations \\(T(n)\\) can be ignored. Based on this principle, the following counting simplification techniques can be summarized.
- Ignore the constant terms in \\(T(n)\\). Since none of them are related to \\(n\\), they have no effect on the time complexity.
- omits all coefficients. For example, loops \\(2n\\) times, \\(5n + 1\\) times, etc., can be simplified and notated as \\(n\\) times because the coefficients before \\(n\\) have no effect on the time complexity.
- Use multiplication when loops are nested. The total number of operations is equal to the product of the number of operations of the outer and inner levels of the loop, and each level of the loop can still be nested by applying the techniques in points
1.and2.respectively.
Given a function, we can use the above trick to count the number of operations.
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef algorithm(n: int):\n a = 1 # +0 (trick 1)\n a = a + n # +0 (trick 1)\n # +n (technique 2)\n for i in range(5 * n + 1):\n print(0)\n # +n*n (technique 3)\n for i in range(2 * n):\n for j in range(n + 1):\n print(0)\nvoid algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n cout << 0 << endl;\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n cout << 0 << endl;\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n System.out.println(0);\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n System.out.println(0);\n }\n }\n}\nvoid Algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n Console.WriteLine(0);\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n Console.WriteLine(0);\n }\n }\n}\nfunc algorithm(n int) {\n a := 1 // +0 (trick 1)\n a = a + n // +0 (trick 1)\n // +n (technique 2)\n for i := 0; i < 5 * n + 1; i++ {\n fmt.Println(0)\n }\n // +n*n (technique 3)\n for i := 0; i < 2 * n; i++ {\n for j := 0; j < n + 1; j++ {\n fmt.Println(0)\n }\n }\n}\nfunc algorithm(n: Int) {\n var a = 1 // +0 (trick 1)\n a = a + n // +0 (trick 1)\n // +n (technique 2)\n for _ in 0 ..< (5 * n + 1) {\n print(0)\n }\n // +n*n (technique 3)\n for _ in 0 ..< (2 * n) {\n for _ in 0 ..< (n + 1) {\n print(0)\n }\n }\n}\nfunction algorithm(n) {\n let a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (let i = 0; i < 5 * n + 1; i++) {\n console.log(0);\n }\n // +n*n (technique 3)\n for (let i = 0; i < 2 * n; i++) {\n for (let j = 0; j < n + 1; j++) {\n console.log(0);\n }\n }\n}\nfunction algorithm(n: number): void {\n let a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (let i = 0; i < 5 * n + 1; i++) {\n console.log(0);\n }\n // +n*n (technique 3)\n for (let i = 0; i < 2 * n; i++) {\n for (let j = 0; j < n + 1; j++) {\n console.log(0);\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n print(0);\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n print(0);\n }\n }\n}\nfn algorithm(n: i32) {\n let mut a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n\n // +n (technique 2)\n for i in 0..(5 * n + 1) {\n println!(\"{}\", 0);\n }\n\n // +n*n (technique 3)\n for i in 0..(2 * n) {\n for j in 0..(n + 1) {\n println!(\"{}\", 0);\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n printf(\"%d\", 0);\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n printf(\"%d\", 0);\n }\n }\n}\nfn algorithm(n: usize) void {\n var a: i32 = 1; // +0 (trick 1)\n a = a + @as(i32, @intCast(n)); // +0 (trick 1)\n\n // +n (technique 2)\n for(0..(5 * n + 1)) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n\n // +n*n (technique 3)\n for(0..(2 * n)) |_| {\n for(0..(n + 1)) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n }\n}\nThe following equations show the statistical results before and after using the above technique, both of which were introduced with a time complexity of \\(O(n^2)\\) .
\\[ \\begin{aligned} T(n) & = 2n(n + 1) + (5n + 1) + 2 & \\text{complete statistics (-.-|||)} \\newline & = 2n^2 + 7n + 3 \\newline T(n) & = n^2 + n & \\text{Lazy Stats (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2-step-2-judging-the-asymptotic-upper-bounds","title":"2. \u00a0 Step 2: Judging The Asymptotic Upper Bounds","text":"The time complexity is determined by the highest order term in the polynomial \\(T(n)\\). This is because as \\(n\\) tends to infinity, the highest order term will play a dominant role and the effects of all other terms can be ignored.
The Table 2-2 shows some examples, some of which have exaggerated values to emphasize the conclusion that \"the coefficients can't touch the order\". As \\(n\\) tends to infinity, these constants become irrelevant.
Table 2-2 \u00a0 Time complexity corresponding to different number of operations
number of operations \\(T(n)\\) time complexity \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234-common-types","title":"2.3.4 \u00a0 Common Types","text":"Let the input data size be \\(n\\) , the common types of time complexity are shown in the Figure 2-9 (in descending order).
\\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n \\log n) < O(n^2) < O(2^n) < O(n!) \\newline \\text{constant order} < \\text{logarithmic order} < \\text{linear order} < \\text{linear logarithmic order} < \\text{square order} < \\text{exponential order} < \\text{multiplication order} \\end{aligned} \\]Figure 2-9 \u00a0 Common time complexity types
"},{"location":"chapter_computational_complexity/time_complexity/#1-constant-order-o1","title":"1. \u00a0 Constant Order \\(O(1)\\)","text":"The number of operations of the constant order is independent of the input data size \\(n\\), i.e., it does not change with \\(n\\).
In the following function, although the number of operations
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.pysizemay be large, the time complexity is still \\(O(1)\\) because it is independent of the input data size \\(n\\) :
time_complexity.cppdef constant(n: int) -> int:\n \"\"\"\u5e38\u6570\u9636\"\"\"\n count = 0\n size = 100000\n for _ in range(size):\n count += 1\n return count\n
time_complexity.java/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.cs/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.go/* \u5e38\u6570\u9636 */\nint Constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.swift/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\n count := 0\n size := 100000\n for i := 0; i < size; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -> Int {\n var count = 0\n let size = 100_000\n for _ in 0 ..< size {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n let count = 0;\n const size = 100000;\n for (let i = 0; i < size; i++) count++;\n return count;\n}\n
time_complexity.dart/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\n let count = 0;\n const size = 100000;\n for (let i = 0; i < size; i++) count++;\n return count;\n}\n
time_complexity.rs/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (var i = 0; i < size; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -> i32 {\n _ = n;\n let mut count = 0;\n let size = 100_000;\n for _ in 0..size {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n int i = 0;\n for (int i = 0; i < size; i++) {\n count++;\n }\n return count;\n}\n
"},{"location":"chapter_computational_complexity/time_complexity/#2-linear-order-on","title":"2. \u00a0 Linear Order \\(O(N)\\)","text":"// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n _ = n;\n var count: i32 = 0;\n const size: i32 = 100_000;\n var i: i32 = 0;\n while(i<size) : (i += 1) {\n count += 1;\n }\n return count;\n}\nThe number of operations in a linear order grows in linear steps relative to the input data size \\(n\\). Linear orders are usually found in single level loops:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef linear(n: int) -> int:\n \"\"\"\u7ebf\u6027\u9636\"\"\"\n count = 0\n for _ in range(n):\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\n count := 0\n for i := 0; i < n; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -> Int {\n var count = 0\n for _ in 0 ..< n {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n let count = 0;\n for (let i = 0; i < n; i++) count++;\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\n let count = 0;\n for (let i = 0; i < n; i++) count++;\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (var i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -> i32 {\n let mut count = 0;\n for _ in 0..n {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\n var count: i32 = 0;\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n count += 1;\n }\n return count;\n}\nThe time complexity of operations such as traversing an array and traversing a linked list is \\(O(n)\\) , where \\(n\\) is the length of the array or linked list:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef array_traversal(nums: list[int]) -> int:\n \"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\n count = 0\n # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for num in nums:\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector<int> &nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int num : nums) {\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int num : nums) {\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n foreach (int num in nums) {\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\n count := 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for range nums {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -> Int {\n var count = 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for _ in nums {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (let i = 0; i < nums.length; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (let i = 0; i < nums.length; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List<int> nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (var _num in nums) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &[i32]) -> i32 {\n let mut count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for _ in nums {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\n var count: i32 = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (nums) |_| {\n count += 1;\n }\n return count;\n}\nIt is worth noting that Input data size \\(n\\) needs to be determined specifically according to the type of input data. For example, in the first example, the variable \\(n\\) is the input data size; in the second example, the array length \\(n\\) is the data size.
"},{"location":"chapter_computational_complexity/time_complexity/#3-squared-order-on2","title":"3. \u00a0 Squared Order \\(O(N^2)\\)","text":"The number of operations in the square order grows in square steps with respect to the size of the input data \\(n\\). The squared order is usually found in nested loops, where both the outer and inner levels are \\(O(n)\\) and therefore overall \\(O(n^2)\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef quadratic(n: int) -> int:\n \"\"\"\u5e73\u65b9\u9636\"\"\"\n count = 0\n # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for i in range(n):\n for j in range(n):\n count += 1\n return count\n
time_complexity.java/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.cs/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.go/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.swift/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\n count := 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for i := 0; i < n; i++ {\n for j := 0; j < n; j++ {\n count++\n }\n }\n return count\n}\n
time_complexity.js/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -> Int {\n var count = 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for _ in 0 ..< n {\n for _ in 0 ..< n {\n count += 1\n }\n }\n return count\n}\n
time_complexity.ts/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.dart/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.rs/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.c/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -> i32 {\n let mut count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for _ in 0..n {\n for _ in 0..n {\n count += 1;\n }\n }\n count\n}\n
time_complexity.zig/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\n var count: i32 = 0;\n var i: i32 = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n while (i < n) : (i += 1) {\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n count += 1;\n }\n }\n return count;\n}\nThe Figure 2-10 compares the three time complexities of constant order, linear order and squared order.
Figure 2-10 \u00a0 Time complexity of constant, linear and quadratic orders
Taking bubble sort as an example, the outer loop executes \\(n - 1\\) times, and the inner loop executes \\(n-1\\), \\(n-2\\), \\(\\dots\\), \\(2\\), \\(1\\) times, which averages out to \\(n / 2\\) times, resulting in a time complexity of \\(O((n - 1) n / 2) = O(n^2)\\) .
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef bubble_sort(nums: list[int]) -> int:\n \"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\n count = 0 # \u8ba1\u6570\u5668\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in range(len(nums) - 1, 0, -1):\n # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j in range(i):\n if nums[j] > nums[j + 1]:\n # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n tmp: int = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n count += 3 # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n return count\n
time_complexity.java/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector<int> &nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.size() - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.cs/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.go/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.Length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.swift/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n count := 0 // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := len(nums) - 1; i > 0; i-- {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j := 0; j < i; j++ {\n if nums[j] > nums[j+1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n tmp := nums[j]\n nums[j] = nums[j+1]\n nums[j+1] = tmp\n count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count\n}\n
time_complexity.js/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -> Int {\n var count = 0 // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0 ..< i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count\n}\n
time_complexity.ts/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n let count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.dart/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n let count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.rs/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List<int> nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (var i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (var j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.c/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &mut [i32]) -> i32 {\n let mut count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in (1..nums.len()).rev() {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0..i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n count\n}\n
time_complexity.zig/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = n - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
"},{"location":"chapter_computational_complexity/time_complexity/#235-exponential-order-o2n","title":"2.3.5 \u00a0 Exponential Order \\(O(2^N)\\)","text":"// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n var count: i32 = 0; // \u8ba1\u6570\u5668 \n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n while (i > 0) : (i -= 1) {\n var j: usize = 0;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n while (j < i) : (j += 1) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n var tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\nCell division in biology is a typical example of exponential growth: the initial state is \\(1\\) cells, after one round of division it becomes \\(2\\), after two rounds of division it becomes \\(4\\), and so on, after \\(n\\) rounds of division there are \\(2^n\\) cells.
The Figure 2-11 and the following code simulate the process of cell division with a time complexity of \\(O(2^n)\\) .
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef exponential(n: int) -> int:\n \"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n count = 0\n base = 1\n # \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in range(n):\n for _ in range(base):\n count += 1\n base *= 2\n # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n
time_complexity.java/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.cs/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.go/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\n int count = 0, bas = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < bas; j++) {\n count++;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.swift/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\n count, base := 0, 1\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for i := 0; i < n; i++ {\n for j := 0; j < base; j++ {\n count++\n }\n base *= 2\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n}\n
time_complexity.js/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -> Int {\n var count = 0\n var base = 1\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in 0 ..< n {\n for _ in 0 ..< base {\n count += 1\n }\n base *= 2\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n}\n
time_complexity.ts/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\n let count = 0,\n base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.dart/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\n let count = 0,\n base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.rs/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (var i = 0; i < n; i++) {\n for (var j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.c/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -> i32 {\n let mut count = 0;\n let mut base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in 0..n {\n for _ in 0..base {\n count += 1\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n count\n}\n
time_complexity.zig/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0;\n int bas = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < bas; j++) {\n count++;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\n var count: i32 = 0;\n var bas: i32 = 1;\n var i: i32 = 0;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n while (i < n) : (i += 1) {\n var j: i32 = 0;\n while (j < bas) : (j += 1) {\n count += 1;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\nFigure 2-11 \u00a0 time complexity of exponential order
In practical algorithms, exponential orders are often found in recursion functions. For example, in the following code, it recursively splits in two and stops after \\(n\\) splits:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef exp_recur(n: int) -> int:\n \"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n == 1:\n return 1\n return exp_recur(n - 1) + exp_recur(n - 1) + 1\n
time_complexity.java/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.cs/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.go/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\n if (n == 1) return 1;\n return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n
time_complexity.swift/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\n if n == 1 {\n return 1\n }\n return expRecur(n-1) + expRecur(n-1) + 1\n}\n
time_complexity.js/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -> Int {\n if n == 1 {\n return 1\n }\n return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n
time_complexity.ts/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\n if (n === 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.dart/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\n if (n === 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.rs/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.c/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -> i32 {\n if n == 1 {\n return 1;\n }\n exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n
time_complexity.zig/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\n if (n == 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\nExponential order grows very rapidly and is more common in exhaustive methods (brute force search, backtracking, etc.). For problems with large data sizes, exponential order is unacceptable and usually requires the use of algorithms such as dynamic programming or greedy algorithms to solve.
"},{"location":"chapter_computational_complexity/time_complexity/#1-logarithmic-order-olog-n","title":"1. \u00a0 Logarithmic Order \\(O(\\Log N)\\)","text":"In contrast to the exponential order, the logarithmic order reflects the \"each round is reduced to half\" case. Let the input data size be \\(n\\), and since each round is reduced to half, the number of loops is \\(\\log_2 n\\), which is the inverse function of \\(2^n\\).
The Figure 2-12 and the code below simulate the process of \"reducing each round to half\" with a time complexity of \\(O(\\log_2 n)\\), which is abbreviated as \\(O(\\log n)\\).
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef logarithmic(n: float) -> int:\n \"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n count = 0\n while n > 1:\n n = n / 2\n count += 1\n return count\n
time_complexity.java/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n /= 2;\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\n count := 0\n for n > 1 {\n n = n / 2\n count++\n }\n return count\n}\n
time_complexity.js/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -> Int {\n var count = 0\n var n = n\n while n > 1 {\n n = n / 2\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\n let count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\n let count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(num n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: f32) -> i32 {\n let mut count = 0;\n while n > 1.0 {\n n = n / 2.0;\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\n var count: i32 = 0;\n var n_var = n;\n while (n_var > 1)\n {\n n_var = n_var / 2;\n count +=1;\n }\n return count;\n}\nFigure 2-12 \u00a0 time complexity of logarithmic order
Similar to the exponential order, the logarithmic order is often found in recursion functions. The following code forms a recursion tree of height \\(\\log_2 n\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef log_recur(n: float) -> int:\n \"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n <= 1:\n return 0\n return log_recur(n / 2) + 1\n
time_complexity.java/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.cs/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.go/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(float n) {\n if (n <= 1) return 0;\n return LogRecur(n / 2) + 1;\n}\n
time_complexity.swift/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\n if n <= 1 {\n return 0\n }\n return logRecur(n/2) + 1\n}\n
time_complexity.js/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -> Int {\n if n <= 1 {\n return 0\n }\n return logRecur(n: n / 2) + 1\n}\n
time_complexity.ts/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.dart/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.rs/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(num n) {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.c/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: f32) -> i32 {\n if n <= 1.0 {\n return 0;\n }\n log_recur(n / 2.0) + 1\n}\n
time_complexity.zig/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\nLogarithmic order is often found in algorithms based on the divide and conquer strategy, which reflects the algorithmic ideas of \"dividing one into many\" and \"simplifying the complexity into simplicity\". It grows slowly and is the second most desirable time complexity after constant order.
What is the base of \\(O(\\log n)\\)?
To be precise, the corresponding time complexity of \"one divided into \\(m\\)\" is \\(O(\\log_m n)\\) . And by using the logarithmic permutation formula, we can get equal time complexity with different bases:
\\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\]That is, the base \\(m\\) can be converted without affecting the complexity. Therefore we usually omit the base \\(m\\) and write the logarithmic order directly as \\(O(\\log n)\\).
"},{"location":"chapter_computational_complexity/time_complexity/#2-linear-logarithmic-order-on-log-n","title":"2. \u00a0 Linear Logarithmic Order \\(O(N \\Log N)\\)","text":"The linear logarithmic order is often found in nested loops, and the time complexity of the two levels of loops is \\(O(\\log n)\\) and \\(O(n)\\) respectively. The related code is as follows:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef linear_log_recur(n: float) -> int:\n \"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\n if n <= 1:\n return 1\n count: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n for _ in range(n):\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(float n) {\n if (n <= 1) return 1;\n int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\n if n <= 1 {\n return 1\n }\n count := linearLogRecur(n/2) + linearLogRecur(n/2)\n for i := 0.0; i < n; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -> Int {\n if n <= 1 {\n return 1\n }\n var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n for _ in stride(from: 0, to: n, by: 1) {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\n if (n <= 1) return 1;\n let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (let i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\n if (n <= 1) return 1;\n let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (let i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(num n) {\n if (n <= 1) return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (var i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: f32) -> i32 {\n if n <= 1.0 {\n return 1;\n }\n let mut count = linear_log_recur(n / 2.0) + linear_log_recur(n / 2.0);\n for _ in 0 ..n as i32 {\n count += 1;\n }\n return count\n}\n
time_complexity.zig/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\n if (n <= 1) return 1;\n var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n var i: f32 = 0;\n while (i < n) : (i += 1) {\n count += 1;\n }\n return count;\n}\nThe Figure 2-13 shows how the linear logarithmic order is generated. The total number of operations at each level of the binary tree is \\(n\\) , and the tree has a total of \\(\\log_2 n + 1\\) levels, resulting in a time complexity of \\(O(n\\log n)\\) .
Figure 2-13 \u00a0 Time complexity of linear logarithmic order
Mainstream sorting algorithms typically have a time complexity of \\(O(n \\log n)\\) , such as quick sort, merge sort, heap sort, etc.
"},{"location":"chapter_computational_complexity/time_complexity/#3-the-factorial-order-on","title":"3. \u00a0 The Factorial Order \\(O(N!)\\)","text":"The factorial order corresponds to the mathematical \"permutations problem\". Given \\(n\\) elements that do not repeat each other, find all possible permutations of them, the number of permutations being:
\\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\]Factorials are usually implemented using recursion. As shown in the Figure 2-14 and in the code below, the first level splits \\(n\\), the second level splits \\(n - 1\\), and so on, until the splitting stops at the \\(n\\)th level:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef factorial_recur(n: int) -> int:\n \"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n == 0:\n return 1\n count = 0\n # \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in range(n):\n count += factorial_recur(n - 1)\n return count\n
time_complexity.java/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.cs/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.go/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\n if (n == 0) return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += FactorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.swift/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\n if n == 0 {\n return 1\n }\n count := 0\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for i := 0; i < n; i++ {\n count += factorialRecur(n - 1)\n }\n return count\n}\n
time_complexity.js/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -> Int {\n if n == 0 {\n return 1\n }\n var count = 0\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in 0 ..< n {\n count += factorialRecur(n: n - 1)\n }\n return count\n}\n
time_complexity.ts/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\n if (n === 0) return 1;\n let count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (let i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.dart/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\n if (n === 0) return 1;\n let count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (let i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.rs/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0) return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (var i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.c/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -> i32 {\n if n == 0 {\n return 1;\n }\n let mut count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in 0..n {\n count += factorial_recur(n - 1);\n }\n count\n}\n
time_complexity.zig/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\n if (n == 0) return 1;\n var count: i32 = 0;\n var i: i32 = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n while (i < n) : (i += 1) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\nFigure 2-14 \u00a0 Time complexity of the factorial order
Note that since there is always \\(n! > 2^n\\) when \\(n \\geq 4\\), the factorial order grows faster than the exponential order, and is also unacceptable when \\(n\\) is large.
"},{"location":"chapter_computational_complexity/time_complexity/#236-worst-best-average-time-complexity","title":"2.3.6 \u00a0 Worst, Best, Average Time Complexity","text":"The time efficiency of algorithms is often not fixed, but is related to the distribution of the input data. Suppose an array
numsof length \\(n\\) is input, wherenumsconsists of numbers from \\(1\\) to \\(n\\), each of which occurs only once; however, the order of the elements is randomly upset, and the goal of the task is to return the index of element \\(1\\). We can draw the following conclusion.- When
nums = [? , ? , ... , 1], i.e., when the end element is \\(1\\), a complete traversal of the array is required, to reach the worst time complexity \\(O(n)\\) . - When
nums = [1, ? , ? , ...], i.e., when the first element is \\(1\\) , there is no need to continue traversing the array no matter how long it is, reaching the optimal time complexity \\(\\Omega(1)\\) .
The \"worst time complexity\" corresponds to the asymptotic upper bound of the function and is denoted by the large \\(O\\) notation. Correspondingly, the \"optimal time complexity\" corresponds to the asymptotic lower bound of the function and is denoted in \\(\\Omega\\) notation:
PythonC++JavaC#GoSwiftJSTSDartRustCZig worst_best_time_complexity.py
worst_best_time_complexity.cppdef random_numbers(n: int) -> list[int]:\n \"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n # \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\n nums = [i for i in range(1, n + 1)]\n # \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n random.shuffle(nums)\n return nums\n\ndef find_one(nums: list[int]) -> int:\n \"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n for i in range(len(nums)):\n # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1:\n return i\n return -1\n
worst_best_time_complexity.java/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector<int> randomNumbers(int n) {\n vector<int> nums(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\n unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector<int> &nums) {\n for (int i = 0; i < nums.size(); i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.cs/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\n Integer[] nums = new Integer[n];\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n Collections.shuffle(Arrays.asList(nums));\n // Integer[] -> int[]\n int[] res = new int[n];\n for (int i = 0; i < n; i++) {\n res[i] = nums[i];\n }\n return res;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n for (int i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.go/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\n int[] nums = new int[n];\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (int i = 0; i < nums.Length; i++) {\n int index = new Random().Next(i, nums.Length);\n (nums[i], nums[index]) = (nums[index], nums[i]);\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n for (int i = 0; i < nums.Length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.swift/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\n nums := make([]int, n)\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for i := 0; i < n; i++ {\n nums[i] = i + 1\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n rand.Shuffle(len(nums), func(i, j int) {\n nums[i], nums[j] = nums[j], nums[i]\n })\n return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n for i := 0; i < len(nums); i++ {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return i\n }\n }\n return -1\n}\n
worst_best_time_complexity.js/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -> [Int] {\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n var nums = Array(1 ... n)\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle()\n return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -> Int {\n for i in nums.indices {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return i\n }\n }\n return -1\n}\n
worst_best_time_complexity.ts/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\n const nums = Array(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (let i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (let i = 0; i < n; i++) {\n const r = Math.floor(Math.random() * (i + 1));\n const temp = nums[i];\n nums[i] = nums[r];\n nums[r] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n for (let i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] === 1) {\n return i;\n }\n }\n return -1;\n}\n
worst_best_time_complexity.dart/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\n const nums = Array(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (let i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (let i = 0; i < n; i++) {\n const r = Math.floor(Math.random() * (i + 1));\n const temp = nums[i];\n nums[i] = nums[r];\n nums[r] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n for (let i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] === 1) {\n return i;\n }\n }\n return -1;\n}\n
worst_best_time_complexity.rs/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList<int> randomNumbers(int n) {\n final nums = List.filled(n, 0);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (var i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle();\n\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List<int> nums) {\n for (var i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1) return i;\n }\n\n return -1;\n}\n
worst_best_time_complexity.c/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -> Vec<i32> {\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n let mut nums = (1..=n).collect::<Vec<i32>>();\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle(&mut thread_rng());\n nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &[i32]) -> Option<usize> {\n for i in 0..nums.len() {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return Some(i);\n }\n }\n None\n}\n
worst_best_time_complexity.zig/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n // \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\n int *nums = (int *)malloc(n * sizeof(int));\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (int i = n - 1; i > 0; i--) {\n int j = rand() % (i + 1);\n int temp = nums[i];\n nums[i] = nums[j];\n nums[j] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n for (int i = 0; i < n; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\n var nums: [n]i32 = undefined;\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (&nums, 0..) |*num, i| {\n num.* = @as(i32, @intCast(i)) + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n const rand = std.crypto.random;\n rand.shuffle(i32, &nums);\n return nums;\n}\n\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n for (nums, 0..) |num, i| {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (num == 1) return @intCast(i);\n }\n return -1;\n}\nIt is worth stating that we rarely use the optimal time complexity in practice because it is usually only attainable with a small probability and may be somewhat misleading. whereas the worst time complexity is more practical because it gives a safe value for efficiency and allows us to use the algorithm with confidence.
From the above examples, it can be seen that the worst or best time complexity only occurs in \"special data distributions\", and the probability of these cases may be very small, which does not truly reflect the efficiency of the algorithm. In contrast, the average time complexity of can reflect the efficiency of the algorithm under random input data, which is denoted by the \\(\\Theta\\) notation.
For some algorithms, we can simply derive the average case under a random data distribution. For example, in the above example, since the input array is scrambled, the probability of an element \\(1\\) appearing at any index is equal, so the average number of loops of the algorithm is half of the length of the array \\(n / 2\\) , and the average time complexity is \\(\\Theta(n / 2) = \\Theta(n)\\) .
However, for more complex algorithms, calculating the average time complexity is often difficult because it is hard to analyze the overall mathematical expectation given the data distribution. In this case, we usually use the worst time complexity as a criterion for the efficiency of the algorithm.
Why do you rarely see the \\(\\Theta\\) symbol?
Perhaps because the \\(O\\) symbol is so catchy, we often use it to denote average time complexity. However, this practice is not standardized in the strict sense. In this book and other sources, if you encounter a statement like \"average time complexity \\(O(n)\\)\", please understand it as \\(\\Theta(n)\\).
"},{"location":"chapter_data_structure/","title":"Data Structure","text":"Abstract
Data structures are like a solid and varied framework.
It provides a blueprint for the orderly organization of data upon which algorithms can come alive.
"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"Classification Of Data Structures","text":"Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be divided into two categories: logical structure and physical structure.
"},{"location":"chapter_data_structure/classification_of_data_structure/#logical-structures-linear-and-non-linear","title":"Logical Structures: Linear And Non-linear","text":"Logical structures reveal logical relationships between data elements. In arrays and linked lists, data are arranged in sequential order, reflecting the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between ancestors and descendants; and graphs are composed of nodes and edges, reflecting the complex network relationship.
As shown in the figure below, logical structures can further be divided into \"linear data structure\" and \"non-linear data structure\". Linear data structures are more intuitive, meaning that the data are arranged linearly in terms of logical relationships; non-linear data structures, on the other hand, are arranged non-linearly.
- Linear data structures: arrays, linked lists, stacks, queues, hash tables.
- Nonlinear data structures: trees, heaps, graphs, hash tables.
Non-linear data structures can be further divided into tree and graph structures.
- Linear structures: arrays, linked lists, queues, stacks, hash tables, with one-to-one sequential relationship between elements.
- Tree structure: tree, heap, hash table, with one-to-many relationship between elements.
- Graph: graph with many-to-many relationship between elements.
When an algorithm is running, the data being processed is stored in memory. The figure below shows a computer memory module where each black square represents a memory space. We can think of the memory as a giant Excel sheet in which each cell can store data of a certain size.
The system accesses the data at the target location by means of a memory address. As shown in the figure below, the computer assigns a unique identifier to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data in memory.
Tip
It is worth noting that comparing memory to the Excel sheet is a simplified analogy. The actual memory working mechanism is more complicated, involving the concepts of address, space, memory management, cache mechanism, virtual and physical memory.
Memory is a shared resource for all programs, and when a block of memory is occupied by one program, it cannot be used by other programs at the same time. Therefore, considering memory resources is crucial in designing data structures and algorithms. For example, the algorithm's peak memory usage should not exceed the remaining free memory of the system; if there is a lack of contiguous memory blocks, then the data structure chosen must be able to be stored in non-contiguous memory blocks.
As shown in the figure below, Physical structure reflects the way data is stored in computer memory and it can be divided into consecutive space storage (arrays) and distributed space storage (linked lists). The physical structure determines how data is accessed, updated, added, deleted, etc. Logical and physical structure complement each other in terms of time efficiency and space efficiency.
It is worth stating that all data structures are implemented based on arrays, linked lists, or a combination of the two. For example, stacks and queues can be implemented using both arrays and linked lists; and implementations of hash tables may contain both arrays and linked lists.
- Array-based structures: stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays of dimension \\(\\geq 3\\)), and so on.
- Linked list-based structures: stacks, queues, hash tables, trees, heaps, graphs, etc.
Data structures based on arrays are also known as \"static data structures\", which means that such structures' length remains constant after initialization. In contrast, data structures based on linked lists are called \"dynamic data structures\", meaning that their length can be adjusted during program execution after initialization.
Tip
If you find it difficult to understand the physical structure, it is recommended that you read the next chapter, \"Arrays and Linked Lists,\" before reviewing this section.
"},{"location":"chapter_introduction/","title":"Chapter 1. \u00a0 Introduction to Algorithms","text":"Abstract
A graceful maiden dances, intertwined with the data, her skirt swaying to the melody of algorithms.
She invites you to a dance, follow her steps, and enter the world of algorithms full of logic and beauty.
"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 1.1 \u00a0 Algorithms Are Everywhere
- 1.2 \u00a0 What Is Algorithms
- 1.3 \u00a0 Summary
When we hear the word \"algorithm\", we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which is ubiquitous in our daily lives.
Before we formally discuss algorithms, an interesting fact is worth sharing: you have already learned many algorithms unconsciously and have become accustomed to applying them in your daily life. Below, I will give a few specific examples to prove this point.
Example 1: Looking Up a Dictionary. In a standard dictionary, each word corresponds to a phonetic transcription and the dictionary is organized alphabetically based on these transcriptions. Let's say we're looking for a word that begins with the letter \\(r\\). This is typically done in the following way:
- Open the dictionary around its midpoint and note the first letter on that page, assuming it to be \\(m\\).
- Given the sequence of words following the initial letter \\(m\\), estimate where words starting with the letter \\(r\\) might be located within the alphabetical order.
- Iterate steps
1.and2.until you find the page where the word begins with the letter \\(r\\).
Figure 1-1 \u00a0 Dictionary search step
The skill of looking up a dictionary, essential for elementary school students, is actually the renowned binary search algorithm. Through the lens of data structures, we can view the dictionary as a sorted \"array\"; while from an algorithmic perspective, the series of operations in looking up a dictionary can be seen as \"binary search\".
Example 2: Organizing Playing Cards. When playing cards, we need to arrange the cards in ascending order each game, as shown in the following process.
- Divide the playing cards into \"ordered\" and \"unordered\" parts, assuming initially that the leftmost card is already ordered.
- Take out a card from the unordered part and insert it into the correct position in the ordered part; once completed, the leftmost two cards will be in an ordered sequence.
- Continue the loop described in step
2., each iteration involving insertion of one card from the unordered segment into the ordered portion, until all cards are appropriately ordered.
Figure 1-2 \u00a0 Playing cards sorting process
The above method of organizing playing cards is essentially the \"insertion sort\" algorithm, which is very efficient for small datasets. Many programming languages' sorting library functions include insertion sort.
Example 3: Making Change. Suppose we buy goods worth \\(69\\) yuan at a supermarket and give the cashier \\(100\\) yuan, then the cashier needs to give us \\(31\\) yuan in change. They would naturally complete the thought process as shown below.
- The options are currencies smaller than \\(31\\), including \\(1\\), \\(5\\), \\(10\\), and \\(20\\).
- Take out the largest \\(20\\) from the options, leaving \\(31 - 20 = 11\\).
- Take out the largest \\(10\\) from the remaining options, leaving \\(11 - 10 = 1\\).
- Take out the largest \\(1\\) from the remaining options, leaving \\(1 - 1 = 0\\).
- Complete the change-making, with the solution being \\(20 + 10 + 1 = 31\\).
Figure 1-3 \u00a0 Change making process
In the aforementioned steps, at each stage, we make the optimal choice (utilizing the highest denomination possible), ultimately deriving at a feasible change-making approach. From the perspective of data structures and algorithms, this approach is essentially a \"greedy\" algorithm.
From preparing a dish to traversing interstellar realms, virtually every problem-solving endeavor relies on algorithms. The emergence of computers enables us to store data structures in memory and write code to call CPUs and GPUs to execute algorithms. Consequently, we can transfer real-life predicaments to computers, efficiently addressing a myriad of complex issues.
Tip
If concepts such as data structures, algorithms, arrays, and binary search still seem somewhat obsecure, I encourage you to continue reading. This book will gently guide you into the realm of understanding data structures and algorithms.
"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 Summary","text":"- Algorithms are ubiquitous in daily life and are not as inaccessible and complex as they might seem. In fact, we have already unconsciously learned many algorithms to solve various problems in life.
- The principle of looking up a word in a dictionary is consistent with the binary search algorithm. The binary search algorithm embodies the important algorithmic concept of divide and conquer.
- The process of organizing playing cards is very similar to the insertion sort algorithm. The insertion sort algorithm is suitable for sorting small datasets.
- The steps of making change in currency essentially follow the greedy algorithm, where each step involves making the best possible choice at the moment.
- An algorithm is a set of instructions or steps used to solve a specific problem within a finite amount of time, while a data structure is the way data is organized and stored in a computer.
- Data structures and algorithms are closely linked. Data structures are the foundation of algorithms, and algorithms are the stage to utilize the functions of data structures.
- We can liken data structures and algorithms to building blocks. The blocks represent data, the shape and connection method of the blocks represent data structures, and the steps of assembling the blocks correspond to algorithms.
An \"algorithm\" is a set of instructions or steps to solve a specific problem within a finite amount of time. It has the following characteristics:
- The problem is clearly defined, including unambiguous definitions of input and output.
- The algorithm is feasible, meaning it can be completed within a finite number of steps, time, and memory space.
- Each step has a definitive meaning. The output is consistently the same under the same inputs and conditions.
A \"data structure\" is a way of organizing and storing data in a computer, with the following design goals:
- Minimize space occupancy to save computer memory.
- Make data operations as fast as possible, covering data access, addition, deletion, updating, etc.
- Provide concise data representation and logical information to enable efficient algorithm execution.
Designing data structures is a balancing act, often requiring trade-offs. If you want to improve in one aspect, you often need to compromise in another. Here are two examples:
- Compared to arrays, linked lists offer more convenience in data addition and deletion but sacrifice data access speed.
- Graphs, compared to linked lists, provide richer logical information but require more memory space.
As shown in the diagram below, data structures and algorithms are highly related and closely integrated, specifically in the following three aspects:
- Data structures are the foundation of algorithms. They provide structured data storage and methods for manipulating data for algorithms.
- Algorithms are the stage where data structures come into play. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved.
- Algorithms can often be implemented based on different data structures, but their execution efficiency can vary greatly. Choosing the right data structure is key.
Figure 1-4 \u00a0 Relationship between data structures and algorithms
Data structures and algorithms can be likened to a set of building blocks, as illustrated in the Figure 1-5 . A building block set includes numerous pieces, accompanied by detailed assembly instructions. Following these instructions step by step allows us to construct an intricate block model.
Figure 1-5 \u00a0 Assembling blocks
The detailed correspondence between the two is shown in the Table 1-1 .
Table 1-1 \u00a0 Comparing Data Structures and Algorithms to Building Blocks
Data Structures and Algorithms Building Blocks Input data Unassembled blocks Data structure Organization of blocks, including shape, size, connections, etc Algorithm A series of steps to assemble the blocks into the desired shape Output data Completed Block modelIt's worth noting that data structures and algorithms are independent of programming languages. For this reason, this book is able to provide implementations in multiple programming languages.
Conventional Abbreviation
In real-life discussions, we often refer to \"Data Structures and Algorithms\" simply as \"Algorithms\". For example, the well-known LeetCode algorithm problems actually test both data structure and algorithm knowledge.
"},{"location":"chapter_preface/","title":"Chapter 0. \u00a0 Preface","text":"Abstract
Algorithms are like a beautiful symphony, with each line of code flowing like a rhythm.
May this book ring softly in your head, leaving a unique and profound melody.
"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 0.1 \u00a0 The Book
- 0.2 \u00a0 How to Read
- 0.3 \u00a0 Summary
The aim of this project is to create an open source, free, novice-friendly introductory tutorial on data structures and algorithms.
- Animated graphs are used throughout the book to structure the knowledge of data structures and algorithms in a way that is clear and easy to understand with a smooth learning curve.
- The source code of the algorithms can be run with a single click, supporting Java, C++, Python, Go, JS, TS, C#, Swift, Rust, Dart, Zig and other languages.
- Readers are encouraged to help each other and make progress in the chapter discussion forums, and questions and comments can usually be answered within two days.
If you are a beginner to algorithms, have never touched an algorithm before, or already have some experience brushing up on data structures and algorithms, and have a vague understanding of data structures and algorithms, repeatedly jumping sideways between what you can and can't do, then this book is just for you!
If you have already accumulated a certain amount of questions and are familiar with most of the question types, then this book can help you review and organize the algorithm knowledge system, and the repository source code can be used as a \"brushing tool library\" or \"algorithm dictionary\".
If you are an algorithm expert, we look forward to receiving your valuable suggestions or participate in the creation together.
precondition
You will need to have at least a basic knowledge of programming in any language and be able to read and write simple code.
"},{"location":"chapter_preface/about_the_book/#012-content-structure","title":"0.1.2 \u00a0 Content Structure","text":"The main contents of the book are shown in the Figure 0-1 .
- Complexity Analysis: dimensions and methods of evaluation of data structures and algorithms. Methods of deriving time complexity, space complexity, common types, examples, etc.
- Data Structures: basic data types, classification methods of data structures. Definition, advantages and disadvantages, common operations, common types, typical applications, implementation methods of data structures such as arrays, linked lists, stacks, queues, hash tables, trees, heaps, graphs, etc.
- Algorithms: definitions, advantages and disadvantages, efficiency, application scenarios, solution steps, sample topics of search, sorting algorithms, divide and conquer, backtracking algorithms, dynamic programming, greedy algorithms, and more.
Figure 0-1 \u00a0 Hello Algo content structure
"},{"location":"chapter_preface/about_the_book/#013-acknowledgements","title":"0.1.3 \u00a0 Acknowledgements","text":"During the creation of this book, I received help from many people, including but not limited to:
- Thank you to my mentor at the company, Dr. Shih Lee, for encouraging me to \"get moving\" during one of our conversations, which strengthened my resolve to write this book.
- I would like to thank my girlfriend Bubbles for being the first reader of this book, and for making many valuable suggestions from the perspective of an algorithm whiz, making this book more suitable for newbies.
- Thanks to Tengbao, Qibao, and Feibao for coming up with a creative name for this book that evokes fond memories of writing the first line of code \"Hello World!\".
- Thanks to Sutong for designing the beautiful cover and logo for this book and patiently revising it many times under my OCD.
- Thanks to @squidfunk for writing layout suggestions and for developing the open source documentation theme Material-for-MkDocs.
During the writing process, I read many textbooks and articles on data structures and algorithms. These works provide excellent models for this book and ensure the accuracy and quality of its contents. I would like to thank all my teachers and predecessors for their outstanding contributions!
This book promotes a hands-on approach to learning, and in this respect is heavily inspired by \"Dive into Deep Learning\". I highly recommend this excellent book to you.
A heartfelt thank you to my parents, it is your constant support and encouragement that gives me the opportunity to do this fun-filled thing.
"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 How To Read","text":"Tip
For the best reading experience, it is recommended that you read through this section.
"},{"location":"chapter_preface/suggestions/#021-conventions-of-style","title":"0.2.1 \u00a0 Conventions Of Style","text":"- Those labeled
*after the title are optional chapters with relatively difficult content. If you have limited time, it is advisable to skip them. - Proper nouns and words and phrases with specific meanings are marked with
\"double quotes\"to avoid ambiguity. - Important proper nouns and their English translations are marked with
\" \"in parentheses, e.g.\"array array\". It is recommended to memorize them for reading the literature. - Bolded text Indicates key content or summary statements, which deserve special attention.
- When it comes to terms that are inconsistent between programming languages, this book follows Python, for example using \\(\\text{None}\\) to mean \"empty\".
- This book partially abandons the specification of annotations in programming languages in exchange for a more compact layout of the content. There are three main types of annotations: title annotations, content annotations, and multi-line annotations.
\"\"\"Header comments for labeling functions, classes, test samples, etc.\"\"\"\"\n\n# Content comments for detailed code solutions\n\n\"\"\"\nmulti-line\nmarginal notes\n\"\"\"\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n
"},{"location":"chapter_preface/suggestions/#022-learn-efficiently-in-animated-graphic-solutions","title":"0.2.2 \u00a0 Learn Efficiently In Animated Graphic Solutions","text":"// Header comments for labeling functions, classes, test samples, etc.\n\n// Content comments for detailed code solutions.\n\n// Multi-line\n// Annotation\nCompared with text, videos and pictures have a higher degree of information density and structure and are easier to understand. In this book, key and difficult knowledge will be presented mainly in the form of animations and graphs, while the text serves as an explanation and supplement to the animations and graphs.
If, while reading the book, you find that a particular paragraph provides an animation or a graphic solution as shown below, please use the figure as the primary source and the text as a supplement and synthesize the two to understand the content.
Figure 0-2 \u00a0 Example animation
"},{"location":"chapter_preface/suggestions/#023-deeper-understanding-in-code-practice","title":"0.2.3 \u00a0 Deeper Understanding In Code Practice","text":"The companion code for this book is hosted in the GitHub repository. As shown in the Figure 0-3 , the source code is accompanied by test samples that can be run with a single click.
If time permits, it is recommended that you refer to the code and knock it through on your own. If you have limited time to study, please read through and run all the code at least once.
The process of writing code is often more rewarding than reading it. Learning by doing is really learning.
Figure 0-3 \u00a0 Running code example
The preliminaries for running the code are divided into three main steps.
Step 1: Install the local programming environment. Please refer to Appendix Tutorial for installation, or skip this step if already installed.
Step 2: Clone or download the code repository. If Git is already installed, you can clone this repository with the following command.
git clone https://github.com/krahets/hello-algo.git\nOf course, you can also in the location shown in the Figure 0-4 , click \"Download ZIP\" directly download the code zip, and then in the local solution.
Figure 0-4 \u00a0 Clone repository with download code
Step 3: Run the source code. As shown in the Figure 0-5 , for the code block labeled with the file name at the top, we can find the corresponding source code file in the
codesfolder of the repository. The source code files can be run with a single click, which will help you save unnecessary debugging time and allow you to focus on what you are learning.Figure 0-5 \u00a0 Code block with corresponding source file
"},{"location":"chapter_preface/suggestions/#024-growing-together-in-questioning-and-discussion","title":"0.2.4 \u00a0 Growing Together In Questioning And Discussion","text":"While reading this book, please don't skip over the points that you didn't learn. Feel free to ask your questions in the comment section. We will be happy to answer them and can usually respond within two days.
As you can see in the Figure 0-6 , each post comes with a comment section at the bottom. I hope you'll pay more attention to the comments section. On the one hand, you can learn about the problems that people encounter, so as to check the gaps and stimulate deeper thinking. On the other hand, we expect you to generously answer other partners' questions, share your insights, and help others improve.
Figure 0-6 \u00a0 Example of comment section
"},{"location":"chapter_preface/suggestions/#025-algorithm-learning-route","title":"0.2.5 \u00a0 Algorithm Learning Route","text":"From a general point of view, we can divide the process of learning data structures and algorithms into three stages.
- Introduction to Algorithms. We need to familiarize ourselves with the characteristics and usage of various data structures and learn about the principles, processes, uses and efficiency of different algorithms.
- Brush up on algorithm questions. It is recommended to start brushing from popular topics, such as Sword to Offer and LeetCode Hot 100, first accumulate at least 100 questions to familiarize yourself with mainstream algorithmic problems. Forgetfulness can be a challenge when first brushing up, but rest assured that this is normal. We can follow the \"Ebbinghaus Forgetting Curve\" to review the questions, and usually after 3-5 rounds of repetitions, we will be able to memorize them.
- Build the knowledge system. In terms of learning, we can read algorithm column articles, solution frameworks and algorithm textbooks to continuously enrich the knowledge system. In terms of brushing, we can try to adopt advanced brushing strategies, such as categorizing by topic, multiple solutions, multiple solutions, etc. Related brushing tips can be found in various communities.
As shown in the Figure 0-7 , this book mainly covers \"Phase 1\" and is designed to help you start Phase 2 and 3 more efficiently.
Figure 0-7 \u00a0 algorithm learning route
"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 Summary","text":"- The main audience of this book is beginners in algorithm. If you already have some basic knowledge, this book can help you systematically review your algorithm knowledge, and the source code in this book can also be used as a \"Coding Toolkit\".
- The book consists of three main sections, Complexity Analysis, Data Structures, and Algorithms, covering most of the topics in the field.
- For newcomers to algorithms, it is crucial to read an introductory book in the beginning stages to avoid many detours or common pitfalls.
- Animations and graphs within the book are usually used to introduce key points and difficult knowledge. These should be given more attention when reading the book.
- Practice is the best way to learn programming. It is highly recommended that you run the source code and type in the code yourself.
- Each chapter in the web version of this book features a discussion forum, and you are welcome to share your questions and insights at any time.
Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code
Dive In Clone Repo Get PDF
The English edition is brewing...
Feel free to engage in Chinese-to-English translation and pull request review! For guidelines, please see #914.
EndorsementsQuote
\"An easy-to-understand book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!\"
\u2014\u2014 Junhui Deng, Professor of Computer Science, Tsinghua University
Quote
\"If I had 'Hello Algo' when I was learning data structures and algorithms, it would have been 10 times easier!\"
\u2014\u2014 Mu Li, Senior Principal Scientist, Amazon
Animated illustrationsEasy to understandSmooth learning curve
\"A picture is worth a thousand words.\"
Off-the-Shelf CodeMulti programming languagesRun with one click
\"Talk is cheap. Show me the code.\"
Learning TogetherDiscussion and questions welcomeReaders progress together
\"Chase the wind and moon, never stopping\"
\"Beyond the plains, there are spring mountains\"
PrefaceTwo years ago, I shared the \"Sword Offer\" series of problem solutions on LeetCode, which received much love and support from many students. During my interactions with readers, the most common question I encountered was \"How to get started with algorithms.\" Gradually, I developed a deep interest in this question.
Blindly solving problems seems to be the most popular method, being simple, direct, and effective. However, problem-solving is like playing a \"Minesweeper\" game, where students with strong self-learning abilities can successfully clear the mines one by one, but those with insufficient foundations may end up bruised from explosions, retreating step by step in frustration. Thoroughly reading textbooks is also common, but for students aiming for job applications, the energy consumed by graduation, resume submissions, and preparing for written tests and interviews makes tackling thick books a daunting challenge.
If you are facing similar troubles, then you are lucky to have found this book. This book is my answer to this question, not necessarily the best solution, but at least an active attempt. Although this book won't directly land you an Offer, it will guide you through the \"knowledge map\" of data structures and algorithms, help you understand the shape, size, and distribution of different \"mines,\" and equip you with various \"demining methods.\" With these skills, I believe you can more comfortably solve problems and read literature, gradually building a complete knowledge system.
I deeply agree with Professor Feynman's saying: \"Knowledge isn't free. You have to pay attention.\" In this sense, this book is not entirely \"free.\" To not disappoint the precious \"attention\" you pay to this book, I will do my utmost, investing the greatest \"attention\" to complete the creation of this book.
AuthorYudong Jin(Krahets), Senior Algorithm Engineer in a top tech company, Master's degree from Shanghai Jiao Tong University. The highest-read blogger across the entire LeetCode, his published \"Illustration of Algorithm Data Structures\" has been subscribed to by over 300k.
ContributionThis book is continuously improved with the joint efforts of many contributors from the open-source community. Thanks to each writer who invested their time and energy, listed in the order generated by GitHub:
The code review work for this book was completed by Gonglja, gvenusleo, hpstory, justin\u2010tse, krahets, night-cruise, nuomi1, Reanon, and sjinzh (listed in alphabetical order). Thanks to them for their time and effort, ensuring the standardization and uniformity of the code in various languages.
GongljaC, C++ gvenusleoDart hpstoryC# justin-tseJS, TS krahetsJava, Python night-cruiseRust nuomi1Swift ReanonGo, C sjinzhRust, Zig"},{"location":"chapter_computational_complexity/","title":"Chapter 2. \u00a0 Complexity Analysis","text":"Abstract
Complexity analysis is like a space-time navigator in the vast universe of algorithms.
It guides us in exploring deeper within the the dimensions of time and space, seeking more elegant solutions.
"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 2.1 \u00a0 Evaluating Algorithm Efficiency
- 2.2 \u00a0 Iteration and Recursion
- 2.3 \u00a0 Time Complexity
- 2.4 \u00a0 Space Complexity
- 2.5 \u00a0 Summary
In algorithms, repeatedly performing a task is common and closely related to complexity analysis. Therefore, before introducing time complexity and space complexity, let's first understand how to implement task repetition in programs, focusing on two basic programming control structures: iteration and recursion.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221-iteration","title":"2.2.1 \u00a0 Iteration","text":"\"Iteration\" is a control structure for repeatedly performing a task. In iteration, a program repeats a block of code as long as a certain condition is met, until this condition is no longer satisfied.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for-loop","title":"1. \u00a0 for Loop","text":"The
forloop is one of the most common forms of iteration, suitable for use when the number of iterations is known in advance.The following function implements the sum \\(1 + 2 + \\dots + n\\) using a
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyforloop, with the sum result recorded in the variableres. Note that in Python,range(a, b)corresponds to a \"left-closed, right-open\" interval, covering \\(a, a + 1, \\dots, b-1\\):
iteration.cppdef for_loop(n: int) -> int:\n \"\"\"for \u5faa\u73af\"\"\"\n res = 0\n # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in range(1, n + 1):\n res += i\n return res\n
iteration.java/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; ++i) {\n res += i;\n }\n return res;\n}\n
iteration.cs/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.go/* for \u5faa\u73af */\nint ForLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.swift/* for \u5faa\u73af */\nfunc forLoop(n int) int {\n res := 0\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i := 1; i <= n; i++ {\n res += i\n }\n return res\n}\n
iteration.js/* for \u5faa\u73af */\nfunc forLoop(n: Int) -> Int {\n var res = 0\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in 1 ... n {\n res += i\n }\n return res\n}\n
iteration.ts/* for \u5faa\u73af */\nfunction forLoop(n) {\n let res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.dart/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\n let res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.rs/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.c/* for \u5faa\u73af */\nfn for_loop(n: i32) -> i32 {\n let mut res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in 1..=n {\n res += i;\n }\n res\n} \n
iteration.zig/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\n var res: i32 = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (1..n+1) |i| {\n res = res + @as(i32, @intCast(i));\n }\n return res;\n} \nThe flowchart below represents this sum function.
Figure 2-1 \u00a0 Flowchart of the Sum Function
The number of operations in this sum function is proportional to the input data size \\(n\\), or in other words, it has a \"linear relationship\". This is actually what time complexity describes. This topic will be detailed in the next section.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while-loop","title":"2. \u00a0 while Loop","text":"Similar to the
forloop, thewhileloop is another method to implement iteration. In awhileloop, the program checks the condition in each round; if the condition is true, it continues, otherwise, the loop ends.Below we use a
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pywhileloop to implement the sum \\(1 + 2 + \\dots + n\\):
iteration.cppdef while_loop(n: int) -> int:\n \"\"\"while \u5faa\u73af\"\"\"\n res = 0\n i = 1 # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n:\n res += i\n i += 1 # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n return res\n
iteration.java/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.cs/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.go/* while \u5faa\u73af */\nint WhileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.swift/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\n res := 0\n // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n i := 1\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++\n }\n return res\n}\n
iteration.js/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -> Int {\n var res = 0\n var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n {\n res += i\n i += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res\n}\n
iteration.ts/* while \u5faa\u73af */\nfunction whileLoop(n) {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.dart/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.rs/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.c/* while \u5faa\u73af */\nfn while_loop(n: i32) -> i32 {\n let mut res = 0;\n let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n {\n res += i;\n i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n res\n}\n
iteration.zig/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\n var res: i32 = 0;\n var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += @intCast(i);\n i += 1;\n }\n return res;\n}\nThe
whileloop is more flexible than theforloop. In awhileloop, we can freely design the initialization and update steps of the condition variable.For example, in the following code, the condition variable \\(i\\) is updated twice in each round, which would be inconvenient to implement with a
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyforloop:
iteration.cppdef while_loop_ii(n: int) -> int:\n \"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\n res = 0\n i = 1 # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n # \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n:\n res += i\n # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1\n i *= 2\n return res\n
iteration.java/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.cs/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.go/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, 4, 5...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1; \n i *= 2;\n }\n return res;\n}\n
iteration.swift/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n res := 0\n // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n i := 1\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n for i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++\n i *= 2\n }\n return res\n}\n
iteration.js/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -> Int {\n var res = 0\n var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1\n i *= 2\n }\n return res\n}\n
iteration.ts/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.dart/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.rs/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.c/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -> i32 {\n let mut res = 0;\n let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1;\n i *= 2;\n }\n res\n}\n
iteration.zig/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n// while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n var res: i32 = 0;\n var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += @intCast(i);\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1;\n i *= 2;\n }\n return res;\n}\nOverall,
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-nested-loops","title":"3. \u00a0 Nested Loops","text":"forloops are more concise, whilewhileloops are more flexible. Both can implement iterative structures. Which one to use should be determined based on the specific requirements of the problem.We can nest one loop structure within another. Below is an example using
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyforloops:
iteration.cppdef nested_for_loop(n: int) -> str:\n \"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\n res = \"\"\n # \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in range(1, n + 1):\n # \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in range(1, n + 1):\n res += f\"({i}, {j}), \"\n return res\n
iteration.java/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\n ostringstream res;\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; ++i) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; ++j) {\n res << \"(\" << i << \", \" << j << \"), \";\n }\n }\n return res.str();\n}\n
iteration.cs/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n StringBuilder res = new StringBuilder();\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res.append(\"(\" + i + \", \" + j + \"), \");\n }\n }\n return res.toString();\n}\n
iteration.go/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\n StringBuilder res = new();\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res.Append($\"({i}, {j}), \");\n }\n }\n return res.ToString();\n}\n
iteration.swift/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\n res := \"\"\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i := 1; i <= n; i++ {\n for j := 1; j <= n; j++ {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n res += fmt.Sprintf(\"(%d, %d), \", i, j)\n }\n }\n return res\n}\n
iteration.js/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -> String {\n var res = \"\"\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in 1 ... n {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in 1 ... n {\n res.append(\"(\\(i), \\(j)), \")\n }\n }\n return res\n}\n
iteration.ts/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\n let res = '';\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (let j = 1; j <= n; j++) {\n res += `(${i}, ${j}), `;\n }\n }\n return res;\n}\n
iteration.dart/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\n let res = '';\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (let j = 1; j <= n; j++) {\n res += `(${i}, ${j}), `;\n }\n }\n return res;\n}\n
iteration.rs/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n String res = \"\";\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res += \"($i, $j), \";\n }\n }\n return res;\n}\n
iteration.c/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -> String {\n let mut res = vec![];\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in 1..=n {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in 1..=n {\n res.push(format!(\"({}, {}), \", i, j));\n }\n }\n res.join(\"\")\n}\n
iteration.zig/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n // n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\n int size = n * n * 26 + 1;\n char *res = malloc(size * sizeof(char));\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n char tmp[26];\n snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n strncat(res, tmp, size - strlen(res) - 1);\n }\n }\n return res;\n}\n// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n var res = std.ArrayList(u8).init(allocator);\n defer res.deinit();\n var buffer: [20]u8 = undefined;\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (1..n+1) |i| {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (1..n+1) |j| {\n var _str = try std.fmt.bufPrint(&buffer, \"({d}, {d}), \", .{i, j});\n try res.appendSlice(_str);\n }\n }\n return res.toOwnedSlice();\n}\nThe flowchart below represents this nested loop.
Figure 2-2 \u00a0 Flowchart of the Nested Loop
In this case, the number of operations in the function is proportional to \\(n^2\\), or the algorithm's running time and the input data size \\(n\\) have a \"quadratic relationship\".
We can continue adding nested loops, each nesting is a \"dimensional escalation,\" which will increase the time complexity to \"cubic,\" \"quartic,\" and so on.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222-recursion","title":"2.2.2 \u00a0 Recursion","text":"\"Recursion\" is an algorithmic strategy that solves problems by having a function call itself. It mainly consists of two phases.
- Recursion: The program continuously calls itself, usually with smaller or more simplified parameters, until reaching a \"termination condition.\"
- Return: Upon triggering the \"termination condition,\" the program begins to return from the deepest recursive function, aggregating the results of each layer.
From an implementation perspective, recursive code mainly includes three elements.
- Termination Condition: Determines when to switch from \"recursion\" to \"return.\"
- Recursive Call: Corresponds to \"recursion,\" where the function calls itself, usually with smaller or more simplified parameters.
- Return Result: Corresponds to \"return,\" where the result of the current recursion level is returned to the previous layer.
Observe the following code, where calling the function
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyrecur(n)completes the computation of \\(1 + 2 + \\dots + n\\):
recursion.cppdef recur(n: int) -> int:\n \"\"\"\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if n == 1:\n return 1\n # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n res = recur(n - 1)\n # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n
recursion.java/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.cs/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.go/* \u9012\u5f52 */\nint Recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = Recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.swift/* \u9012\u5f52 */\nfunc recur(n int) int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n res := recur(n - 1)\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n}\n
recursion.js/* \u9012\u5f52 */\nfunc recur(n: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n let res = recur(n: n - 1)\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n}\n
recursion.ts/* \u9012\u5f52 */\nfunction recur(n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n const res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.dart/* \u9012\u5f52 */\nfunction recur(n: number): number {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n const res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.rs/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.c/* \u9012\u5f52 */\nfn recur(n: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1;\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n let res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n n + res\n}\n
recursion.zig/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1) {\n return 1;\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n var res: i32 = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\nThe Figure 2-3 shows the recursive process of this function.
Figure 2-3 \u00a0 Recursive Process of the Sum Function
Although iteration and recursion can achieve the same results from a computational standpoint, they represent two entirely different paradigms of thinking and solving problems.
- Iteration: Solves problems \"from the bottom up.\" It starts with the most basic steps, then repeatedly adds or accumulates these steps until the task is complete.
- Recursion: Solves problems \"from the top down.\" It breaks down the original problem into smaller sub-problems, each of which has the same form as the original problem. These sub-problems are then further decomposed into even smaller sub-problems, stopping at the base case (whose solution is known).
Taking the sum function as an example, let's define the problem as \\(f(n) = 1 + 2 + \\dots + n\\).
- Iteration: In a loop, simulate the summing process, iterating from \\(1\\) to \\(n\\), performing the sum operation in each round, to obtain \\(f(n)\\).
- Recursion: Break down the problem into sub-problems \\(f(n) = n + f(n-1)\\), continuously (recursively) decomposing until reaching the base case \\(f(1) = 1\\) and then stopping.
Each time a recursive function calls itself, the system allocates memory for the newly initiated function to store local variables, call addresses, and other information. This leads to two main consequences.
- The function's context data is stored in a memory area called \"stack frame space\" and is only released after the function returns. Therefore, recursion generally consumes more memory space than iteration.
- Recursive calls introduce additional overhead. Hence, recursion is usually less time-efficient than loops.
As shown in the Figure 2-4 , there are \\(n\\) unreturned recursive functions before triggering the termination condition, indicating a recursion depth of \\(n\\).
Figure 2-4 \u00a0 Recursion Call Depth
In practice, the depth of recursion allowed by programming languages is usually limited, and excessively deep recursion can lead to stack overflow errors.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-tail-recursion","title":"2. \u00a0 Tail Recursion","text":"Interestingly, if a function makes its recursive call as the last step before returning, it can be optimized by compilers or interpreters to be as space-efficient as iteration. This scenario is known as \"tail recursion\".
- Regular Recursion: The function needs to perform more code after returning to the previous level, so the system needs to save the context of the previous call.
- Tail Recursion: The recursive call is the last operation before the function returns, meaning no further actions are required upon returning to the previous level, so the system doesn't need to save the context of the previous level's function.
For example, in calculating \\(1 + 2 + \\dots + n\\), we can make the result variable
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyresa parameter of the function, thereby achieving tail recursion:
recursion.cppdef tail_recur(n, res):\n \"\"\"\u5c3e\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if n == 0:\n return res\n # \u5c3e\u9012\u5f52\u8c03\u7528\n return tail_recur(n - 1, res + n)\n
recursion.java/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.cs/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.go/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return TailRecur(n - 1, res + n);\n}\n
recursion.swift/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n-1, res+n)\n}\n
recursion.js/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n: n - 1, res: res + n)\n}\n
recursion.ts/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.dart/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.rs/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.c/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res;\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n tail_recur(n - 1, res + n)\n}\n
recursion.zig/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0) {\n return res;\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\nThe execution process of tail recursion is shown in the following figure. Comparing regular recursion and tail recursion, the point of the summation operation is different.
- Regular Recursion: The summation operation occurs during the \"return\" phase, requiring another summation after each layer returns.
- Tail Recursion: The summation operation occurs during the \"recursion\" phase, and the \"return\" phase only involves returning through each layer.
Figure 2-5 \u00a0 Tail Recursion Process
Tip
Note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if the function is in the form of tail recursion, it may still encounter stack overflow issues.
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-recursion-tree","title":"3. \u00a0 Recursion Tree","text":"When dealing with algorithms related to \"divide and conquer\", recursion often offers a more intuitive approach and more readable code than iteration. Take the \"Fibonacci sequence\" as an example.
Question
Given a Fibonacci sequence \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\), find the \\(n\\)th number in the sequence.
Let the \\(n\\)th number of the Fibonacci sequence be \\(f(n)\\), it's easy to deduce two conclusions:
- The first two numbers of the sequence are \\(f(1) = 0\\) and \\(f(2) = 1\\).
- Each number in the sequence is the sum of the two preceding ones, that is, \\(f(n) = f(n - 1) + f(n - 2)\\).
Using the recursive relation, and considering the first two numbers as termination conditions, we can write the recursive code. Calling
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyfib(n)will yield the \\(n\\)th number of the Fibonacci sequence:
recursion.cppdef fib(n: int) -> int:\n \"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 or n == 2:\n return n - 1\n # \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n res = fib(n - 1) + fib(n - 2)\n # \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n
recursion.java/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.cs/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.go/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = Fib(n - 1) + Fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.swift/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n res := fib(n-1) + fib(n-2)\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n}\n
recursion.js/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n let res = fib(n: n - 1) + fib(n: n - 2)\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n}\n
recursion.ts/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n === 1 || n === 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n const res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.dart/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n === 1 || n === 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n const res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.rs/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.c/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1;\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n let res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c\n res\n}\n
recursion.zig/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 or n == 2) {\n return n - 1;\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n var res: i32 = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\nObserving the above code, we see that it recursively calls two functions within itself, meaning that one call generates two branching calls. As illustrated below, this continuous recursive calling eventually creates a \"recursion tree\" with a depth of \\(n\\).
Figure 2-6 \u00a0 Fibonacci Sequence Recursion Tree
Fundamentally, recursion embodies the paradigm of \"breaking down a problem into smaller sub-problems.\" This divide-and-conquer strategy is crucial.
- From an algorithmic perspective, many important strategies like searching, sorting, backtracking, divide-and-conquer, and dynamic programming directly or indirectly use this way of thinking.
- From a data structure perspective, recursion is naturally suited for dealing with linked lists, trees, and graphs, as they are well suited for analysis using the divide-and-conquer approach.
Summarizing the above content, the following table shows the differences between iteration and recursion in terms of implementation, performance, and applicability.
Table: Comparison of Iteration and Recursion Characteristics
Iteration Recursion Approach Loop structure Function calls itself Time Efficiency Generally higher efficiency, no function call overhead Each function call generates overhead Memory Usage Typically uses a fixed size of memory space Accumulative function calls can use a substantial amount of stack frame space Suitable Problems Suitable for simple loop tasks, intuitive and readable code Suitable for problem decomposition, like trees, graphs, divide-and-conquer, backtracking, etc., concise and clear code structureTip
If you find the following content difficult to understand, consider revisiting it after reading the \"Stack\" chapter.
So, what is the intrinsic connection between iteration and recursion? Taking the above recursive function as an example, the summation operation occurs during the recursion's \"return\" phase. This means that the initially called function is actually the last to complete its summation operation, mirroring the \"last in, first out\" principle of a stack.
In fact, recursive terms like \"call stack\" and \"stack frame space\" hint at the close relationship between recursion and stacks.
- Recursion: When a function is called, the system allocates a new stack frame on the \"call stack\" for that function, storing local variables, parameters, return addresses, and other data.
- Return: When a function completes execution and returns, the corresponding stack frame is removed from the \"call stack,\" restoring the execution environment of the previous function.
Therefore, we can use an explicit stack to simulate the behavior of the call stack, thus transforming recursion into an iterative form:
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py
recursion.cppdef for_loop_recur(n: int) -> int:\n \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n # \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack = []\n res = 0\n # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in range(n, 0, -1):\n # \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.append(i)\n # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while stack:\n # \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop()\n # res = 1+2+3+...+n\n return res\n
recursion.java/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack<int> stack;\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.empty()) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.top();\n stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.cs/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n Stack<Integer> stack = new Stack<>();\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.isEmpty()) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.go/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n Stack<int> stack = new();\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.Push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.Count > 0) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.Pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.swift/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack := list.New()\n res := 0\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i := n; i > 0; i-- {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.PushBack(i)\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n for stack.Len() != 0 {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.Back().Value.(int)\n stack.Remove(stack.Back())\n }\n // res = 1+2+3+...+n\n return res\n}\n
recursion.js/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -> Int {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n var stack: [Int] = []\n var res = 0\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in stride(from: n, to: 0, by: -1) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.append(i)\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while !stack.isEmpty {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.removeLast()\n }\n // res = 1+2+3+...+n\n return res\n}\n
recursion.ts/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n const stack = [];\n let res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (let i = 1; i <= n; i++) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.length) { \n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.dart/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \n const stack: number[] = [];\n let res: number = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (let i = 1; i <= n; i++) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.length) { \n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.rs/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n List<int> stack = [];\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.add(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.isEmpty) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.removeLast();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.c/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -> i32 {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n let mut stack = Vec::new();\n let mut res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in (1..=n).rev() {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while !stack.is_empty() {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop().unwrap();\n }\n // res = 1+2+3+...+n\n res\n}\n
recursion.zig/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n int stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\n int top = -1; // \u6808\u9876\u7d22\u5f15\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack[1 + top++] = i;\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (top >= 0) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack[top--];\n }\n // res = 1+2+3+...+n\n return res;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n var stack: [n]i32 = undefined;\n var res: i32 = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n var i: usize = n;\n while (i > 0) {\n stack[i - 1] = @intCast(i);\n i -= 1;\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n var index: usize = n;\n while (index > 0) {\n index -= 1;\n res += stack[index];\n }\n // res = 1+2+3+...+n\n return res;\n}\nObserving the above code, when recursion is transformed into iteration, the code becomes more complex. Although iteration and recursion can often be transformed into each other, it's not always advisable to do so for two reasons:
- The transformed code may become harder to understand and less readable.
- For some complex problems, simulating the behavior of the system's call stack can be quite challenging.
In summary, choosing between iteration and recursion depends on the nature of the specific problem. In programming practice, weighing the pros and cons of each and choosing the appropriate method for the situation is essential.
"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 Algorithm Efficiency Assessment","text":"In algorithm design, we pursue the following two objectives in sequence.
- Finding a Solution to the Problem: The algorithm should reliably find the correct solution within the stipulated range of inputs.
- Seeking the Optimal Solution: For the same problem, multiple solutions might exist, and we aim to find the most efficient algorithm possible.
In other words, under the premise of being able to solve the problem, algorithm efficiency has become the main criterion for evaluating the merits of an algorithm, which includes the following two dimensions.
- Time Efficiency: The speed at which an algorithm runs.
- Space Efficiency: The size of the memory space occupied by an algorithm.
In short, our goal is to design data structures and algorithms that are both fast and memory-efficient. Effectively assessing algorithm efficiency is crucial because only then can we compare various algorithms and guide the process of algorithm design and optimization.
There are mainly two methods of efficiency assessment: actual testing and theoretical estimation.
"},{"location":"chapter_computational_complexity/performance_evaluation/#211-actual-testing","title":"2.1.1 \u00a0 Actual Testing","text":"Suppose we have algorithms
AandB, both capable of solving the same problem, and we need to compare their efficiencies. The most direct method is to use a computer to run these two algorithms and monitor and record their runtime and memory usage. This assessment method reflects the actual situation but has significant limitations.On one hand, it's difficult to eliminate interference from the testing environment. Hardware configurations can affect algorithm performance. For example, algorithm
Amight run faster thanBon one computer, but the opposite result may occur on another computer with different configurations. This means we would need to test on a variety of machines to calculate average efficiency, which is impractical.On the other hand, conducting a full test is very resource-intensive. As the volume of input data changes, the efficiency of the algorithms may vary. For example, with smaller data volumes, algorithm
"},{"location":"chapter_computational_complexity/performance_evaluation/#212-theoretical-estimation","title":"2.1.2 \u00a0 Theoretical Estimation","text":"Amight run faster thanB, but the opposite might be true with larger data volumes. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires significant computational resources.Due to the significant limitations of actual testing, we can consider evaluating algorithm efficiency solely through calculations. This estimation method is known as \"asymptotic complexity analysis,\" or simply \"complexity analysis.\"
Complexity analysis reflects the relationship between the time and space resources required for algorithm execution and the size of the input data. It describes the trend of growth in the time and space required by the algorithm as the size of the input data increases. This definition might sound complex, but we can break it down into three key points to understand it better.
- \"Time and space resources\" correspond to \"time complexity\" and \"space complexity,\" respectively.
- \"As the size of input data increases\" means that complexity reflects the relationship between algorithm efficiency and the volume of input data.
- \"The trend of growth in time and space\" indicates that complexity analysis focuses not on the specific values of runtime or space occupied but on the \"rate\" at which time or space grows.
Complexity analysis overcomes the disadvantages of actual testing methods, reflected in the following aspects:
- It is independent of the testing environment and applicable to all operating platforms.
- It can reflect algorithm efficiency under different data volumes, especially in the performance of algorithms with large data volumes.
Tip
If you're still confused about the concept of complexity, don't worry. We will introduce it in detail in subsequent chapters.
Complexity analysis provides us with a \"ruler\" to measure the time and space resources needed to execute an algorithm and compare the efficiency between different algorithms.
Complexity is a mathematical concept and may be abstract and challenging for beginners. From this perspective, complexity analysis might not be the best content to introduce first. However, when discussing the characteristics of a particular data structure or algorithm, it's hard to avoid analyzing its speed and space usage.
In summary, it's recommended that you establish a preliminary understanding of complexity analysis before diving deep into data structures and algorithms, so that you can carry out simple complexity analyses of algorithms.
"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 Space Complexity","text":"\"Space complexity\" is used to measure the growth trend of the memory space occupied by an algorithm as the amount of data increases. This concept is very similar to time complexity, except that \"running time\" is replaced with \"occupied memory space\".
"},{"location":"chapter_computational_complexity/space_complexity/#241-space-related-to-algorithms","title":"2.4.1 \u00a0 Space Related to Algorithms","text":"The memory space used by an algorithm during its execution mainly includes the following types.
- Input Space: Used to store the input data of the algorithm.
- Temporary Space: Used to store variables, objects, function contexts, and other data during the algorithm's execution.
- Output Space: Used to store the output data of the algorithm.
Generally, the scope of space complexity statistics includes both \"Temporary Space\" and \"Output Space\".
Temporary space can be further divided into three parts.
- Temporary Data: Used to save various constants, variables, objects, etc., during the algorithm's execution.
- Stack Frame Space: Used to save the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns.
- Instruction Space: Used to store compiled program instructions, which are usually negligible in actual statistics.
When analyzing the space complexity of a program, we typically count the Temporary Data, Stack Frame Space, and Output Data, as shown in the Figure 2-15 .
Figure 2-15 \u00a0 Space Types Used in Algorithms
The relevant code is as follows:
PythonC++JavaC#GoSwiftJSTSDartRustCZigclass Node:\n \"\"\"Classes\"\"\"\"\n def __init__(self, x: int):\n self.val: int = x # node value\n self.next: Node | None = None # reference to the next node\n\ndef function() -> int:\n \"\"\"\"Functions\"\"\"\"\"\n # Perform certain operations...\n return 0\n\ndef algorithm(n) -> int: # input data\n A = 0 # temporary data (constant, usually in uppercase)\n b = 0 # temporary data (variable)\n node = Node(0) # temporary data (object)\n c = function() # Stack frame space (call function)\n return A + b + c # output data\n/* Structures */\nstruct Node {\n int val;\n Node *next;\n Node(int x) : val(x), next(nullptr) {}\n};\n\n/* Functions */\nint func() {\n // Perform certain operations...\n return 0;\n}\n\nint algorithm(int n) { // input data\n const int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n Node* node = new Node(0); // temporary data (object)\n int c = func(); // stack frame space (call function)\n return a + b + c; // output data\n}\n/* Classes */\nclass Node {\n int val;\n Node next;\n Node(int x) { val = x; }\n}\n\n/* Functions */\nint function() {\n // Perform certain operations...\n return 0;\n}\n\nint algorithm(int n) { // input data\n final int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n Node node = new Node(0); // temporary data (object)\n int c = function(); // stack frame space (call function)\n return a + b + c; // output data\n}\n/* Classes */\nclass Node {\n int val;\n Node next;\n Node(int x) { val = x; }\n}\n\n/* Functions */\nint Function() {\n // Perform certain operations...\n return 0;\n}\n\nint Algorithm(int n) { // input data\n const int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n Node node = new(0); // temporary data (object)\n int c = Function(); // stack frame space (call function)\n return a + b + c; // output data\n}\n/* Structures */\ntype node struct {\n val int\n next *node\n}\n\n/* Create node structure */\nfunc newNode(val int) *node {\n return &node{val: val}\n}\n\n/* Functions */\nfunc function() int {\n // Perform certain operations...\n return 0\n}\n\nfunc algorithm(n int) int { // input data\n const a = 0 // temporary data (constant)\n b := 0 // temporary storage of data (variable)\n newNode(0) // temporary data (object)\n c := function() // stack frame space (call function)\n return a + b + c // output data\n}\n/* Classes */\nclass Node {\n var val: Int\n var next: Node?\n\n init(x: Int) {\n val = x\n }\n}\n\n/* Functions */\nfunc function() -> Int {\n // Perform certain operations...\n return 0\n}\n\nfunc algorithm(n: Int) -> Int { // input data\n let a = 0 // temporary data (constant)\n var b = 0 // temporary data (variable)\n let node = Node(x: 0) // temporary data (object)\n let c = function() // stack frame space (call function)\n return a + b + c // output data\n}\n/* Classes */\nclass Node {\n val;\n next;\n constructor(val) {\n this.val = val === undefined ? 0 : val; // node value\n this.next = null; // reference to the next node\n }\n}\n\n/* Functions */\nfunction constFunc() {\n // Perform certain operations\n return 0;\n}\n\nfunction algorithm(n) { // input data\n const a = 0; // temporary data (constant)\n let b = 0; // temporary data (variable)\n const node = new Node(0); // temporary data (object)\n const c = constFunc(); // Stack frame space (calling function)\n return a + b + c; // output data\n}\n/* Classes */\nclass Node {\n val: number;\n next: Node | null;\n constructor(val?: number) {\n this.val = val === undefined ? 0 : val; // node value\n this.next = null; // reference to the next node\n }\n}\n\n/* Functions */\nfunction constFunc(): number {\n // Perform certain operations\n return 0;\n}\n\nfunction algorithm(n: number): number { // input data\n const a = 0; // temporary data (constant)\n let b = 0; // temporary data (variable)\n const node = new Node(0); // temporary data (object)\n const c = constFunc(); // Stack frame space (calling function)\n return a + b + c; // output data\n}\n/* Classes */\nclass Node {\n int val;\n Node next;\n Node(this.val, [this.next]);\n}\n\n/* Functions */\nint function() {\n // Perform certain operations...\n return 0;\n}\n\nint algorithm(int n) { // input data\n const int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n Node node = Node(0); // temporary data (object)\n int c = function(); // stack frame space (call function)\n return a + b + c; // output data\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Structures */\nstruct Node {\n val: i32,\n next: Option<Rc<RefCell<Node>>>,\n}\n\n/* Creating a Node structure */\nimpl Node {\n fn new(val: i32) -> Self {\n Self { val: val, next: None }\n }\n}\n\n/* Functions */\nfn function() -> i32 { \n // Perform certain operations...\n return 0;\n}\n\nfn algorithm(n: i32) -> i32 { // input data\n const a: i32 = 0; // temporary data (constant)\n let mut b = 0; // temporary data (variable)\n let node = Node::new(0); // temporary data (object)\n let c = function(); // stack frame space (call function)\n return a + b + c; // output data\n}\n/* Functions */\nint func() {\n // Perform certain operations...\n return 0;\n}\n\nint algorithm(int n) { // input data\n const int a = 0; // temporary data (constant)\n int b = 0; // temporary data (variable)\n int c = func(); // stack frame space (call function)\n return a + b + c; // output data\n}\n
"},{"location":"chapter_computational_complexity/space_complexity/#242-calculation-method","title":"2.4.2 \u00a0 Calculation Method","text":"\nThe method for calculating space complexity is roughly similar to that of time complexity, with the only change being the shift of the statistical object from \"number of operations\" to \"size of used space\".
However, unlike time complexity, we usually only focus on the worst-case space complexity. This is because memory space is a hard requirement, and we must ensure that there is enough memory space reserved under all input data.
Consider the following code, the term \"worst-case\" in worst-case space complexity has two meanings.
- Based on the worst input data: When \\(n < 10\\), the space complexity is \\(O(1)\\); but when \\(n > 10\\), the initialized array
numsoccupies \\(O(n)\\) space, thus the worst-case space complexity is \\(O(n)\\). - Based on the peak memory used during the algorithm's execution: For example, before executing the last line, the program occupies \\(O(1)\\) space; when initializing the array
nums, the program occupies \\(O(n)\\) space, hence the worst-case space complexity is \\(O(n)\\).
def algorithm(n: int):\n a = 0 # O(1)\n b = [0] * 10000 # O(1)\n if n > 10:\n nums = [0] * n # O(n)\nvoid algorithm(int n) {\n int a = 0; // O(1)\n vector<int> b(10000); // O(1)\n if (n > 10)\n vector<int> nums(n); // O(n)\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n int[] b = new int[10000]; // O(1)\n if (n > 10)\n int[] nums = new int[n]; // O(n)\n}\nvoid Algorithm(int n) {\n int a = 0; // O(1)\n int[] b = new int[10000]; // O(1)\n if (n > 10) {\n int[] nums = new int[n]; // O(n)\n }\n}\nfunc algorithm(n int) {\n a := 0 // O(1)\n b := make([]int, 10000) // O(1)\n var nums []int\n if n > 10 {\n nums := make([]int, n) // O(n)\n }\n fmt.Println(a, b, nums)\n}\nfunc algorithm(n: Int) {\n let a = 0 // O(1)\n let b = Array(repeating: 0, count: 10000) // O(1)\n if n > 10 {\n let nums = Array(repeating: 0, count: n) // O(n)\n }\n}\nfunction algorithm(n) {\n const a = 0; // O(1)\n const b = new Array(10000); // O(1)\n if (n > 10) {\n const nums = new Array(n); // O(n)\n }\n}\nfunction algorithm(n: number): void {\n const a = 0; // O(1)\n const b = new Array(10000); // O(1)\n if (n > 10) {\n const nums = new Array(n); // O(n)\n }\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n List<int> b = List.filled(10000, 0); // O(1)\n if (n > 10) {\n List<int> nums = List.filled(n, 0); // O(n)\n }\n}\nfn algorithm(n: i32) {\n let a = 0; // O(1)\n let b = [0; 10000]; // O(1)\n if n > 10 {\n let nums = vec![0; n as usize]; // O(n)\n }\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n int b[10000]; // O(1)\n if (n > 10)\n int nums[n] = {0}; // O(n)\n}\n\nIn recursive functions, stack frame space must be taken into count. Consider the following code:
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef function() -> int:\n # Perform certain operations\n return 0\n\ndef loop(n: int):\n \"\"\"Loop O(1)\"\"\"\"\"\n for _ in range(n):\n function()\n\ndef recur(n: int) -> int:\n \"\"\"Recursion O(n)\"\"\"\"\"\n if n == 1: return\n return recur(n - 1)\nint func() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nint function() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nint Function() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid Loop(int n) {\n for (int i = 0; i < n; i++) {\n Function();\n }\n}\n/* Recursion O(n) */\nint Recur(int n) {\n if (n == 1) return 1;\n return Recur(n - 1);\n}\nfunc function() int {\n // Perform certain operations\n return 0\n}\n\n/* Cycle O(1) */\nfunc loop(n int) {\n for i := 0; i < n; i++ {\n function()\n }\n}\n\n/* Recursion O(n) */\nfunc recur(n int) {\n if n == 1 {\n return\n }\n recur(n - 1)\n}\n@discardableResult\nfunc function() -> Int {\n // Perform certain operations\n return 0\n}\n\n/* Cycle O(1) */\nfunc loop(n: Int) {\n for _ in 0 ..< n {\n function()\n }\n}\n\n/* Recursion O(n) */\nfunc recur(n: Int) {\n if n == 1 {\n return\n }\n recur(n: n - 1)\n}\nfunction constFunc() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nfunction loop(n) {\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n/* Recursion O(n) */\nfunction recur(n) {\n if (n === 1) return;\n return recur(n - 1);\n}\nfunction constFunc(): number {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nfunction loop(n: number): void {\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n/* Recursion O(n) */\nfunction recur(n: number): void {\n if (n === 1) return;\n return recur(n - 1);\n}\nint function() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nfn function() -> i32 {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nfn loop(n: i32) {\n for i in 0..n {\n function();\n }\n}\n/* Recursion O(n) */\nvoid recur(n: i32) {\n if n == 1 {\n return;\n }\n recur(n - 1);\n}\nint func() {\n // Perform certain operations\n return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\n\nThe time complexity of both
loop()andrecur()functions is \\(O(n)\\), but their space complexities differ.- The
loop()function callsfunction()\\(n\\) times in a loop, where each iteration'sfunction()returns and releases its stack frame space, so the space complexity remains \\(O(1)\\). - The recursive function
recur()will have \\(n\\) instances of unreturnedrecur()existing simultaneously during its execution, thus occupying \\(O(n)\\) stack frame space.
Let the size of the input data be \\(n\\), the following chart displays common types of space complexities (arranged from low to high).
\\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n^2) < O(2^n) \\newline \\text{Constant Order} < \\text{Logarithmic Order} < \\text{Linear Order} < \\text{Quadratic Order} < \\text{Exponential Order} \\end{aligned} \\]Figure 2-16 \u00a0 Common Types of Space Complexity
"},{"location":"chapter_computational_complexity/space_complexity/#1-constant-order-o1","title":"1. \u00a0 Constant Order \\(O(1)\\)","text":"Constant order is common in constants, variables, objects that are independent of the size of input data \\(n\\).
Note that memory occupied by initializing variables or calling functions in a loop, which is released upon entering the next cycle, does not accumulate over space, thus the space complexity remains \\(O(1)\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef function() -> int:\n \"\"\"\u51fd\u6570\"\"\"\n # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n\ndef constant(n: int):\n \"\"\"\u5e38\u6570\u9636\"\"\"\n # \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n a = 0\n nums = [0] * 10000\n node = ListNode(0)\n # \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in range(n):\n c = 0\n # \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in range(n):\n function()\n
space_complexity.java/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const int a = 0;\n int b = 0;\n vector<int> nums(10000);\n ListNode node(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n
space_complexity.cs/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n final int a = 0;\n int b = 0;\n int[] nums = new int[10000];\n ListNode node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n
space_complexity.go/* \u51fd\u6570 */\nint Function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n int a = 0;\n int b = 0;\n int[] nums = new int[10000];\n ListNode node = new(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n Function();\n }\n}\n
space_complexity.swift/* \u51fd\u6570 */\nfunc function() int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0\n b := 0\n nums := make([]int, 10000)\n ListNode := newNode(0)\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n var c int\n for i := 0; i < n; i++ {\n c = 0\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for i := 0; i < n; i++ {\n function()\n }\n fmt.Println(a, b, nums, c, ListNode)\n}\n
space_complexity.js/* \u51fd\u6570 */\n@discardableResult\nfunc function() -> Int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n let a = 0\n var b = 0\n let nums = Array(repeating: 0, count: 10000)\n let node = ListNode(x: 0)\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in 0 ..< n {\n let c = 0\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in 0 ..< n {\n function()\n }\n}\n
space_complexity.ts/* \u51fd\u6570 */\nfunction constFunc() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0;\n const b = 0;\n const nums = new Array(10000);\n const node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n const c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n
space_complexity.dart/* \u51fd\u6570 */\nfunction constFunc(): number {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0;\n const b = 0;\n const nums = new Array(10000);\n const node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n const c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n
space_complexity.rs/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n final int a = 0;\n int b = 0;\n List<int> nums = List.filled(10000, 0);\n ListNode node = ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (var i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (var i = 0; i < n; i++) {\n function();\n }\n}\n
space_complexity.c/* \u51fd\u6570 */\nfn function() ->i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const A: i32 = 0;\n let b = 0;\n let nums = vec![0; 10000];\n let node = ListNode::new(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for i in 0..n {\n let c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for i in 0..n {\n function();\n }\n}\n
space_complexity.zig/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const int a = 0;\n int b = 0;\n int nums[1000];\n ListNode *node = newListNode(0);\n free(node);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n
"},{"location":"chapter_computational_complexity/space_complexity/#2-linear-order-on","title":"2. \u00a0 Linear Order \\(O(n)\\)","text":"// \u51fd\u6570\nfn function() i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a: i32 = 0;\n var b: i32 = 0;\n var nums = [_]i32{0}**10000;\n var node = inc.ListNode(i32){.val = 0};\n var i: i32 = 0;\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n while (i < n) : (i += 1) {\n var c: i32 = 0;\n _ = c;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n i = 0;\n while (i < n) : (i += 1) {\n _ = function();\n }\n _ = a;\n _ = b;\n _ = nums;\n _ = node;\n}\nLinear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to \\(n\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef linear(n: int):\n \"\"\"\u7ebf\u6027\u9636\"\"\"\n # \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n nums = [0] * n\n # \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n hmap = dict[int, str]()\n for i in range(n):\n hmap[i] = str(i)\n
space_complexity.java/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n vector<int> nums(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n vector<ListNode> nodes;\n for (int i = 0; i < n; i++) {\n nodes.push_back(ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n unordered_map<int, string> map;\n for (int i = 0; i < n; i++) {\n map[i] = to_string(i);\n }\n}\n
space_complexity.cs/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int[] nums = new int[n];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n nodes.add(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Map<Integer, String> map = new HashMap<>();\n for (int i = 0; i < n; i++) {\n map.put(i, String.valueOf(i));\n }\n}\n
space_complexity.go/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int[] nums = new int[n];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = [];\n for (int i = 0; i < n; i++) {\n nodes.Add(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Dictionary<int, string> map = [];\n for (int i = 0; i < n; i++) {\n map.Add(i, i.ToString());\n }\n}\n
space_complexity.swift/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n _ = make([]int, n)\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var nodes []*node\n for i := 0; i < n; i++ {\n nodes = append(nodes, newNode(i))\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n m := make(map[int]string, n)\n for i := 0; i < n; i++ {\n m[i] = strconv.Itoa(i)\n }\n}\n
space_complexity.js/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n let nums = Array(repeating: 0, count: n)\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let nodes = (0 ..< n).map { ListNode(x: $0) }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, \"\\($0)\") })\n}\n
space_complexity.ts/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n const nums = new Array(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const nodes = [];\n for (let i = 0; i < n; i++) {\n nodes.push(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const map = new Map();\n for (let i = 0; i < n; i++) {\n map.set(i, i.toString());\n }\n}\n
space_complexity.dart/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n const nums = new Array(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const nodes: ListNode[] = [];\n for (let i = 0; i < n; i++) {\n nodes.push(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const map = new Map();\n for (let i = 0; i < n; i++) {\n map.set(i, i.toString());\n }\n}\n
space_complexity.rs/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n List<int> nums = List.filled(n, 0);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = [];\n for (var i = 0; i < n; i++) {\n nodes.add(ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Map<int, String> map = HashMap();\n for (var i = 0; i < n; i++) {\n map.putIfAbsent(i, () => i.toString());\n }\n}\n
space_complexity.c/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n let mut nums = vec![0; n as usize];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let mut nodes = Vec::new();\n for i in 0..n {\n nodes.push(ListNode::new(i))\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let mut map = HashMap::new();\n for i in 0..n {\n map.insert(i, i.to_string());\n }\n}\n
space_complexity.zig/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n int key;\n int val;\n UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int *nums = malloc(sizeof(int) * n);\n free(nums);\n\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n ListNode **nodes = malloc(sizeof(ListNode *) * n);\n for (int i = 0; i < n; i++) {\n nodes[i] = newListNode(i);\n }\n // \u5185\u5b58\u91ca\u653e\n for (int i = 0; i < n; i++) {\n free(nodes[i]);\n }\n free(nodes);\n\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n HashTable *h = NULL;\n for (int i = 0; i < n; i++) {\n HashTable *tmp = malloc(sizeof(HashTable));\n tmp->key = i;\n tmp->val = i;\n HASH_ADD_INT(h, key, tmp);\n }\n\n // \u5185\u5b58\u91ca\u653e\n HashTable *curr, *tmp;\n HASH_ITER(hh, h, curr, tmp) {\n HASH_DEL(h, curr);\n free(curr);\n }\n}\n// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n var nums = [_]i32{0}**n;\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n defer nodes.deinit();\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n try nodes.append(i);\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n defer map.deinit();\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n defer std.heap.page_allocator.free(string);\n try map.put(i, string);\n }\n _ = nums;\n}\nAs shown below, this function's recursive depth is \\(n\\), meaning there are \\(n\\) instances of unreturned
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.pylinear_recur()function, using \\(O(n)\\) size of stack frame space:
space_complexity.cppdef linear_recur(n: int):\n \"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n print(\"\u9012\u5f52 n =\", n)\n if n == 1:\n return\n linear_recur(n - 1)\n
space_complexity.java/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n cout << \"\u9012\u5f52 n = \" << n << endl;\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n
space_complexity.cs/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n System.out.println(\"\u9012\u5f52 n = \" + n);\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n
space_complexity.go/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\n Console.WriteLine(\"\u9012\u5f52 n = \" + n);\n if (n == 1) return;\n LinearRecur(n - 1);\n}\n
space_complexity.swift/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\n fmt.Println(\"\u9012\u5f52 n =\", n)\n if n == 1 {\n return\n }\n spaceLinearRecur(n - 1)\n}\n
space_complexity.js/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\n print(\"\u9012\u5f52 n = \\(n)\")\n if n == 1 {\n return\n }\n linearRecur(n: n - 1)\n}\n
space_complexity.ts/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\n console.log(`\u9012\u5f52 n = ${n}`);\n if (n === 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.dart/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\n console.log(`\u9012\u5f52 n = ${n}`);\n if (n === 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.rs/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n print('\u9012\u5f52 n = $n');\n if (n == 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.c/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\n println!(\"\u9012\u5f52 n = {}\", n);\n if n == 1 {return};\n linear_recur(n - 1);\n}\n
space_complexity.zig/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n printf(\"\u9012\u5f52 n = %d\\r\\n\", n);\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\n std.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\n if (n == 1) return;\n linearRecur(n - 1);\n}\nFigure 2-17 \u00a0 Recursive Function Generating Linear Order Space Complexity
"},{"location":"chapter_computational_complexity/space_complexity/#3-quadratic-order-on2","title":"3. \u00a0 Quadratic Order \\(O(n^2)\\)","text":"Quadratic order is common in matrices and graphs, where the number of elements is quadratic to \\(n\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef quadratic(n: int):\n \"\"\"\u5e73\u65b9\u9636\"\"\"\n # \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n num_matrix = [[0] * n for _ in range(n)]\n
space_complexity.java/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n vector<vector<int>> numMatrix;\n for (int i = 0; i < n; i++) {\n vector<int> tmp;\n for (int j = 0; j < n; j++) {\n tmp.push_back(0);\n }\n numMatrix.push_back(tmp);\n }\n}\n
space_complexity.cs/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n int[][] numMatrix = new int[n][n];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<Integer>> numList = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n List<Integer> tmp = new ArrayList<>();\n for (int j = 0; j < n; j++) {\n tmp.add(0);\n }\n numList.add(tmp);\n }\n}\n
space_complexity.go/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n int[,] numMatrix = new int[n, n];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numList = [];\n for (int i = 0; i < n; i++) {\n List<int> tmp = [];\n for (int j = 0; j < n; j++) {\n tmp.Add(0);\n }\n numList.Add(tmp);\n }\n}\n
space_complexity.swift/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n numMatrix := make([][]int, n)\n for i := 0; i < n; i++ {\n numMatrix[i] = make([]int, n)\n }\n}\n
space_complexity.js/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n
space_complexity.ts/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numMatrix = Array(n)\n .fill(null)\n .map(() => Array(n).fill(null));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numList = [];\n for (let i = 0; i < n; i++) {\n const tmp = [];\n for (let j = 0; j < n; j++) {\n tmp.push(0);\n }\n numList.push(tmp);\n }\n}\n
space_complexity.dart/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numMatrix = Array(n)\n .fill(null)\n .map(() => Array(n).fill(null));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numList = [];\n for (let i = 0; i < n; i++) {\n const tmp = [];\n for (let j = 0; j < n; j++) {\n tmp.push(0);\n }\n numList.push(tmp);\n }\n}\n
space_complexity.rs/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numMatrix = List.generate(n, (_) => List.filled(n, 0));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numList = [];\n for (var i = 0; i < n; i++) {\n List<int> tmp = [];\n for (int j = 0; j < n; j++) {\n tmp.add(0);\n }\n numList.add(tmp);\n }\n}\n
space_complexity.c/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n let num_matrix = vec![vec![0; n as usize]; n as usize];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n let mut num_list = Vec::new();\n for i in 0..n {\n let mut tmp = Vec::new();\n for j in 0..n {\n tmp.push(0);\n }\n num_list.push(tmp);\n }\n}\n
space_complexity.zig/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n int **numMatrix = malloc(sizeof(int *) * n);\n for (int i = 0; i < n; i++) {\n int *tmp = malloc(sizeof(int) * n);\n for (int j = 0; j < n; j++) {\n tmp[j] = 0;\n }\n numMatrix[i] = tmp;\n }\n\n // \u5185\u5b58\u91ca\u653e\n for (int i = 0; i < n; i++) {\n free(numMatrix[i]);\n }\n free(numMatrix);\n}\n// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n defer nodes.deinit();\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n defer tmp.deinit();\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n try tmp.append(0);\n }\n try nodes.append(tmp);\n }\n}\nAs shown below, the recursive depth of this function is \\(n\\), and in each recursive call, an array is initialized with lengths \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\), averaging \\(n/2\\), thus overall occupying \\(O(n^2)\\) space:
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef quadratic_recur(n: int) -> int:\n \"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n <= 0:\n return 0\n # \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n nums = [0] * n\n return quadratic_recur(n - 1)\n
space_complexity.java/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n vector<int> nums(n);\n cout << \"\u9012\u5f52 n = \" << n << \" \u4e2d\u7684 nums \u957f\u5ea6 = \" << nums.size() << endl;\n return quadraticRecur(n - 1);\n}\n
space_complexity.cs/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n int[] nums = new int[n];\n System.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\n return quadraticRecur(n - 1);\n}\n
space_complexity.go/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\n if (n <= 0) return 0;\n int[] nums = new int[n];\n Console.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\n return QuadraticRecur(n - 1);\n}\n
space_complexity.swift/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n if n <= 0 {\n return 0\n }\n nums := make([]int, n)\n fmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\n return spaceQuadraticRecur(n - 1)\n}\n
space_complexity.js/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -> Int {\n if n <= 0 {\n return 0\n }\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n let nums = Array(repeating: 0, count: n)\n print(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\n return quadraticRecur(n: n - 1)\n}\n
space_complexity.ts/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\n if (n <= 0) return 0;\n const nums = new Array(n);\n console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n return quadraticRecur(n - 1);\n}\n
space_complexity.dart/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\n if (n <= 0) return 0;\n const nums = new Array(n);\n console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n return quadraticRecur(n - 1);\n}\n
space_complexity.rs/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0) return 0;\n List<int> nums = List.filled(n, 0);\n print('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\n return quadraticRecur(n - 1);\n}\n
space_complexity.c/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -> i32 {\n if n <= 0 {return 0};\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n let nums = vec![0; n as usize];\n println!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\n return quadratic_recur(n - 1);\n}\n
space_complexity.zig/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n int *nums = malloc(sizeof(int) * n);\n printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\n int res = quadraticRecur(n - 1);\n free(nums);\n return res;\n}\n// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n if (n <= 0) return 0;\n var nums = [_]i32{0}**n;\n std.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\n return quadraticRecur(n - 1);\n}\nFigure 2-18 \u00a0 Recursive Function Generating Quadratic Order Space Complexity
"},{"location":"chapter_computational_complexity/space_complexity/#4-exponential-order-o2n","title":"4. \u00a0 Exponential Order \\(O(2^n)\\)","text":"Exponential order is common in binary trees. Observe the below image, a \"full binary tree\" with \\(n\\) levels has \\(2^n - 1\\) nodes, occupying \\(O(2^n)\\) space:
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef build_tree(n: int) -> TreeNode | None:\n \"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\n if n == 0:\n return None\n root = TreeNode(0)\n root.left = build_tree(n - 1)\n root.right = build_tree(n - 1)\n return root\n
space_complexity.java/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n if (n == 0)\n return nullptr;\n TreeNode *root = new TreeNode(0);\n root->left = buildTree(n - 1);\n root->right = buildTree(n - 1);\n return root;\n}\n
space_complexity.cs/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\n if (n == 0)\n return null;\n TreeNode root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.go/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\n if (n == 0) return null;\n TreeNode root = new(0) {\n left = BuildTree(n - 1),\n right = BuildTree(n - 1)\n };\n return root;\n}\n
space_complexity.swift/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\n if n == 0 {\n return nil\n }\n root := newTreeNode(0)\n root.left = buildTree(n - 1)\n root.right = buildTree(n - 1)\n return root\n}\n
space_complexity.js/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -> TreeNode? {\n if n == 0 {\n return nil\n }\n let root = TreeNode(x: 0)\n root.left = buildTree(n: n - 1)\n root.right = buildTree(n: n - 1)\n return root\n}\n
space_complexity.ts/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\n if (n === 0) return null;\n const root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.dart/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n if (n === 0) return null;\n const root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.rs/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\n if (n == 0) return null;\n TreeNode root = TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.c/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -> Option<Rc<RefCell<TreeNode>>> {\n if n == 0 {return None};\n let root = TreeNode::new(0);\n root.borrow_mut().left = build_tree(n - 1);\n root.borrow_mut().right = build_tree(n - 1);\n return Some(root);\n}\n
space_complexity.zig/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n if (n == 0)\n return NULL;\n TreeNode *root = newTreeNode(0);\n root->left = buildTree(n - 1);\n root->right = buildTree(n - 1);\n return root;\n}\n// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n if (n == 0) return null;\n const root = try mem_allocator.create(inc.TreeNode(i32));\n root.init(0);\n root.left = try buildTree(mem_allocator, n - 1);\n root.right = try buildTree(mem_allocator, n - 1);\n return root;\n}\nFigure 2-19 \u00a0 Full Binary Tree Generating Exponential Order Space Complexity
"},{"location":"chapter_computational_complexity/space_complexity/#5-logarithmic-order-olog-n","title":"5. \u00a0 Logarithmic Order \\(O(\\log n)\\)","text":"Logarithmic order is common in divide-and-conquer algorithms. For example, in merge sort, an array of length \\(n\\) is recursively divided in half each round, forming a recursion tree of height \\(\\log n\\), using \\(O(\\log n)\\) stack frame space.
Another example is converting a number to a string. Given a positive integer \\(n\\), its number of digits is \\(\\log_{10} n + 1\\), corresponding to the length of the string, thus the space complexity is \\(O(\\log_{10} n + 1) = O(\\log n)\\).
"},{"location":"chapter_computational_complexity/space_complexity/#244-balancing-time-and-space","title":"2.4.4 \u00a0 Balancing Time and Space","text":"Ideally, we aim for both time complexity and space complexity to be optimal. However, in practice, optimizing both simultaneously is often difficult.
Lowering time complexity usually comes at the cost of increased space complexity, and vice versa. The approach of sacrificing memory space to improve algorithm speed is known as \"space-time tradeoff\"; the reverse is known as \"time-space tradeoff\".
The choice depends on which aspect we value more. In most cases, time is more precious than space, so \"space-time tradeoff\" is often the more common strategy. Of course, controlling space complexity is also very important when dealing with large volumes of data.
"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 Summary","text":""},{"location":"chapter_computational_complexity/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"Algorithm Efficiency Assessment
- Time efficiency and space efficiency are the two main criteria for assessing the merits of an algorithm.
- We can assess algorithm efficiency through actual testing, but it's challenging to eliminate the influence of the test environment, and it consumes substantial computational resources.
- Complexity analysis can overcome the disadvantages of actual testing. Its results are applicable across all operating platforms and can reveal the efficiency of algorithms at different data scales.
Time Complexity
- Time complexity measures the trend of an algorithm's running time with the increase in data volume, effectively assessing algorithm efficiency. However, it can fail in certain cases, such as with small input data volumes or when time complexities are the same, making it challenging to precisely compare the efficiency of algorithms.
- Worst-case time complexity is denoted using big O notation, representing the asymptotic upper bound, reflecting the growth level of the number of operations \\(T(n)\\) as \\(n\\) approaches infinity.
- Calculating time complexity involves two steps: first counting the number of operations, then determining the asymptotic upper bound.
- Common time complexities, arranged from low to high, include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n \\log n)\\), \\(O(n^2)\\), \\(O(2^n)\\), and \\(O(n!)\\), among others.
- The time complexity of some algorithms is not fixed and depends on the distribution of input data. Time complexities are divided into worst, best, and average cases. The best case is rarely used because input data generally needs to meet strict conditions to achieve the best case.
- Average time complexity reflects the efficiency of an algorithm under random data inputs, closely resembling the algorithm's performance in actual applications. Calculating average time complexity requires accounting for the distribution of input data and the subsequent mathematical expectation.
Space Complexity
- Space complexity, similar to time complexity, measures the trend of memory space occupied by an algorithm with the increase in data volume.
- The relevant memory space used during the algorithm's execution can be divided into input space, temporary space, and output space. Generally, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions.
- We usually focus only on the worst-case space complexity, which means calculating the space complexity of the algorithm under the worst input data and at the worst moment of operation.
- Common space complexities, arranged from low to high, include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n^2)\\), and \\(O(2^n)\\), among others.
Is the space complexity of tail recursion \\(O(1)\\)?
Theoretically, the space complexity of a tail-recursive function can be optimized to \\(O(1)\\). However, most programming languages (such as Java, Python, C++, Go, C#) do not support automatic optimization of tail recursion, so it's generally considered to have a space complexity of \\(O(n)\\).
What is the difference between the terms 'function' and 'method'?
A \"function\" can be executed independently, with all parameters passed explicitly. A \"method\" is associated with an object and is implicitly passed to the object calling it, able to operate on the data contained within an instance of a class.
Here are some examples from common programming languages:
- C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with these structures are equivalent to methods in other programming languages.
- Java and C# are object-oriented programming languages where code blocks (methods) are typically part of a class. Static methods behave like functions because they are bound to the class and cannot access specific instance variables.
- C++ and Python support both procedural programming (functions) and object-oriented programming (methods).
Does the 'Common Types of Space Complexity' figure reflect the absolute size of occupied space?
No, the figure shows space complexities, which reflect growth trends, not the absolute size of the occupied space.
If you take \\(n = 8\\), you might find that the values of each curve don't correspond to their functions. This is because each curve includes a constant term, intended to compress the value range into a visually comfortable range.
In practice, since we usually don't know the \"constant term\" complexity of each method, it's generally not possible to choose the best solution for \\(n = 8\\) based solely on complexity. However, for \\(n = 8^5\\), it's much easier to choose, as the growth trend becomes dominant.
"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 Time Complexity","text":"Time complexity is a concept used to measure how the run time of an algorithm increases with the size of the input data. Understanding time complexity is crucial for accurately assessing the efficiency of an algorithm.
- Determining the Running Platform: This includes hardware configuration, programming language, system environment, etc., all of which can affect the efficiency of code execution.
- Evaluating the Run Time for Various Computational Operations: For instance, an addition operation
+might take 1 ns, a multiplication operation*might take 10 ns, a print operationprint()might take 5 ns, etc. - Counting All the Computational Operations in the Code: Summing the execution times of all these operations gives the total run time.
For example, consider the following code with an input size of \\(n\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig# Under an operating platform\ndef algorithm(n: int):\n a = 2 # 1 ns\n a = a + 1 # 1 ns\n a = a * 2 # 10 ns\n # Cycle n times\n for _ in range(n): # 1 ns\n print(0) # 5 ns\n// Under a particular operating platform\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n cout << 0 << endl; // 5 ns\n }\n}\n// Under a particular operating platform\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n System.out.println(0); // 5 ns\n }\n}\n// Under a particular operating platform\nvoid Algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n Console.WriteLine(0); // 5 ns\n }\n}\n// Under a particular operating platform\nfunc algorithm(n int) {\n a := 2 // 1 ns\n a = a + 1 // 1 ns\n a = a * 2 // 10 ns\n // Loop n times\n for i := 0; i < n; i++ { // 1 ns\n fmt.Println(a) // 5 ns\n }\n}\n// Under a particular operating platform\nfunc algorithm(n: Int) {\n var a = 2 // 1 ns\n a = a + 1 // 1 ns\n a = a * 2 // 10 ns\n // Loop n times\n for _ in 0 ..< n { // 1 ns\n print(0) // 5 ns\n }\n}\n// Under a particular operating platform\nfunction algorithm(n) {\n var a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for(let i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n console.log(0); // 5 ns\n }\n}\n// Under a particular operating platform\nfunction algorithm(n: number): void {\n var a: number = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for(let i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n console.log(0); // 5 ns\n }\n}\n// Under a particular operating platform\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n print(0); // 5 ns\n }\n}\n// Under a particular operating platform\nfn algorithm(n: i32) {\n let mut a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for _ in 0..n { // 1 ns for each round i++\n println!(\"{}\", 0); // 5 ns\n }\n}\n// Under a particular operating platform\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // Loop n times\n for (int i = 0; i < n; i++) { // 1 ns , every round i++ is executed\n printf(\"%d\", 0); // 5 ns\n }\n}\n// Under a particular operating platform\nfn algorithm(n: usize) void {\n var a: i32 = 2; // 1 ns\n a += 1; // 1 ns\n a *= 2; // 10 ns\n // Loop n times\n for (0..n) |_| { // 1 ns\n std.debug.print(\"{}\\n\", .{0}); // 5 ns\n }\n}\nUsing the above method, the run time of the algorithm can be calculated as \\((6n + 12)\\) ns:
\\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\]However, in practice, counting the run time of an algorithm is neither practical nor reasonable. First, we don't want to tie the estimated time to the running platform, as algorithms need to run on various platforms. Second, it's challenging to know the run time for each type of operation, making the estimation process difficult.
"},{"location":"chapter_computational_complexity/time_complexity/#231-assessing-time-growth-trend","title":"2.3.1 \u00a0 Assessing Time Growth Trend","text":"Time complexity analysis does not count the algorithm's run time, but rather the growth trend of the run time as the data volume increases.
Let's understand this concept of \"time growth trend\" with an example. Assume the input data size is \\(n\\), and consider three algorithms
PythonC++JavaC#GoSwiftJSTSDartRustCZigA,B, andC:# Time complexity of algorithm A: constant order\ndef algorithm_A(n: int):\n print(0)\n# Time complexity of algorithm B: linear order\ndef algorithm_B(n: int):\n for _ in range(n):\n print(0)\n# Time complexity of algorithm C: constant order\ndef algorithm_C(n: int):\n for _ in range(1000000):\n print(0)\n// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n cout << 0 << endl;\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n cout << 0 << endl;\n }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n cout << 0 << endl;\n }\n}\n// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n System.out.println(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n System.out.println(0);\n }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n System.out.println(0);\n }\n}\n// Time complexity of algorithm A: constant order\nvoid AlgorithmA(int n) {\n Console.WriteLine(0);\n}\n// Time complexity of algorithm B: linear order\nvoid AlgorithmB(int n) {\n for (int i = 0; i < n; i++) {\n Console.WriteLine(0);\n }\n}\n// Time complexity of algorithm C: constant order\nvoid AlgorithmC(int n) {\n for (int i = 0; i < 1000000; i++) {\n Console.WriteLine(0);\n }\n}\n// Time complexity of algorithm A: constant order\nfunc algorithm_A(n int) {\n fmt.Println(0)\n}\n// Time complexity of algorithm B: linear order\nfunc algorithm_B(n int) {\n for i := 0; i < n; i++ {\n fmt.Println(0)\n }\n}\n// Time complexity of algorithm C: constant order\nfunc algorithm_C(n int) {\n for i := 0; i < 1000000; i++ {\n fmt.Println(0)\n }\n}\n// Time complexity of algorithm A: constant order\nfunc algorithmA(n: Int) {\n print(0)\n}\n\n// Time complexity of algorithm B: linear order\nfunc algorithmB(n: Int) {\n for _ in 0 ..< n {\n print(0)\n }\n}\n\n// Time complexity of algorithm C: constant order\nfunc algorithmC(n: Int) {\n for _ in 0 ..< 1000000 {\n print(0)\n }\n}\n// Time complexity of algorithm A: constant order\nfunction algorithm_A(n) {\n console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n) {\n for (let i = 0; i < n; i++) {\n console.log(0);\n }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n) {\n for (let i = 0; i < 1000000; i++) {\n console.log(0);\n }\n}\n// Time complexity of algorithm A: constant order\nfunction algorithm_A(n: number): void {\n console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n: number): void {\n for (let i = 0; i < n; i++) {\n console.log(0);\n }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n: number): void {\n for (let i = 0; i < 1000000; i++) {\n console.log(0);\n }\n}\n// Time complexity of algorithm A: constant order\nvoid algorithmA(int n) {\n print(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithmB(int n) {\n for (int i = 0; i < n; i++) {\n print(0);\n }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithmC(int n) {\n for (int i = 0; i < 1000000; i++) {\n print(0);\n }\n}\n// Time complexity of algorithm A: constant order\nfn algorithm_A(n: i32) {\n println!(\"{}\", 0);\n}\n// Time complexity of algorithm B: linear order\nfn algorithm_B(n: i32) {\n for _ in 0..n {\n println!(\"{}\", 0);\n }\n}\n// Time complexity of algorithm C: constant order\nfn algorithm_C(n: i32) {\n for _ in 0..1000000 {\n println!(\"{}\", 0);\n }\n}\n// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n printf(\"%d\", 0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n printf(\"%d\", 0);\n }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n printf(\"%d\", 0);\n }\n}\n// Time complexity of algorithm A: constant order\nfn algorithm_A(n: usize) void {\n _ = n;\n std.debug.print(\"{}\\n\", .{0});\n}\n// Time complexity of algorithm B: linear order\nfn algorithm_B(n: i32) void {\n for (0..n) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n}\n// Time complexity of algorithm C: constant order\nfn algorithm_C(n: i32) void {\n _ = n;\n for (0..1000000) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n}\nThe following figure shows the time complexities of these three algorithms.
- Algorithm
Ahas just one print operation, and its run time does not grow with \\(n\\). Its time complexity is considered \"constant order.\" - Algorithm
Binvolves a print operation looping \\(n\\) times, and its run time grows linearly with \\(n\\). Its time complexity is \"linear order.\" - Algorithm
Chas a print operation looping 1,000,000 times. Although it takes a long time, it is independent of the input data size \\(n\\). Therefore, the time complexity ofCis the same asA, which is \"constant order.\"
Figure 2-7 \u00a0 Time Growth Trend of Algorithms A, B, and C
Compared to directly counting the run time of an algorithm, what are the characteristics of time complexity analysis?
- Time complexity effectively assesses algorithm efficiency. For instance, algorithm
Bhas linearly growing run time, which is slower than algorithmAwhen \\(n > 1\\) and slower thanCwhen \\(n > 1,000,000\\). In fact, as long as the input data size \\(n\\) is sufficiently large, a \"constant order\" complexity algorithm will always be better than a \"linear order\" one, demonstrating the essence of time growth trend. - Time complexity analysis is more straightforward. Obviously, the running platform and the types of computational operations are irrelevant to the trend of run time growth. Therefore, in time complexity analysis, we can simply treat the execution time of all computational operations as the same \"unit time,\" simplifying the \"computational operation run time count\" to a \"computational operation count.\" This significantly reduces the complexity of estimation.
- Time complexity has its limitations. For example, although algorithms
AandChave the same time complexity, their actual run times can be quite different. Similarly, even though algorithmBhas a higher time complexity thanC, it is clearly superior when the input data size \\(n\\) is small. In these cases, it's difficult to judge the efficiency of algorithms based solely on time complexity. Nonetheless, despite these issues, complexity analysis remains the most effective and commonly used method for evaluating algorithm efficiency.
Consider a function with an input size of \\(n\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef algorithm(n: int):\n a = 1 # +1\n a = a + 1 # +1\n a = a * 2 # +1\n # Cycle n times\n for i in range(n): # +1\n print(0) # +1\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n cout << 0 << endl; // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n System.out.println(0); // +1\n }\n}\nvoid Algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n Console.WriteLine(0); // +1\n }\n}\nfunc algorithm(n int) {\n a := 1 // +1\n a = a + 1 // +1\n a = a * 2 // +1\n // Loop n times\n for i := 0; i < n; i++ { // +1\n fmt.Println(a) // +1\n }\n}\nfunc algorithm(n: Int) {\n var a = 1 // +1\n a = a + 1 // +1\n a = a * 2 // +1\n // Loop n times\n for _ in 0 ..< n { // +1\n print(0) // +1\n }\n}\nfunction algorithm(n) {\n var a = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // Loop n times\n for(let i = 0; i < n; i++){ // +1 (execute i ++ every round)\n console.log(0); // +1\n }\n}\nfunction algorithm(n: number): void{\n var a: number = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // Loop n times\n for(let i = 0; i < n; i++){ // +1 (execute i ++ every round)\n console.log(0); // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n print(0); // +1\n }\n}\nfn algorithm(n: i32) {\n let mut a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n\n // Loop n times\n for _ in 0..n { // +1 (execute i ++ every round)\n println!(\"{}\", 0); // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // Loop n times\n for (int i = 0; i < n; i++) { // +1 (execute i ++ every round)\n printf(\"%d\", 0); // +1\n }\n} \nfn algorithm(n: usize) void {\n var a: i32 = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // Loop n times\n for (0..n) |_| { // +1 (execute i ++ every round)\n std.debug.print(\"{}\\n\", .{0}); // +1\n }\n}\nGiven a function that represents the number of operations of an algorithm as a function of the input size \\(n\\), denoted as \\(T(n)\\), consider the following example:
\\[ T(n) = 3 + 2n \\]Since \\(T(n)\\) is a linear function, its growth trend is linear, and therefore, its time complexity is of linear order, denoted as \\(O(n)\\). This mathematical notation, known as \"big-O notation,\" represents the \"asymptotic upper bound\" of the function \\(T(n)\\).
In essence, time complexity analysis is about finding the asymptotic upper bound of the \"number of operations \\(T(n)\\)\". It has a precise mathematical definition.
Asymptotic Upper Bound
If there exist positive real numbers \\(c\\) and \\(n_0\\) such that for all \\(n > n_0\\), \\(T(n) \\leq c \\cdot f(n)\\), then \\(f(n)\\) is considered an asymptotic upper bound of \\(T(n)\\), denoted as \\(T(n) = O(f(n))\\).
As illustrated below, calculating the asymptotic upper bound involves finding a function \\(f(n)\\) such that, as \\(n\\) approaches infinity, \\(T(n)\\) and \\(f(n)\\) have the same growth order, differing only by a constant factor \\(c\\).
Figure 2-8 \u00a0 Asymptotic Upper Bound of a Function
"},{"location":"chapter_computational_complexity/time_complexity/#233-calculation-method","title":"2.3.3 \u00a0 Calculation Method","text":"While the concept of asymptotic upper bound might seem mathematically dense, you don't need to fully grasp it right away. Let's first understand the method of calculation, which can be practiced and comprehended over time.
Once \\(f(n)\\) is determined, we obtain the time complexity \\(O(f(n))\\). But how do we determine the asymptotic upper bound \\(f(n)\\)? This process generally involves two steps: counting the number of operations and determining the asymptotic upper bound.
"},{"location":"chapter_computational_complexity/time_complexity/#1-step-1-counting-the-number-of-operations","title":"1. \u00a0 Step 1: Counting the Number of Operations","text":"This step involves going through the code line by line. However, due to the presence of the constant \\(c\\) in \\(c \\cdot f(n)\\), all coefficients and constant terms in \\(T(n)\\) can be ignored. This principle allows for simplification techniques in counting operations.
- Ignore constant terms in \\(T(n)\\), as they do not affect the time complexity being independent of \\(n\\).
- Omit all coefficients. For example, looping \\(2n\\), \\(5n + 1\\) times, etc., can be simplified to \\(n\\) times since the coefficient before \\(n\\) does not impact the time complexity.
- Use multiplication for nested loops. The total number of operations equals the product of the number of operations in each loop, applying the simplification techniques from points 1 and 2 for each loop level.
Given a function, we can use these techniques to count operations:
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef algorithm(n: int):\n a = 1 # +0 (trick 1)\n a = a + n # +0 (trick 1)\n # +n (technique 2)\n for i in range(5 * n + 1):\n print(0)\n # +n*n (technique 3)\n for i in range(2 * n):\n for j in range(n + 1):\n print(0)\nvoid algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n cout << 0 << endl;\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n cout << 0 << endl;\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n System.out.println(0);\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n System.out.println(0);\n }\n }\n}\nvoid Algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n Console.WriteLine(0);\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n Console.WriteLine(0);\n }\n }\n}\nfunc algorithm(n int) {\n a := 1 // +0 (trick 1)\n a = a + n // +0 (trick 1)\n // +n (technique 2)\n for i := 0; i < 5 * n + 1; i++ {\n fmt.Println(0)\n }\n // +n*n (technique 3)\n for i := 0; i < 2 * n; i++ {\n for j := 0; j < n + 1; j++ {\n fmt.Println(0)\n }\n }\n}\nfunc algorithm(n: Int) {\n var a = 1 // +0 (trick 1)\n a = a + n // +0 (trick 1)\n // +n (technique 2)\n for _ in 0 ..< (5 * n + 1) {\n print(0)\n }\n // +n*n (technique 3)\n for _ in 0 ..< (2 * n) {\n for _ in 0 ..< (n + 1) {\n print(0)\n }\n }\n}\nfunction algorithm(n) {\n let a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (let i = 0; i < 5 * n + 1; i++) {\n console.log(0);\n }\n // +n*n (technique 3)\n for (let i = 0; i < 2 * n; i++) {\n for (let j = 0; j < n + 1; j++) {\n console.log(0);\n }\n }\n}\nfunction algorithm(n: number): void {\n let a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (let i = 0; i < 5 * n + 1; i++) {\n console.log(0);\n }\n // +n*n (technique 3)\n for (let i = 0; i < 2 * n; i++) {\n for (let j = 0; j < n + 1; j++) {\n console.log(0);\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n print(0);\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n print(0);\n }\n }\n}\nfn algorithm(n: i32) {\n let mut a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n\n // +n (technique 2)\n for i in 0..(5 * n + 1) {\n println!(\"{}\", 0);\n }\n\n // +n*n (technique 3)\n for i in 0..(2 * n) {\n for j in 0..(n + 1) {\n println!(\"{}\", 0);\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0 (trick 1)\n a = a + n; // +0 (trick 1)\n // +n (technique 2)\n for (int i = 0; i < 5 * n + 1; i++) {\n printf(\"%d\", 0);\n }\n // +n*n (technique 3)\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n printf(\"%d\", 0);\n }\n }\n}\nfn algorithm(n: usize) void {\n var a: i32 = 1; // +0 (trick 1)\n a = a + @as(i32, @intCast(n)); // +0 (trick 1)\n\n // +n (technique 2)\n for(0..(5 * n + 1)) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n\n // +n*n (technique 3)\n for(0..(2 * n)) |_| {\n for(0..(n + 1)) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n }\n}\nThe formula below shows the counting results before and after simplification, both leading to a time complexity of \\(O(n^2)\\):
\\[ \\begin{aligned} T(n) & = 2n(n + 1) + (5n + 1) + 2 & \\text{Complete Count (-.-|||)} \\newline & = 2n^2 + 7n + 3 \\newline T(n) & = n^2 + n & \\text{Simplified Count (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2-step-2-determining-the-asymptotic-upper-bound","title":"2. \u00a0 Step 2: Determining the Asymptotic Upper Bound","text":"The time complexity is determined by the highest order term in \\(T(n)\\). This is because, as \\(n\\) approaches infinity, the highest order term dominates, rendering the influence of other terms negligible.
The following table illustrates examples of different operation counts and their corresponding time complexities. Some exaggerated values are used to emphasize that coefficients cannot alter the order of growth. When \\(n\\) becomes very large, these constants become insignificant.
Table: Time Complexity for Different Operation Counts
Operation Count \\(T(n)\\) Time Complexity \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234-common-types-of-time-complexity","title":"2.3.4 \u00a0 Common Types of Time Complexity","text":"Let's consider the input data size as \\(n\\). The common types of time complexities are illustrated below, arranged from lowest to highest:
\\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n \\log n) < O(n^2) < O(2^n) < O(n!) \\newline \\text{Constant Order} < \\text{Logarithmic Order} < \\text{Linear Order} < \\text{Linear-Logarithmic Order} < \\text{Quadratic Order} < \\text{Exponential Order} < \\text{Factorial Order} \\end{aligned} \\]Figure 2-9 \u00a0 Common Types of Time Complexity
"},{"location":"chapter_computational_complexity/time_complexity/#1-constant-order-o1","title":"1. \u00a0 Constant Order \\(O(1)\\)","text":"Constant order means the number of operations is independent of the input data size \\(n\\). In the following function, although the number of operations
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.pysizemight be large, the time complexity remains \\(O(1)\\) as it's unrelated to \\(n\\):
time_complexity.cppdef constant(n: int) -> int:\n \"\"\"\u5e38\u6570\u9636\"\"\"\n count = 0\n size = 100000\n for _ in range(size):\n count += 1\n return count\n
time_complexity.java/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.cs/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.go/* \u5e38\u6570\u9636 */\nint Constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.swift/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\n count := 0\n size := 100000\n for i := 0; i < size; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -> Int {\n var count = 0\n let size = 100_000\n for _ in 0 ..< size {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n let count = 0;\n const size = 100000;\n for (let i = 0; i < size; i++) count++;\n return count;\n}\n
time_complexity.dart/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\n let count = 0;\n const size = 100000;\n for (let i = 0; i < size; i++) count++;\n return count;\n}\n
time_complexity.rs/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (var i = 0; i < size; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -> i32 {\n _ = n;\n let mut count = 0;\n let size = 100_000;\n for _ in 0..size {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n int i = 0;\n for (int i = 0; i < size; i++) {\n count++;\n }\n return count;\n}\n
"},{"location":"chapter_computational_complexity/time_complexity/#2-linear-order-on","title":"2. \u00a0 Linear Order \\(O(n)\\)","text":"// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n _ = n;\n var count: i32 = 0;\n const size: i32 = 100_000;\n var i: i32 = 0;\n while(i<size) : (i += 1) {\n count += 1;\n }\n return count;\n}\nLinear order indicates the number of operations grows linearly with the input data size \\(n\\). Linear order commonly appears in single-loop structures:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef linear(n: int) -> int:\n \"\"\"\u7ebf\u6027\u9636\"\"\"\n count = 0\n for _ in range(n):\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\n count := 0\n for i := 0; i < n; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -> Int {\n var count = 0\n for _ in 0 ..< n {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n let count = 0;\n for (let i = 0; i < n; i++) count++;\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\n let count = 0;\n for (let i = 0; i < n; i++) count++;\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (var i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -> i32 {\n let mut count = 0;\n for _ in 0..n {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\n var count: i32 = 0;\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n count += 1;\n }\n return count;\n}\nOperations like array traversal and linked list traversal have a time complexity of \\(O(n)\\), where \\(n\\) is the length of the array or list:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef array_traversal(nums: list[int]) -> int:\n \"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\n count = 0\n # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for num in nums:\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector<int> &nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int num : nums) {\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int num : nums) {\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n foreach (int num in nums) {\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\n count := 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for range nums {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -> Int {\n var count = 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for _ in nums {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (let i = 0; i < nums.length; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (let i = 0; i < nums.length; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List<int> nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (var _num in nums) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &[i32]) -> i32 {\n let mut count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for _ in nums {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\n var count: i32 = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (nums) |_| {\n count += 1;\n }\n return count;\n}\nIt's important to note that the input data size \\(n\\) should be determined based on the type of input data. For example, in the first example, \\(n\\) represents the input data size, while in the second example, the length of the array \\(n\\) is the data size.
"},{"location":"chapter_computational_complexity/time_complexity/#3-quadratic-order-on2","title":"3. \u00a0 Quadratic Order \\(O(n^2)\\)","text":"Quadratic order means the number of operations grows quadratically with the input data size \\(n\\). Quadratic order typically appears in nested loops, where both the outer and inner loops have a time complexity of \\(O(n)\\), resulting in an overall complexity of \\(O(n^2)\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef quadratic(n: int) -> int:\n \"\"\"\u5e73\u65b9\u9636\"\"\"\n count = 0\n # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for i in range(n):\n for j in range(n):\n count += 1\n return count\n
time_complexity.java/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.cs/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.go/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.swift/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\n count := 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for i := 0; i < n; i++ {\n for j := 0; j < n; j++ {\n count++\n }\n }\n return count\n}\n
time_complexity.js/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -> Int {\n var count = 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for _ in 0 ..< n {\n for _ in 0 ..< n {\n count += 1\n }\n }\n return count\n}\n
time_complexity.ts/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.dart/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.rs/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.c/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -> i32 {\n let mut count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for _ in 0..n {\n for _ in 0..n {\n count += 1;\n }\n }\n count\n}\n
time_complexity.zig/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\n var count: i32 = 0;\n var i: i32 = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n while (i < n) : (i += 1) {\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n count += 1;\n }\n }\n return count;\n}\nThe following image compares constant order, linear order, and quadratic order time complexities.
Figure 2-10 \u00a0 Constant, Linear, and Quadratic Order Time Complexities
For instance, in bubble sort, the outer loop runs \\(n - 1\\) times, and the inner loop runs \\(n-1\\), \\(n-2\\), ..., \\(2\\), \\(1\\) times, averaging \\(n / 2\\) times, resulting in a time complexity of \\(O((n - 1) n / 2) = O(n^2)\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef bubble_sort(nums: list[int]) -> int:\n \"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\n count = 0 # \u8ba1\u6570\u5668\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in range(len(nums) - 1, 0, -1):\n # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j in range(i):\n if nums[j] > nums[j + 1]:\n # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n tmp: int = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n count += 3 # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n return count\n
time_complexity.java/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector<int> &nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.size() - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.cs/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.go/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.Length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.swift/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n count := 0 // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := len(nums) - 1; i > 0; i-- {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j := 0; j < i; j++ {\n if nums[j] > nums[j+1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n tmp := nums[j]\n nums[j] = nums[j+1]\n nums[j+1] = tmp\n count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count\n}\n
time_complexity.js/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -> Int {\n var count = 0 // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0 ..< i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count\n}\n
time_complexity.ts/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n let count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.dart/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n let count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.rs/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List<int> nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (var i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (var j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.c/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &mut [i32]) -> i32 {\n let mut count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in (1..nums.len()).rev() {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0..i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n count\n}\n
time_complexity.zig/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = n - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
"},{"location":"chapter_computational_complexity/time_complexity/#4-exponential-order-o2n","title":"4. \u00a0 Exponential Order \\(O(2^n)\\)","text":"// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n var count: i32 = 0; // \u8ba1\u6570\u5668 \n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n while (i > 0) : (i -= 1) {\n var j: usize = 0;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n while (j < i) : (j += 1) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n var tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\nBiological \"cell division\" is a classic example of exponential order growth: starting with one cell, it becomes two after one division, four after two divisions, and so on, resulting in \\(2^n\\) cells after \\(n\\) divisions.
The following image and code simulate the cell division process, with a time complexity of \\(O(2^n)\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef exponential(n: int) -> int:\n \"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n count = 0\n base = 1\n # \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in range(n):\n for _ in range(base):\n count += 1\n base *= 2\n # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n
time_complexity.java/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.cs/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.go/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\n int count = 0, bas = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < bas; j++) {\n count++;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.swift/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\n count, base := 0, 1\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for i := 0; i < n; i++ {\n for j := 0; j < base; j++ {\n count++\n }\n base *= 2\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n}\n
time_complexity.js/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -> Int {\n var count = 0\n var base = 1\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in 0 ..< n {\n for _ in 0 ..< base {\n count += 1\n }\n base *= 2\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n}\n
time_complexity.ts/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\n let count = 0,\n base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.dart/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\n let count = 0,\n base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.rs/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (var i = 0; i < n; i++) {\n for (var j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.c/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -> i32 {\n let mut count = 0;\n let mut base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in 0..n {\n for _ in 0..base {\n count += 1\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n count\n}\n
time_complexity.zig/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0;\n int bas = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < bas; j++) {\n count++;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\n var count: i32 = 0;\n var bas: i32 = 1;\n var i: i32 = 0;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n while (i < n) : (i += 1) {\n var j: i32 = 0;\n while (j < bas) : (j += 1) {\n count += 1;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\nFigure 2-11 \u00a0 Exponential Order Time Complexity
In practice, exponential order often appears in recursive functions. For example, in the code below, it recursively splits into two halves, stopping after \\(n\\) divisions:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef exp_recur(n: int) -> int:\n \"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n == 1:\n return 1\n return exp_recur(n - 1) + exp_recur(n - 1) + 1\n
time_complexity.java/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.cs/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.go/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\n if (n == 1) return 1;\n return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n
time_complexity.swift/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\n if n == 1 {\n return 1\n }\n return expRecur(n-1) + expRecur(n-1) + 1\n}\n
time_complexity.js/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -> Int {\n if n == 1 {\n return 1\n }\n return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n
time_complexity.ts/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\n if (n === 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.dart/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\n if (n === 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.rs/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.c/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -> i32 {\n if n == 1 {\n return 1;\n }\n exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n
time_complexity.zig/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\n if (n == 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\nExponential order growth is extremely rapid and is commonly seen in exhaustive search methods (brute force, backtracking, etc.). For large-scale problems, exponential order is unacceptable, often requiring dynamic programming or greedy algorithms as solutions.
"},{"location":"chapter_computational_complexity/time_complexity/#5-logarithmic-order-olog-n","title":"5. \u00a0 Logarithmic Order \\(O(\\log n)\\)","text":"In contrast to exponential order, logarithmic order reflects situations where \"the size is halved each round.\" Given an input data size \\(n\\), since the size is halved each round, the number of iterations is \\(\\log_2 n\\), the inverse function of \\(2^n\\).
The following image and code simulate the \"halving each round\" process, with a time complexity of \\(O(\\log_2 n)\\), commonly abbreviated as \\(O(\\log n)\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef logarithmic(n: float) -> int:\n \"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n count = 0\n while n > 1:\n n = n / 2\n count += 1\n return count\n
time_complexity.java/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n /= 2;\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\n count := 0\n for n > 1 {\n n = n / 2\n count++\n }\n return count\n}\n
time_complexity.js/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -> Int {\n var count = 0\n var n = n\n while n > 1 {\n n = n / 2\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\n let count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\n let count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(num n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: f32) -> i32 {\n let mut count = 0;\n while n > 1.0 {\n n = n / 2.0;\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\n var count: i32 = 0;\n var n_var = n;\n while (n_var > 1)\n {\n n_var = n_var / 2;\n count +=1;\n }\n return count;\n}\nFigure 2-12 \u00a0 Logarithmic Order Time Complexity
Like exponential order, logarithmic order also frequently appears in recursive functions. The code below forms a recursive tree of height \\(\\log_2 n\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef log_recur(n: float) -> int:\n \"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n <= 1:\n return 0\n return log_recur(n / 2) + 1\n
time_complexity.java/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.cs/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.go/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(float n) {\n if (n <= 1) return 0;\n return LogRecur(n / 2) + 1;\n}\n
time_complexity.swift/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\n if n <= 1 {\n return 0\n }\n return logRecur(n/2) + 1\n}\n
time_complexity.js/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -> Int {\n if n <= 1 {\n return 0\n }\n return logRecur(n: n / 2) + 1\n}\n
time_complexity.ts/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.dart/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.rs/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(num n) {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.c/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: f32) -> i32 {\n if n <= 1.0 {\n return 0;\n }\n log_recur(n / 2.0) + 1\n}\n
time_complexity.zig/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\nLogarithmic order is typical in algorithms based on the divide-and-conquer strategy, embodying the \"split into many\" and \"simplify complex problems\" approach. It's slow-growing and is the most ideal time complexity after constant order.
What is the base of \\(O(\\log n)\\)?
Technically, \"splitting into \\(m\\)\" corresponds to a time complexity of \\(O(\\log_m n)\\). Using the logarithm base change formula, we can equate different logarithmic complexities:
\\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\]This means the base \\(m\\) can be changed without affecting the complexity. Therefore, we often omit the base \\(m\\) and simply denote logarithmic order as \\(O(\\log n)\\).
"},{"location":"chapter_computational_complexity/time_complexity/#6-linear-logarithmic-order-on-log-n","title":"6. \u00a0 Linear-Logarithmic Order \\(O(n \\log n)\\)","text":"Linear-logarithmic order often appears in nested loops, with the complexities of the two loops being \\(O(\\log n)\\) and \\(O(n)\\) respectively. The related code is as follows:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef linear_log_recur(n: float) -> int:\n \"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\n if n <= 1:\n return 1\n count: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n for _ in range(n):\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(float n) {\n if (n <= 1) return 1;\n int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\n if n <= 1 {\n return 1\n }\n count := linearLogRecur(n/2) + linearLogRecur(n/2)\n for i := 0.0; i < n; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -> Int {\n if n <= 1 {\n return 1\n }\n var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n for _ in stride(from: 0, to: n, by: 1) {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\n if (n <= 1) return 1;\n let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (let i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\n if (n <= 1) return 1;\n let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (let i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(num n) {\n if (n <= 1) return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (var i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: f32) -> i32 {\n if n <= 1.0 {\n return 1;\n }\n let mut count = linear_log_recur(n / 2.0) + linear_log_recur(n / 2.0);\n for _ in 0 ..n as i32 {\n count += 1;\n }\n return count\n}\n
time_complexity.zig/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\n if (n <= 1) return 1;\n var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n var i: f32 = 0;\n while (i < n) : (i += 1) {\n count += 1;\n }\n return count;\n}\nThe image below demonstrates how linear-logarithmic order is generated. Each level of a binary tree has \\(n\\) operations, and the tree has \\(\\log_2 n + 1\\) levels, resulting in a time complexity of \\(O(n \\log n)\\).
Figure 2-13 \u00a0 Linear-Logarithmic Order Time Complexity
Mainstream sorting algorithms typically have a time complexity of \\(O(n \\log n)\\), such as quicksort, mergesort, and heapsort.
"},{"location":"chapter_computational_complexity/time_complexity/#7-factorial-order-on","title":"7. \u00a0 Factorial Order \\(O(n!)\\)","text":"Factorial order corresponds to the mathematical problem of \"full permutation.\" Given \\(n\\) distinct elements, the total number of possible permutations is:
\\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\]Factorials are typically implemented using recursion. As shown in the image and code below, the first level splits into \\(n\\) branches, the second level into \\(n - 1\\) branches, and so on, stopping after the \\(n\\)th level:
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef factorial_recur(n: int) -> int:\n \"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n == 0:\n return 1\n count = 0\n # \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in range(n):\n count += factorial_recur(n - 1)\n return count\n
time_complexity.java/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.cs/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.go/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\n if (n == 0) return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += FactorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.swift/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\n if n == 0 {\n return 1\n }\n count := 0\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for i := 0; i < n; i++ {\n count += factorialRecur(n - 1)\n }\n return count\n}\n
time_complexity.js/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -> Int {\n if n == 0 {\n return 1\n }\n var count = 0\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in 0 ..< n {\n count += factorialRecur(n: n - 1)\n }\n return count\n}\n
time_complexity.ts/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\n if (n === 0) return 1;\n let count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (let i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.dart/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\n if (n === 0) return 1;\n let count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (let i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.rs/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0) return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (var i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.c/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -> i32 {\n if n == 0 {\n return 1;\n }\n let mut count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in 0..n {\n count += factorial_recur(n - 1);\n }\n count\n}\n
time_complexity.zig/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\n if (n == 0) return 1;\n var count: i32 = 0;\n var i: i32 = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n while (i < n) : (i += 1) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\nFigure 2-14 \u00a0 Factorial Order Time Complexity
Note that factorial order grows even faster than exponential order; it's unacceptable for larger \\(n\\) values.
"},{"location":"chapter_computational_complexity/time_complexity/#235-worst-best-and-average-time-complexities","title":"2.3.5 \u00a0 Worst, Best, and Average Time Complexities","text":"The time efficiency of an algorithm is often not fixed but depends on the distribution of the input data. Assume we have an array
numsof length \\(n\\), consisting of numbers from \\(1\\) to \\(n\\), each appearing only once, but in a randomly shuffled order. The task is to return the index of the element \\(1\\). We can draw the following conclusions:- When
nums = [?, ?, ..., 1], that is, when the last element is \\(1\\), it requires a complete traversal of the array, achieving the worst-case time complexity of \\(O(n)\\). - When
nums = [1, ?, ?, ...], that is, when the first element is \\(1\\), no matter the length of the array, no further traversal is needed, achieving the best-case time complexity of \\(\\Omega(1)\\).
The \"worst-case time complexity\" corresponds to the asymptotic upper bound, denoted by the big \\(O\\) notation. Correspondingly, the \"best-case time complexity\" corresponds to the asymptotic lower bound, denoted by \\(\\Omega\\):
PythonC++JavaC#GoSwiftJSTSDartRustCZig worst_best_time_complexity.py
worst_best_time_complexity.cppdef random_numbers(n: int) -> list[int]:\n \"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n # \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\n nums = [i for i in range(1, n + 1)]\n # \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n random.shuffle(nums)\n return nums\n\ndef find_one(nums: list[int]) -> int:\n \"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n for i in range(len(nums)):\n # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1:\n return i\n return -1\n
worst_best_time_complexity.java/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector<int> randomNumbers(int n) {\n vector<int> nums(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\n unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector<int> &nums) {\n for (int i = 0; i < nums.size(); i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.cs/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\n Integer[] nums = new Integer[n];\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n Collections.shuffle(Arrays.asList(nums));\n // Integer[] -> int[]\n int[] res = new int[n];\n for (int i = 0; i < n; i++) {\n res[i] = nums[i];\n }\n return res;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n for (int i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.go/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\n int[] nums = new int[n];\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (int i = 0; i < nums.Length; i++) {\n int index = new Random().Next(i, nums.Length);\n (nums[i], nums[index]) = (nums[index], nums[i]);\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n for (int i = 0; i < nums.Length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.swift/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\n nums := make([]int, n)\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for i := 0; i < n; i++ {\n nums[i] = i + 1\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n rand.Shuffle(len(nums), func(i, j int) {\n nums[i], nums[j] = nums[j], nums[i]\n })\n return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n for i := 0; i < len(nums); i++ {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return i\n }\n }\n return -1\n}\n
worst_best_time_complexity.js/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -> [Int] {\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n var nums = Array(1 ... n)\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle()\n return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -> Int {\n for i in nums.indices {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return i\n }\n }\n return -1\n}\n
worst_best_time_complexity.ts/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\n const nums = Array(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (let i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (let i = 0; i < n; i++) {\n const r = Math.floor(Math.random() * (i + 1));\n const temp = nums[i];\n nums[i] = nums[r];\n nums[r] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n for (let i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] === 1) {\n return i;\n }\n }\n return -1;\n}\n
worst_best_time_complexity.dart/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\n const nums = Array(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (let i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (let i = 0; i < n; i++) {\n const r = Math.floor(Math.random() * (i + 1));\n const temp = nums[i];\n nums[i] = nums[r];\n nums[r] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n for (let i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] === 1) {\n return i;\n }\n }\n return -1;\n}\n
worst_best_time_complexity.rs/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList<int> randomNumbers(int n) {\n final nums = List.filled(n, 0);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (var i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle();\n\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List<int> nums) {\n for (var i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1) return i;\n }\n\n return -1;\n}\n
worst_best_time_complexity.c/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -> Vec<i32> {\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n let mut nums = (1..=n).collect::<Vec<i32>>();\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle(&mut thread_rng());\n nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &[i32]) -> Option<usize> {\n for i in 0..nums.len() {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return Some(i);\n }\n }\n None\n}\n
worst_best_time_complexity.zig/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n // \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\n int *nums = (int *)malloc(n * sizeof(int));\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (int i = n - 1; i > 0; i--) {\n int j = rand() % (i + 1);\n int temp = nums[i];\n nums[i] = nums[j];\n nums[j] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n for (int i = 0; i < n; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\n var nums: [n]i32 = undefined;\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (&nums, 0..) |*num, i| {\n num.* = @as(i32, @intCast(i)) + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n const rand = std.crypto.random;\n rand.shuffle(i32, &nums);\n return nums;\n}\n\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n for (nums, 0..) |num, i| {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (num == 1) return @intCast(i);\n }\n return -1;\n}\nIt's important to note that the best-case time complexity is rarely used in practice, as it is usually only achievable under very low probabilities and might be misleading. The worst-case time complexity is more practical as it provides a safety value for efficiency, allowing us to confidently use the algorithm.
From the above example, it's clear that both the worst-case and best-case time complexities only occur under \"special data distributions,\" which may have a small probability of occurrence and may not accurately reflect the algorithm's run efficiency. In contrast, the average time complexity can reflect the algorithm's efficiency under random input data, denoted by the \\(\\Theta\\) notation.
For some algorithms, we can simply estimate the average case under a random data distribution. For example, in the aforementioned example, since the input array is shuffled, the probability of element \\(1\\) appearing at any index is equal. Therefore, the average number of loops for the algorithm is half the length of the array \\(n / 2\\), giving an average time complexity of \\(\\Theta(n / 2) = \\Theta(n)\\).
However, calculating the average time complexity for more complex algorithms can be quite difficult, as it's challenging to analyze the overall mathematical expectation under the data distribution. In such cases, we usually use the worst-case time complexity as the standard for judging the efficiency of the algorithm.
Why is the \\(\\Theta\\) symbol rarely seen?
Possibly because the \\(O\\) notation is more commonly spoken, it is often used to represent the average time complexity. However, strictly speaking, this practice is not accurate. In this book and other materials, if you encounter statements like \"average time complexity \\(O(n)\\)\", please understand it directly as \\(\\Theta(n)\\).
"},{"location":"chapter_data_structure/","title":"Data Structure","text":"Abstract
Data structures resemble a stable and diverse framework.
They serve as a blueprint for organizing data orderly, enabling algorithms to come to life upon this foundation.
"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"Classification Of Data Structures","text":"Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be divided into two categories: logical structure and physical structure.
"},{"location":"chapter_data_structure/classification_of_data_structure/#logical-structures-linear-and-non-linear","title":"Logical Structures: Linear And Non-linear","text":"Logical structures reveal logical relationships between data elements. In arrays and linked lists, data are arranged in sequential order, reflecting the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between ancestors and descendants; and graphs are composed of nodes and edges, reflecting the complex network relationship.
As shown in the figure below, logical structures can further be divided into \"linear data structure\" and \"non-linear data structure\". Linear data structures are more intuitive, meaning that the data are arranged linearly in terms of logical relationships; non-linear data structures, on the other hand, are arranged non-linearly.
- Linear data structures: arrays, linked lists, stacks, queues, hash tables.
- Nonlinear data structures: trees, heaps, graphs, hash tables.
Non-linear data structures can be further divided into tree and graph structures.
- Linear structures: arrays, linked lists, queues, stacks, hash tables, with one-to-one sequential relationship between elements.
- Tree structure: tree, heap, hash table, with one-to-many relationship between elements.
- Graph: graph with many-to-many relationship between elements.
When an algorithm is running, the data being processed is stored in memory. The figure below shows a computer memory module where each black square represents a memory space. We can think of the memory as a giant Excel sheet in which each cell can store data of a certain size.
The system accesses the data at the target location by means of a memory address. As shown in the figure below, the computer assigns a unique identifier to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data in memory.
Tip
It is worth noting that comparing memory to the Excel sheet is a simplified analogy. The actual memory working mechanism is more complicated, involving the concepts of address, space, memory management, cache mechanism, virtual and physical memory.
Memory is a shared resource for all programs, and when a block of memory is occupied by one program, it cannot be used by other programs at the same time. Therefore, considering memory resources is crucial in designing data structures and algorithms. For example, the algorithm's peak memory usage should not exceed the remaining free memory of the system; if there is a lack of contiguous memory blocks, then the data structure chosen must be able to be stored in non-contiguous memory blocks.
As shown in the figure below, Physical structure reflects the way data is stored in computer memory and it can be divided into consecutive space storage (arrays) and distributed space storage (linked lists). The physical structure determines how data is accessed, updated, added, deleted, etc. Logical and physical structure complement each other in terms of time efficiency and space efficiency.
It is worth stating that all data structures are implemented based on arrays, linked lists, or a combination of the two. For example, stacks and queues can be implemented using both arrays and linked lists; and implementations of hash tables may contain both arrays and linked lists.
- Array-based structures: stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays of dimension \\(\\geq 3\\)), and so on.
- Linked list-based structures: stacks, queues, hash tables, trees, heaps, graphs, etc.
Data structures based on arrays are also known as \"static data structures\", which means that such structures' length remains constant after initialization. In contrast, data structures based on linked lists are called \"dynamic data structures\", meaning that their length can be adjusted during program execution after initialization.
Tip
If you find it difficult to understand the physical structure, it is recommended that you read the next chapter, \"Arrays and Linked Lists,\" before reviewing this section.
"},{"location":"chapter_introduction/","title":"Chapter 1. \u00a0 Introduction to Algorithms","text":"Abstract
A graceful maiden dances, intertwined with the data, her skirt swaying to the melody of algorithms.
She invites you to a dance, follow her steps, and enter the world of algorithms full of logic and beauty.
"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 1.1 \u00a0 Algorithms Are Everywhere
- 1.2 \u00a0 What Is Algorithms
- 1.3 \u00a0 Summary
When we hear the word \"algorithm,\" we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.
Before formally discussing algorithms, there's an interesting fact worth sharing: you have already unconsciously learned many algorithms and have become accustomed to applying them in your daily life. Here, I will give a few specific examples to prove this point.
Example 1: Looking Up a Dictionary. In an English dictionary, words are listed alphabetically. Suppose we're searching for a word that starts with the letter \\(r\\). This is typically done in the following way:
- Open the dictionary to about halfway and check the first letter on the page, let's say the letter is \\(m\\).
- Since \\(r\\) comes after \\(m\\) in the alphabet, we can ignore the first half of the dictionary and focus on the latter half.
- Repeat steps
1.and2.until you find the page where the word starts with \\(r\\).
Figure 1-1 \u00a0 Process of Looking Up a Dictionary
This essential skill for elementary students, looking up a dictionary, is actually the famous \"Binary Search\" algorithm. From a data structure perspective, we can consider the dictionary as a sorted \"array\"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as \"Binary Search.\"
Example 2: Organizing Playing Cards. When playing cards, we need to arrange the cards in our hand in ascending order, as shown in the following process.
- Divide the playing cards into \"ordered\" and \"unordered\" sections, assuming initially the leftmost card is already in order.
- Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order.
- Continue to repeat step
2.until all cards are in order.
Figure 1-2 \u00a0 Playing Cards Sorting Process
The above method of organizing playing cards is essentially the \"Insertion Sort\" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.
Example 3: Making Change. Suppose we buy goods worth \\(69\\) yuan at a supermarket and give the cashier \\(100\\) yuan, then the cashier needs to give us \\(31\\) yuan in change. They would naturally complete the thought process as shown below.
- The options are currencies smaller than \\(31\\), including \\(1\\), \\(5\\), \\(10\\), and \\(20\\).
- Take out the largest \\(20\\) from the options, leaving \\(31 - 20 = 11\\).
- Take out the largest \\(10\\) from the remaining options, leaving \\(11 - 10 = 1\\).
- Take out the largest \\(1\\) from the remaining options, leaving \\(1 - 1 = 0\\).
- Complete the change-making, with the solution being \\(20 + 10 + 1 = 31\\).
Figure 1-3 \u00a0 Change making process
In the above steps, we make the best choice at each step (using the largest denomination possible), ultimately resulting in a feasible change-making plan. From the perspective of data structures and algorithms, this method is essentially a \"Greedy\" algorithm.
From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers, solving various complex issues more efficiently.
Tip
If concepts such as data structures, algorithms, arrays, and binary search still seem somewhat obsecure, I encourage you to continue reading. This book will gently guide you into the realm of understanding data structures and algorithms.
"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 Summary","text":"- Algorithms are ubiquitous in daily life and are not as inaccessible and complex as they might seem. In fact, we have already unconsciously learned many algorithms to solve various problems in life.
- The principle of looking up a word in a dictionary is consistent with the binary search algorithm. The binary search algorithm embodies the important algorithmic concept of divide and conquer.
- The process of organizing playing cards is very similar to the insertion sort algorithm. The insertion sort algorithm is suitable for sorting small datasets.
- The steps of making change in currency essentially follow the greedy algorithm, where each step involves making the best possible choice at the moment.
- An algorithm is a set of instructions or steps used to solve a specific problem within a finite amount of time, while a data structure is the way data is organized and stored in a computer.
- Data structures and algorithms are closely linked. Data structures are the foundation of algorithms, and algorithms are the stage to utilize the functions of data structures.
- We can liken data structures and algorithms to building blocks. The blocks represent data, the shape and connection method of the blocks represent data structures, and the steps of assembling the blocks correspond to algorithms.
An \"algorithm\" is a set of instructions or steps to solve a specific problem within a finite amount of time. It has the following characteristics:
- The problem is clearly defined, including unambiguous definitions of input and output.
- The algorithm is feasible, meaning it can be completed within a finite number of steps, time, and memory space.
- Each step has a definitive meaning. The output is consistently the same under the same inputs and conditions.
A \"data structure\" is a way of organizing and storing data in a computer, with the following design goals:
- Minimize space occupancy to save computer memory.
- Make data operations as fast as possible, covering data access, addition, deletion, updating, etc.
- Provide concise data representation and logical information to enable efficient algorithm execution.
Designing data structures is a balancing act, often requiring trade-offs. If you want to improve in one aspect, you often need to compromise in another. Here are two examples:
- Compared to arrays, linked lists offer more convenience in data addition and deletion but sacrifice data access speed.
- Graphs, compared to linked lists, provide richer logical information but require more memory space.
As shown in the Figure 1-4 , data structures and algorithms are highly related and closely integrated, specifically in the following three aspects:
- Data structures are the foundation of algorithms. They provide structured data storage and methods for manipulating data for algorithms.
- Algorithms are the stage where data structures come into play. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved.
- Algorithms can often be implemented based on different data structures, but their execution efficiency can vary greatly. Choosing the right data structure is key.
Figure 1-4 \u00a0 Relationship between data structures and algorithms
Data structures and algorithms can be likened to a set of building blocks, as illustrated in the Figure 1-5 . A building block set includes numerous pieces, accompanied by detailed assembly instructions. Following these instructions step by step allows us to construct an intricate block model.
Figure 1-5 \u00a0 Assembling blocks
The detailed correspondence between the two is shown in the Table 1-1 .
Table 1-1 \u00a0 Comparing Data Structures and Algorithms to Building Blocks
Data Structures and Algorithms Building Blocks Input data Unassembled blocks Data structure Organization of blocks, including shape, size, connections, etc Algorithm A series of steps to assemble the blocks into the desired shape Output data Completed Block modelIt's worth noting that data structures and algorithms are independent of programming languages. For this reason, this book is able to provide implementations in multiple programming languages.
Conventional Abbreviation
In real-life discussions, we often refer to \"Data Structures and Algorithms\" simply as \"Algorithms\". For example, the well-known LeetCode algorithm problems actually test both data structure and algorithm knowledge.
"},{"location":"chapter_preface/","title":"Chapter 0. \u00a0 Preface","text":"Abstract
Algorithms are like a beautiful symphony, with each line of code flowing like a rhythm.
May this book ring softly in your mind, leaving a unique and profound melody.
"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 0.1 \u00a0 The Book
- 0.2 \u00a0 How to Read
- 0.3 \u00a0 Summary
This open-source project aims to create a free, and beginner-friendly crash course on data structures and algorithms.
- Using animated illustrations, it delivers structured insights into data structures and algorithmic concepts, ensuring comprehensibility and a smooth learning curve.
- Run code with just one click, supporting Java, C++, Python, Go, JS, TS, C#, Swift, Rust, Dart, Zig and other languages.
- Readers are encouraged to engage with each other in the discussion area for each section, questions and comments are usually answered within two days.
If you are new to algorithms with limited exposure, or you have accumulated some experience in algorithms, but you only have a vague understanding of data structures and algorithms, and you are constantly jumping between \"yep\" and \"hmm\", then this book is for you!
If you have already accumulated a certain amount of problem-solving experience, and are familiar with most types of problems, then this book can help you review and organize your algorithm knowledge system. The repository's source code can be used as a \"problem-solving toolkit\" or an \"algorithm cheat sheet\".
If you are an algorithm expert, we look forward to receiving your valuable suggestions, or join us and collaborate.
Prerequisites
You should know how to write and read simple code in at least one programming language.
"},{"location":"chapter_preface/about_the_book/#012-content-structure","title":"0.1.2 \u00a0 Content Structure","text":"The main content of the book is shown in the following figure.
- Complexity Analysis: explores aspects and methods for evaluating data structures and algorithms. Covers methods of deriving time complexity and space complexity, along with common types and examples.
- Data Structures: focuses on fundamental data types, classification methods, definitions, pros and cons, common operations, types, applications, and implementation methods of data structures such as array, linked list, stack, queue, hash table, tree, heap, graph, etc.
- Algorithms: defines algorithms, discusses their pros and cons, efficiency, application scenarios, problem-solving steps, and includes sample questions for various algorithms such as search, sorting, divide and conquer, backtracking, dynamic programming, greedy algorithms, and more.
Figure 0-1 \u00a0 Main Content of the Book
"},{"location":"chapter_preface/about_the_book/#013-acknowledgements","title":"0.1.3 \u00a0 Acknowledgements","text":"Throughout the creation of this book, numerous individuals provided invaluable assistance, including but not limited to:
- Thanks to my mentor at the company, Dr. Xi Li, who encouraged me in a conversation to \"get moving fast,\" which solidified my determination to write this book;
- Thanks to my girlfriend Paopao, as the first reader of this book, for offering many valuable suggestions from the perspective of a beginner in algorithms, making this book more suitable for newbies;
- Thanks to Tengbao, Qibao, and Feibao for coming up with a creative name for this book, evoking everyone's fond memories of writing their first line of code \"Hello World!\";
- Thanks to Xiaoquan for providing professional help in intellectual property, which has played a significant role in the development of this open-source book;
- Thanks to Sutong for designing a beautiful cover and logo for this book, and for patiently making multiple revisions under my insistence;
- Thanks to @squidfunk for providing writing and typesetting suggestions, as well as his developed open-source documentation theme Material-for-MkDocs.
Throughout the writing journey, I delved into numerous textbooks and articles on data structures and algorithms. These works served as exemplary models, ensuring the accuracy and quality of this book's content. I extend my gratitude to all who preceded me for their invaluable contributions!
This book advocates a combination of hands-on and minds-on learning, inspired in this regard by \"Dive into Deep Learning\". I highly recommend this excellent book to all readers.
Heartfelt thanks to my parents, whose ongoing support and encouragement have allowed me to do this interesting work.
"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 How To Read","text":"Tip
For the best reading experience, it is recommended that you read through this section.
"},{"location":"chapter_preface/suggestions/#021-conventions-of-style","title":"0.2.1 \u00a0 Conventions Of Style","text":"- Those labeled
*after the title are optional chapters with relatively difficult content. If you have limited time, it is advisable to skip them. - Proper nouns and words and phrases with specific meanings are marked with
\"double quotes\"to avoid ambiguity. - Important proper nouns and their English translations are marked with
\" \"in parentheses, e.g.\"array array\". It is recommended to memorize them for reading the literature. - Bolded text Indicates key content or summary statements, which deserve special attention.
- When it comes to terms that are inconsistent between programming languages, this book follows Python, for example using \\(\\text{None}\\) to mean \"empty\".
- This book partially abandons the specification of annotations in programming languages in exchange for a more compact layout of the content. There are three main types of annotations: title annotations, content annotations, and multi-line annotations.
\"\"\"Header comments for labeling functions, classes, test samples, etc.\"\"\"\"\n\n# Content comments for detailed code solutions\n\n\"\"\"\nmulti-line\nmarginal notes\n\"\"\"\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n/* Header comments for labeling functions, classes, test samples, etc. */\n\n// Content comments for detailed code solutions.\n\n/**\n * multi-line\n * marginal notes\n */\n
"},{"location":"chapter_preface/suggestions/#022-learn-efficiently-in-animated-graphic-solutions","title":"0.2.2 \u00a0 Learn Efficiently In Animated Graphic Solutions","text":"// Header comments for labeling functions, classes, test samples, etc.\n\n// Content comments for detailed code solutions.\n\n// Multi-line\n// Annotation\nCompared with text, videos and pictures have a higher degree of information density and structure and are easier to understand. In this book, key and difficult knowledge will be presented mainly in the form of animations and graphs, while the text serves as an explanation and supplement to the animations and graphs.
If, while reading the book, you find that a particular paragraph provides an animation or a graphic solution as shown below, please use the figure as the primary source and the text as a supplement and synthesize the two to understand the content.
Figure 0-2 \u00a0 Example animation
"},{"location":"chapter_preface/suggestions/#023-deeper-understanding-in-code-practice","title":"0.2.3 \u00a0 Deeper Understanding In Code Practice","text":"The companion code for this book is hosted in the GitHub repository. As shown in the Figure 0-3 , the source code is accompanied by test samples that can be run with a single click.
If time permits, it is recommended that you refer to the code and knock it through on your own. If you have limited time to study, please read through and run all the code at least once.
The process of writing code is often more rewarding than reading it. Learning by doing is really learning.
Figure 0-3 \u00a0 Running code example
The preliminaries for running the code are divided into three main steps.
Step 1: Install the local programming environment. Please refer to Appendix Tutorial for installation, or skip this step if already installed.
Step 2: Clone or download the code repository. If Git is already installed, you can clone this repository with the following command.
git clone https://github.com/krahets/hello-algo.git\nOf course, you can also in the location shown in the Figure 0-4 , click \"Download ZIP\" directly download the code zip, and then in the local solution.
Figure 0-4 \u00a0 Clone repository with download code
Step 3: Run the source code. As shown in the Figure 0-5 , for the code block labeled with the file name at the top, we can find the corresponding source code file in the
codesfolder of the repository. The source code files can be run with a single click, which will help you save unnecessary debugging time and allow you to focus on what you are learning.Figure 0-5 \u00a0 Code block with corresponding source file
"},{"location":"chapter_preface/suggestions/#024-growing-together-in-questioning-and-discussion","title":"0.2.4 \u00a0 Growing Together In Questioning And Discussion","text":"While reading this book, please don't skip over the points that you didn't learn. Feel free to ask your questions in the comment section. We will be happy to answer them and can usually respond within two days.
As you can see in the Figure 0-6 , each post comes with a comment section at the bottom. I hope you'll pay more attention to the comments section. On the one hand, you can learn about the problems that people encounter, so as to check the gaps and stimulate deeper thinking. On the other hand, we expect you to generously answer other partners' questions, share your insights, and help others improve.
Figure 0-6 \u00a0 Example of comment section
"},{"location":"chapter_preface/suggestions/#025-algorithm-learning-route","title":"0.2.5 \u00a0 Algorithm Learning Route","text":"From a general point of view, we can divide the process of learning data structures and algorithms into three stages.
- Introduction to Algorithms. We need to familiarize ourselves with the characteristics and usage of various data structures and learn about the principles, processes, uses and efficiency of different algorithms.
- Brush up on algorithm questions. It is recommended to start brushing from popular topics, such as Sword to Offer and LeetCode Hot 100, first accumulate at least 100 questions to familiarize yourself with mainstream algorithmic problems. Forgetfulness can be a challenge when first brushing up, but rest assured that this is normal. We can follow the \"Ebbinghaus Forgetting Curve\" to review the questions, and usually after 3-5 rounds of repetitions, we will be able to memorize them.
- Build the knowledge system. In terms of learning, we can read algorithm column articles, solution frameworks and algorithm textbooks to continuously enrich the knowledge system. In terms of brushing, we can try to adopt advanced brushing strategies, such as categorizing by topic, multiple solutions, multiple solutions, etc. Related brushing tips can be found in various communities.
As shown in the Figure 0-7 , this book mainly covers \"Phase 1\" and is designed to help you start Phase 2 and 3 more efficiently.
Figure 0-7 \u00a0 algorithm learning route
"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 Summary","text":"- The main audience of this book is beginners in algorithm. If you already have some basic knowledge, this book can help you systematically review your algorithm knowledge, and the source code in this book can also be used as a \"Coding Toolkit\".
- The book consists of three main sections, Complexity Analysis, Data Structures, and Algorithms, covering most of the topics in the field.
- For newcomers to algorithms, it is crucial to read an introductory book in the beginning stages to avoid many detours or common pitfalls.
- Animations and figures within the book are usually used to introduce key points and difficult knowledge. These should be given more attention when reading the book.
- Practice is the best way to learn programming. It is highly recommended that you run the source code and type in the code yourself.
- Each chapter in the web version of this book features a discussion section, and you are welcome to share your questions and insights at any time.
https://www.hello-algo.com/en/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_computational_complexity/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_computational_complexity/iteration_and_recursion/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_computational_complexity/performance_evaluation/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_computational_complexity/space_complexity/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_computational_complexity/summary/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_computational_complexity/time_complexity/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_data_structure/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_data_structure/classification_of_data_structure/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_introduction/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_introduction/algorithms_are_everywhere/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_introduction/summary/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_introduction/what_is_dsa/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_preface/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_preface/about_the_book/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_preface/suggestions/ -2023-12-21 +2023-12-24 daily https://www.hello-algo.com/en/chapter_preface/summary/ -2023-12-21 +2023-12-24 daily \u52a8\u753b\u56fe\u89e3\u3001\u4e00\u952e\u8fd0\u884c\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6559\u7a0b
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\u6211\u6df1\u6df1\u8d5e\u540c\u8d39\u66fc\u6559\u6388\u6240\u8a00\uff1a\u201cKnowledge isn't free. You have to pay attention.\u201d\u4ece\u8fd9\u4e2a\u610f\u4e49\u4e0a\u770b\uff0c\u8fd9\u672c\u4e66\u5e76\u975e\u5b8c\u5168\u201c\u514d\u8d39\u201d\u3002\u4e3a\u4e86\u4e0d\u8f9c\u8d1f\u4f60\u4e3a\u672c\u4e66\u6240\u4ed8\u51fa\u7684\u5b9d\u8d35\u201c\u6ce8\u610f\u529b\u201d\uff0c\u6211\u4f1a\u5c3d\u6211\u6240\u80fd\uff0c\u6295\u5165\u6700\u5927\u7684\u201c\u6ce8\u610f\u529b\u201d\u6765\u5b8c\u6210\u8fd9\u672c\u4e66\u7684\u521b\u4f5c\u3002
\u4f5c\u8005\u9773\u5b87\u680b (Krahets)\uff0c\u5927\u5382\u9ad8\u7ea7\u7b97\u6cd5\u5de5\u7a0b\u5e08\uff0c\u4e0a\u6d77\u4ea4\u901a\u5927\u5b66\u7855\u58eb\u3002\u529b\u6263\uff08LeetCode\uff09\u5168\u7f51\u9605\u8bfb\u91cf\u6700\u9ad8\u535a\u4e3b\uff0c\u53d1\u8868\u7684\u300a\u56fe\u89e3\u7b97\u6cd5\u6570\u636e\u7ed3\u6784\u300b\u5df2\u88ab\u8ba2\u9605 30 \u4e07\u672c\u3002
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GongljaC, C++ gvenusleoDart hpstoryC# justin-tseJS, TS krahetsJava, Python night-cruiseRust nuomi1Swift ReanonGo, C sjinzhRust, Zig"},{"location":"chapter_appendix/","title":"\u7b2c 16 \u7ae0 \u00a0 \u9644\u5f55","text":""},{"location":"chapter_appendix/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5
- 16.2 \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c
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"},{"location":"chapter_appendix/contribution/#1","title":"1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"\u5982\u56fe 16-3 \u6240\u793a\uff0c\u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u201c\u7f16\u8f91\u56fe\u6807\u201d\u3002\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u672c\u6216\u4ee3\u7801\u3002
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- \u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u68c0\u67e5\u5185\u5bb9\u7684\u6b63\u786e\u6027\uff0c\u5e76\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7edf\u4e00\u3002
- \u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u4fee\u6539\u8bf4\u660e\uff0c\u7136\u540e\u70b9\u51fb\u201cPropose file change\u201d\u6309\u94ae\u3002\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002
\u56fe 16-3 \u00a0 \u9875\u9762\u7f16\u8f91\u6309\u952e
\u56fe\u7247\u65e0\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u9700\u8981\u901a\u8fc7\u65b0\u5efa Issue \u6216\u8bc4\u8bba\u7559\u8a00\u6765\u63cf\u8ff0\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5c3d\u5feb\u91cd\u65b0\u7ed8\u5236\u5e76\u66ff\u6362\u56fe\u7247\u3002
"},{"location":"chapter_appendix/contribution/#2","title":"2. \u00a0 \u5185\u5bb9\u521b\u4f5c","text":"\u5982\u679c\u60a8\u6709\u5174\u8da3\u53c2\u4e0e\u6b64\u5f00\u6e90\u9879\u76ee\uff0c\u5305\u62ec\u5c06\u4ee3\u7801\u7ffb\u8bd1\u6210\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u3001\u6269\u5c55\u6587\u7ae0\u5185\u5bb9\u7b49\uff0c\u90a3\u4e48\u9700\u8981\u5b9e\u65bd\u4ee5\u4e0b Pull Request \u5de5\u4f5c\u6d41\u7a0b\u3002
- \u767b\u5f55 GitHub \uff0c\u5c06\u672c\u4e66\u7684\u4ee3\u7801\u4ed3\u5e93 Fork \u5230\u4e2a\u4eba\u8d26\u53f7\u4e0b\u3002
- \u8fdb\u5165\u60a8\u7684 Fork \u4ed3\u5e93\u7f51\u9875\uff0c\u4f7f\u7528
git clone\u547d\u4ee4\u5c06\u4ed3\u5e93\u514b\u9686\u81f3\u672c\u5730\u3002 - \u5728\u672c\u5730\u8fdb\u884c\u5185\u5bb9\u521b\u4f5c\uff0c\u5e76\u8fdb\u884c\u5b8c\u6574\u6d4b\u8bd5\uff0c\u9a8c\u8bc1\u4ee3\u7801\u7684\u6b63\u786e\u6027\u3002
- \u5c06\u672c\u5730\u6240\u505a\u66f4\u6539 Commit \uff0c\u7136\u540e Push \u81f3\u8fdc\u7a0b\u4ed3\u5e93\u3002
- \u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002
\u5728
hello-algo\u6839\u76ee\u5f55\u4e0b\uff0c\u6267\u884c\u4ee5\u4e0b Docker \u811a\u672c\uff0c\u5373\u53ef\u5728http://localhost:8000\u8bbf\u95ee\u672c\u9879\u76ee\uff1adocker-compose up -d\n\u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\uff1a
"},{"location":"chapter_appendix/installation/","title":"16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1611-ide","title":"16.1.1 \u00a0 \u5b89\u88c5 IDE","text":"docker-compose down\n\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u3001\u8f7b\u91cf\u7684 VS Code \u4f5c\u4e3a\u672c\u5730\u96c6\u6210\u5f00\u53d1\u73af\u5883\uff08IDE\uff09\u3002\u8bbf\u95ee VS Code \u5b98\u7f51\uff0c\u6839\u636e\u64cd\u4f5c\u7cfb\u7edf\u9009\u62e9\u76f8\u5e94\u7248\u672c\u7684 VS Code \u8fdb\u884c\u4e0b\u8f7d\u548c\u5b89\u88c5\u3002
\u56fe 16-1 \u00a0 \u4ece\u5b98\u7f51\u4e0b\u8f7d VS Code
VS Code \u62e5\u6709\u5f3a\u5927\u7684\u6269\u5c55\u5305\u751f\u6001\u7cfb\u7edf\uff0c\u652f\u6301\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u7684\u8fd0\u884c\u548c\u8c03\u8bd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5b89\u88c5\u201cPython Extension Pack\u201d\u6269\u5c55\u5305\u4e4b\u540e\uff0c\u5373\u53ef\u8fdb\u884c Python \u4ee3\u7801\u8c03\u8bd5\u3002\u5b89\u88c5\u6b65\u9aa4\u5982\u56fe 16-2 \u6240\u793a\u3002
\u56fe 16-2 \u00a0 \u5b89\u88c5 VS Code \u6269\u5c55\u5305
"},{"location":"chapter_appendix/installation/#1612","title":"16.1.2 \u00a0 \u5b89\u88c5\u8bed\u8a00\u73af\u5883","text":""},{"location":"chapter_appendix/installation/#1-python","title":"1. \u00a0 Python \u73af\u5883","text":"- \u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \uff0c\u9700\u8981 Python 3.10 \u6216\u66f4\u65b0\u7248\u672c\u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
python\uff0c\u5b89\u88c5 Python Extension Pack \u3002 - \uff08\u53ef\u9009\uff09\u5728\u547d\u4ee4\u884c\u8f93\u5165
pip install black\uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002
- Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff1bMacOS \u81ea\u5e26 Clang \uff0c\u65e0\u987b\u5b89\u88c5\u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
c++\uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002 - \uff08\u53ef\u9009\uff09\u6253\u5f00 Settings \u9875\u9762\uff0c\u641c\u7d22
Clang_format_fallback Style\u4ee3\u7801\u683c\u5f0f\u5316\u9009\u9879\uff0c\u8bbe\u7f6e\u4e3a{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }\u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 > JDK 9\uff09\u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
java\uff0c\u5b89\u88c5 Extension Pack for Java \u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 8.0 \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
C# Dev Kit\uff0c\u5b89\u88c5 C# Dev Kit \uff08\u914d\u7f6e\u6559\u7a0b\uff09\u3002 - \u4e5f\u53ef\u4f7f\u7528 Visual Studio\uff08\u5b89\u88c5\u6559\u7a0b\uff09\u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 go \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
go\uff0c\u5b89\u88c5 Go \u3002 - \u6309\u5feb\u6377\u952e
Ctrl + Shift + P\u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9Go: Install/Update Tools\uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
swift\uff0c\u5b89\u88c5 Swift for Visual Studio Code \u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
javascript\uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002 - \uff08\u53ef\u9009\uff09\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
Prettier\uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 Dart \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
dart\uff0c\u5b89\u88c5 Dart \u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
rust\uff0c\u5b89\u88c5 rust-analyzer \u3002
\u8868 16-1 \u5217\u51fa\u4e86\u4e66\u4e2d\u51fa\u73b0\u7684\u91cd\u8981\u672f\u8bed\u3002\u5efa\u8bae\u8bfb\u8005\u540c\u65f6\u8bb0\u4f4f\u5b83\u4eec\u7684\u4e2d\u82f1\u6587\u53eb\u6cd5\uff0c\u4ee5\u4fbf\u9605\u8bfb\u82f1\u6587\u6587\u732e\u3002
\u8868 16-1 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u540d\u8bcd
\u4e2d\u6587 English \u4e2d\u6587 English \u7b97\u6cd5 algorithm \u5c42\u5e8f\u904d\u5386 level-order traversal \u6570\u636e\u7ed3\u6784 data structure \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal \u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 asymptotic complexity analysis \u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal \u65f6\u95f4\u590d\u6742\u5ea6 time complexity \u4e8c\u53c9\u641c\u7d22\u6811 binary search tree \u7a7a\u95f4\u590d\u6742\u5ea6 space complexity \u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 balanced binary search tree \u8fed\u4ee3 iteration \u5e73\u8861\u56e0\u5b50 balance factor \u9012\u5f52 recursion \u5806 heap \u5c3e\u9012\u5f52 tail recursion \u5927\u9876\u5806 max heap \u9012\u5f52\u6811 recursion tree \u5c0f\u9876\u5806 min heap \u5927 \\(O\\) \u8bb0\u53f7 big-\\(O\\) notation \u4f18\u5148\u961f\u5217 priority queue \u6e10\u8fd1\u4e0a\u754c asymptotic upper bound \u5806\u5316 heapify \u539f\u7801 sign-magnitude \u56fe graph \u53cd\u7801 1\u2019s complement \u9876\u70b9 vertex \u8865\u7801 2\u2019s complement \u65e0\u5411\u56fe undirected graph \u6570\u7ec4 array \u6709\u5411\u56fe directed graph \u7d22\u5f15 index \u8fde\u901a\u56fe connected graph \u94fe\u8868 linked list \u975e\u8fde\u901a\u56fe disconnected graph \u94fe\u8868\u8282\u70b9 linked list node, list node \u6709\u6743\u56fe weighted graph \u5217\u8868 list \u90bb\u63a5 adjacency \u52a8\u6001\u6570\u7ec4 dynamic array \u8def\u5f84 path \u786c\u76d8 hard disk \u5165\u5ea6 in-degree \u5185\u5b58 random-access memory (RAM) \u51fa\u5ea6 out-degree \u7f13\u5b58 cache memory \u90bb\u63a5\u77e9\u9635 adjacency matrix \u7f13\u5b58\u672a\u547d\u4e2d cache miss \u90bb\u63a5\u8868 adjacency list \u7f13\u5b58\u547d\u4e2d\u7387 cache hit rate \u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 breadth-first search \u6808 stack \u6df1\u5ea6\u4f18\u5148\u641c\u7d22 depth-first search \u961f\u5217 queue \u4e8c\u5206\u67e5\u627e binary search \u53cc\u5411\u961f\u5217 double-ended queue \u641c\u7d22\u7b97\u6cd5 searching algorithm \u54c8\u5e0c\u8868 hash table \u6392\u5e8f\u7b97\u6cd5 sorting algorithm \u6876 bucket \u9009\u62e9\u6392\u5e8f selection sort \u54c8\u5e0c\u51fd\u6570 hash function \u5192\u6ce1\u6392\u5e8f bubble sort \u54c8\u5e0c\u51b2\u7a81 hash collision \u63d2\u5165\u6392\u5e8f insertion sort \u8d1f\u8f7d\u56e0\u5b50 load factor \u5feb\u901f\u6392\u5e8f quick sort \u94fe\u5f0f\u5730\u5740 separate chaining \u5f52\u5e76\u6392\u5e8f merge sort \u5f00\u653e\u5bfb\u5740 open addressing \u5806\u6392\u5e8f heap sort \u7ebf\u6027\u63a2\u6d4b linear probing \u6876\u6392\u5e8f bucket sort \u61d2\u5220\u9664 lazy deletion \u8ba1\u6570\u6392\u5e8f counting sort \u4e8c\u53c9\u6811 binary tree \u57fa\u6570\u6392\u5e8f radix sort \u6811\u8282\u70b9 tree node \u5206\u6cbb divide and conquer \u5de6\u5b50\u8282\u70b9 left-child node \u6c49\u8bfa\u5854\u95ee\u9898 hanota problem \u53f3\u5b50\u8282\u70b9 right-child node \u56de\u6eaf\u7b97\u6cd5 backtracking algorithm \u7236\u8282\u70b9 parent node \u7ea6\u675f constraint \u5de6\u5b50\u6811 left subtree \u89e3 solution \u53f3\u5b50\u6811 right subtree \u72b6\u6001 state \u6839\u8282\u70b9 root node \u526a\u679d pruning \u53f6\u8282\u70b9 leaf node \u5168\u6392\u5217\u95ee\u9898 permutations problem \u8fb9 edge \u5b50\u96c6\u548c\u95ee\u9898 subset-sum problem \u5c42 level N \u7687\u540e\u95ee\u9898 N-queens problem \u5ea6 degree \u52a8\u6001\u89c4\u5212 dynamic programming \u9ad8\u5ea6 height \u521d\u59cb\u72b6\u6001 initial state \u6df1\u5ea6 depth \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b state-trasition equation \u5b8c\u7f8e\u4e8c\u53c9\u6811 perfect binary tree \u80cc\u5305\u95ee\u9898 knapsack problem \u5b8c\u5168\u4e8c\u53c9\u6811 complete binary tree \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 edit distance problem \u5b8c\u6ee1\u4e8c\u53c9\u6811 full binary tree \u8d2a\u5fc3\u7b97\u6cd5 greedy algorithm \u5e73\u8861\u4e8c\u53c9\u6811 balanced binary tree AVL \u6811 AVL tree \u7ea2\u9ed1\u6811 red-black tree"},{"location":"chapter_array_and_linkedlist/","title":"\u7b2c 4 \u7ae0 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868","text":"Abstract
\u6570\u636e\u7ed3\u6784\u7684\u4e16\u754c\u5982\u540c\u4e00\u5835\u539a\u5b9e\u7684\u7816\u5899\u3002
\u6570\u7ec4\u7684\u7816\u5757\u6574\u9f50\u6392\u5217\uff0c\u9010\u4e2a\u7d27\u8d34\u3002\u94fe\u8868\u7684\u7816\u5757\u5206\u6563\u5404\u5904\uff0c\u8fde\u63a5\u7684\u85e4\u8513\u81ea\u7531\u5730\u7a7f\u68ad\u4e8e\u7816\u7f1d\u4e4b\u95f4\u3002
"},{"location":"chapter_array_and_linkedlist/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 4.1 \u00a0 \u6570\u7ec4
- 4.2 \u00a0 \u94fe\u8868
- 4.3 \u00a0 \u5217\u8868
- 4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *
- 4.5 \u00a0 \u5c0f\u7ed3
\u300c\u6570\u7ec4 array\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5c06\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002\u6211\u4eec\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u8be5\u5143\u7d20\u7684\u300c\u7d22\u5f15 index\u300d\u3002\u56fe 4-1 \u5c55\u793a\u4e86\u6570\u7ec4\u7684\u4e3b\u8981\u672f\u8bed\u548c\u6982\u5ff5\u3002
\u56fe 4-1 \u00a0 \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f
"},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1 \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/array/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u6570\u7ec4","text":"\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9700\u6c42\u9009\u7528\u6570\u7ec4\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u5f0f\uff1a\u65e0\u521d\u59cb\u503c\u3001\u7ed9\u5b9a\u521d\u59cb\u503c\u3002\u5728\u672a\u6307\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4f1a\u5c06\u6570\u7ec4\u5143\u7d20\u521d\u59cb\u5316\u4e3a \\(0\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cpp# \u521d\u59cb\u5316\u6570\u7ec4\narr: list[int] = [0] * 5 # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4] \n
array.java/* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\uff08\u9700\u8981\u624b\u52a8\u91ca\u653e\u7a7a\u95f4\uff09\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n
array.cs/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
array.go/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = [1, 3, 2, 5, 4];\n
array.swift/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n
array.js/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n
array.ts/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n
array.dart/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n
array.rs/* \u521d\u59cb\u5316\u6570\u7ec4 */\nList<int> arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList<int> nums = [1, 3, 2, 5, 4];\n
array.c/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: Vec<i32> = vec![0; 5]; // [0, 0, 0, 0, 0]\nlet nums: Vec<i32> = vec![1, 3, 2, 5, 4];\n
array.zig/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n
"},{"location":"chapter_array_and_linkedlist/array/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"// \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n\u6570\u7ec4\u5143\u7d20\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u8ba1\u7b97\u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u5185\u5b58\u5730\u5740\uff08\u9996\u5143\u7d20\u5185\u5b58\u5730\u5740\uff09\u548c\u67d0\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u56fe 4-2 \u6240\u793a\u7684\u516c\u5f0f\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u8be5\u5143\u7d20\u3002
\u56fe 4-2 \u00a0 \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97
\u89c2\u5bdf\u56fe 4-2 \uff0c\u6211\u4eec\u53d1\u73b0\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \uff0c\u8fd9\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\u4f1a\u66f4\u81ea\u7136\u3002\u4f46\u4ece\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u662f\u5185\u5b58\u5730\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u5b83\u7684\u7d22\u5f15\u4e3a \\(0\\) \u662f\u5408\u7406\u7684\u3002
\u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u5143\u7d20\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef random_access(nums: list[int]) -> int:\n \"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n # \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n random_index = random.randint(0, len(nums) - 1)\n # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n random_num = nums[random_index]\n return random_num\n
array.java/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = rand() % size;\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
array.cs/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int[] nums) {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
array.go/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint RandomAccess(int[] nums) {\n Random random = new();\n // \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = random.Next(nums.Length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
array.swift/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n randomIndex := rand.Intn(len(nums))\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n randomNum = nums[randomIndex]\n return\n}\n
array.js/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -> Int {\n // \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n let randomIndex = nums.indices.randomElement()!\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n let randomNum = nums[randomIndex]\n return randomNum\n}\n
array.ts/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums) {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n const random_index = Math.floor(Math.random() * nums.length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n const random_num = nums[random_index];\n return random_num;\n}\n
array.dart/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n const random_index = Math.floor(Math.random() * nums.length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n const random_num = nums[random_index];\n return random_num;\n}\n
array.rs/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(List<int> nums) {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = Random().nextInt(nums.length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
array.c/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfn random_access(nums: &[i32]) -> i32 {\n // \u5728\u533a\u95f4 [0, nums.len()) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n let random_index = rand::thread_rng().gen_range(0..nums.len());\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n let random_num = nums[random_index];\n random_num\n}\n
array.zig/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = rand() % size;\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#3","title":"3. \u00a0 \u63d2\u5165\u5143\u7d20","text":"// \u968f\u673a\u8bbf\u95ee\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n // \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\n var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n var randomNum = nums[randomIndex];\n return randomNum;\n}\n\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u5b58\u653e\u4efb\u4f55\u6570\u636e\u3002\u5982\u56fe 4-3 \u6240\u793a\uff0c\u5982\u679c\u60f3\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u5219\u9700\u8981\u5c06\u8be5\u5143\u7d20\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u4e4b\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002
\u56fe 4-3 \u00a0 \u6570\u7ec4\u63d2\u5165\u5143\u7d20\u793a\u4f8b
\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u56fa\u5b9a\u7684\uff0c\u56e0\u6b64\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u4f1a\u5bfc\u81f4\u6570\u7ec4\u5c3e\u90e8\u5143\u7d20\u201c\u4e22\u5931\u201d\u3002\u6211\u4eec\u5c06\u8fd9\u4e2a\u95ee\u9898\u7684\u89e3\u51b3\u65b9\u6848\u7559\u5728\u201c\u5217\u8868\u201d\u7ae0\u8282\u4e2d\u8ba8\u8bba\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef insert(nums: list[int], num: int, index: int):\n \"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for i in range(len(nums) - 1, index, -1):\n nums[i] = nums[i - 1]\n # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num\n
array.java/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int i = size - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.cs/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int i = nums.length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.go/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int i = nums.Length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.swift/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for i := len(nums) - 1; i > index; i-- {\n nums[i] = nums[i-1]\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num\n}\n
array.js/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for i in nums.indices.dropFirst(index).reversed() {\n nums[i] = nums[i - 1]\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num\n}\n
array.ts/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (let i = nums.length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.dart/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (let i = nums.length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.rs/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List<int> nums, int _num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (var i = nums.length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 _num \u8d4b\u7ed9 index \u5904\u5143\u7d20\n nums[index] = _num;\n}\n
array.c/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &mut Vec<i32>, num: i32, index: usize) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for i in (index + 1..nums.len()).rev() {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.zig/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int i = size - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#4","title":"4. \u00a0 \u5220\u9664\u5143\u7d20","text":"// \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n var i = nums.len - 1;\n while (i > index) : (i -= 1) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n\u540c\u7406\uff0c\u5982\u56fe 4-4 \u6240\u793a\uff0c\u82e5\u60f3\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002
\u56fe 4-4 \u00a0 \u6570\u7ec4\u5220\u9664\u5143\u7d20\u793a\u4f8b
\u8bf7\u6ce8\u610f\uff0c\u5220\u9664\u5143\u7d20\u5b8c\u6210\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u987b\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef remove(nums: list[int], index: int):\n \"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\"\"\"\n # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for i in range(index, len(nums) - 1):\n nums[i] = nums[i + 1]\n
array.java/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int i = index; i < size - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.cs/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int i = index; i < nums.length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.go/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int i = index; i < nums.Length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.swift/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for i := index; i < len(nums)-1; i++ {\n nums[i] = nums[i+1]\n }\n}\n
array.js/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for i in nums.indices.dropFirst(index).dropLast() {\n nums[i] = nums[i + 1]\n }\n}\n
array.ts/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (let i = index; i < nums.length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.dart/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (let i = index; i < nums.length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.rs/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(List<int> nums, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (var i = index; i < nums.length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.c/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfn remove(nums: &mut Vec<i32>, index: usize) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for i in index..nums.len() - 1 {\n nums[i] = nums[i + 1];\n }\n}\n
array.zig/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int i = index; i < size - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n// \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n var i = index;\n while (i < nums.len - 1) : (i += 1) {\n nums[i] = nums[i + 1];\n }\n}\n\u603b\u7684\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\u3002
- \u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002
- \u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u4e22\u5931\u3002
- \u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u201c\u65e0\u610f\u4e49\u201d\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e2\u53ef\u4ee5\u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u83b7\u53d6\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef traverse(nums: list[int]):\n \"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\n count = 0\n # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for i in range(len(nums)):\n count += nums[i]\n # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for num in nums:\n count += num\n # \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n for i, num in enumerate(nums):\n count += nums[i]\n count += num\n
array.java/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size; i++) {\n count += nums[i];\n }\n}\n
array.cs/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < nums.length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (int num : nums) {\n count += num;\n }\n}\n
array.go/* \u904d\u5386\u6570\u7ec4 */\nvoid Traverse(int[] nums) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < nums.Length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n foreach (int num in nums) {\n count += num;\n }\n}\n
array.swift/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\n count := 0\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for i := 0; i < len(nums); i++ {\n count += nums[i]\n }\n count = 0\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for _, num := range nums {\n count += num\n }\n // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n for i, num := range nums {\n count += nums[i]\n count += num\n }\n}\n
array.js/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\n var count = 0\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for i in nums.indices {\n count += nums[i]\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for num in nums {\n count += num\n }\n}\n
array.ts/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\n let count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (let i = 0; i < nums.length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (const num of nums) {\n count += num;\n }\n}\n
array.dart/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\n let count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (let i = 0; i < nums.length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (const num of nums) {\n count += num;\n }\n}\n
array.rs/* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List<int> nums) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (var i = 0; i < nums.length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (int _num in nums) {\n count += _num;\n }\n // \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\n nums.forEach((_num) {\n count += _num;\n });\n}\n
array.c/* \u904d\u5386\u6570\u7ec4 */\nfn traverse(nums: &[i32]) {\n let mut _count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for i in 0..nums.len() {\n _count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for num in nums {\n _count += num;\n }\n}\n
array.zig/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size; i++) {\n count += nums[i];\n }\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#6","title":"6. \u00a0 \u67e5\u627e\u5143\u7d20","text":"// \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\n var count: i32 = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n var i: i32 = 0;\n while (i < nums.len) : (i += 1) {\n count += nums[i];\n }\n count = 0;\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (nums) |num| {\n count += num;\n }\n}\n\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\u9700\u8981\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u5224\u65ad\u5143\u7d20\u503c\u662f\u5426\u5339\u914d\uff0c\u82e5\u5339\u914d\u5219\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002
\u56e0\u4e3a\u6570\u7ec4\u662f\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u6240\u4ee5\u4e0a\u8ff0\u67e5\u627e\u64cd\u4f5c\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u67e5\u627e\u201d\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef find(nums: list[int], target: int) -> int:\n \"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\n for i in range(len(nums)):\n if nums[i] == target:\n return i\n return -1\n
array.java/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n for (int i = 0; i < size; i++) {\n if (nums[i] == target)\n return i;\n }\n return -1;\n}\n
array.cs/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\n for (int i = 0; i < nums.length; i++) {\n if (nums[i] == target)\n return i;\n }\n return -1;\n}\n
array.go/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\n for (int i = 0; i < nums.Length; i++) {\n if (nums[i] == target)\n return i;\n }\n return -1;\n}\n
array.swift/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\n index = -1\n for i := 0; i < len(nums); i++ {\n if nums[i] == target {\n index = i\n break\n }\n }\n return\n}\n
array.js/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -> Int {\n for i in nums.indices {\n if nums[i] == target {\n return i\n }\n }\n return -1\n}\n
array.ts/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\n for (let i = 0; i < nums.length; i++) {\n if (nums[i] === target) return i;\n }\n return -1;\n}\n
array.dart/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\n for (let i = 0; i < nums.length; i++) {\n if (nums[i] === target) {\n return i;\n }\n }\n return -1;\n}\n
array.rs/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List<int> nums, int target) {\n for (var i = 0; i < nums.length; i++) {\n if (nums[i] == target) return i;\n }\n return -1;\n}\n
array.c/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &[i32], target: i32) -> Option<usize> {\n for i in 0..nums.len() {\n if nums[i] == target {\n return Some(i);\n }\n }\n None\n}\n
array.zig/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n for (int i = 0; i < size; i++) {\n if (nums[i] == target)\n return i;\n }\n return -1;\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#7","title":"7. \u00a0 \u6269\u5bb9\u6570\u7ec4","text":"// \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\n for (nums, 0..) |num, i| {\n if (num == target) return @intCast(i);\n }\n return -1;\n}\n\u5728\u590d\u6742\u7684\u7cfb\u7edf\u73af\u5883\u4e2d\uff0c\u7a0b\u5e8f\u96be\u4ee5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u4ece\u800c\u65e0\u6cd5\u5b89\u5168\u5730\u6269\u5c55\u6570\u7ec4\u5bb9\u91cf\u3002\u56e0\u6b64\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u4e0d\u53ef\u53d8\u7684\u3002
\u5982\u679c\u6211\u4eec\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u91cd\u65b0\u5efa\u7acb\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u590d\u5236\u5230\u65b0\u6570\u7ec4\u3002\u8fd9\u662f\u4e00\u4e2a \\(O(n)\\) \u7684\u64cd\u4f5c\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\u975e\u5e38\u8017\u65f6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef extend(nums: list[int], enlarge: int) -> list[int]:\n \"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n res = [0] * (len(nums) + enlarge)\n # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for i in range(len(nums)):\n res[i] = nums[i]\n # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res\n
array.java/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n int *res = new int[size + enlarge];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < size; i++) {\n res[i] = nums[i];\n }\n // \u91ca\u653e\u5185\u5b58\n delete[] nums;\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.cs/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n int[] res = new int[nums.length + enlarge];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < nums.length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.go/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n int[] res = new int[nums.Length + enlarge];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < nums.Length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.swift/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n res := make([]int, len(nums)+enlarge)\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for i, num := range nums {\n res[i] = num\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res\n}\n
array.js/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -> [Int] {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n var res = Array(repeating: 0, count: nums.count + enlarge)\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for i in nums.indices {\n res[i] = nums[i]\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res\n}\n
array.ts/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n const res = new Array(nums.length + enlarge).fill(0);\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (let i = 0; i < nums.length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.dart/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n const res = new Array(nums.length + enlarge).fill(0);\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (let i = 0; i < nums.length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.rs/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList<int> extend(List<int> nums, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n List<int> res = List.filled(nums.length + enlarge, 0);\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (var i = 0; i < nums.length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.c/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfn extend(nums: Vec<i32>, enlarge: usize) -> Vec<i32> {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n let mut res: Vec<i32> = vec![0; nums.len() + enlarge];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\n for i in 0..nums.len() {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n res\n}\n
array.zig/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < size; i++) {\n res[i] = nums[i];\n }\n // \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\n for (int i = size; i < size + enlarge; i++) {\n res[i] = 0;\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2 \u00a0 \u6570\u7ec4\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"// \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n var res = try mem_allocator.alloc(i32, nums.len + enlarge);\n @memset(res, 0);\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n std.mem.copy(i32, res, nums);\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u4e14\u5143\u7d20\u7c7b\u578b\u76f8\u540c\u3002\u8fd9\u79cd\u505a\u6cd5\u5305\u542b\u4e30\u5bcc\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u7cfb\u7edf\u53ef\u4ee5\u5229\u7528\u8fd9\u4e9b\u4fe1\u606f\u6765\u4f18\u5316\u6570\u636e\u7ed3\u6784\u7684\u64cd\u4f5c\u6548\u7387\u3002
- \u7a7a\u95f4\u6548\u7387\u9ad8\uff1a\u6570\u7ec4\u4e3a\u6570\u636e\u5206\u914d\u4e86\u8fde\u7eed\u7684\u5185\u5b58\u5757\uff0c\u65e0\u987b\u989d\u5916\u7684\u7ed3\u6784\u5f00\u9500\u3002
- \u652f\u6301\u968f\u673a\u8bbf\u95ee\uff1a\u6570\u7ec4\u5141\u8bb8\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u4efb\u4f55\u5143\u7d20\u3002
- \u7f13\u5b58\u5c40\u90e8\u6027\uff1a\u5f53\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u4ed6\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002
\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u662f\u4e00\u628a\u53cc\u5203\u5251\uff0c\u5176\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002
- \u63d2\u5165\u4e0e\u5220\u9664\u6548\u7387\u4f4e\uff1a\u5f53\u6570\u7ec4\u4e2d\u5143\u7d20\u8f83\u591a\u65f6\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u9700\u8981\u79fb\u52a8\u5927\u91cf\u7684\u5143\u7d20\u3002
- \u957f\u5ea6\u4e0d\u53ef\u53d8\uff1a\u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u5c31\u56fa\u5b9a\u4e86\uff0c\u6269\u5bb9\u6570\u7ec4\u9700\u8981\u5c06\u6240\u6709\u6570\u636e\u590d\u5236\u5230\u65b0\u6570\u7ec4\uff0c\u5f00\u9500\u5f88\u5927\u3002
- \u7a7a\u95f4\u6d6a\u8d39\uff1a\u5982\u679c\u6570\u7ec4\u5206\u914d\u7684\u5927\u5c0f\u8d85\u8fc7\u5b9e\u9645\u6240\u9700\uff0c\u90a3\u4e48\u591a\u4f59\u7684\u7a7a\u95f4\u5c31\u88ab\u6d6a\u8d39\u4e86\u3002
\u6570\u7ec4\u662f\u4e00\u79cd\u57fa\u7840\u4e14\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\uff0c\u65e2\u9891\u7e41\u5e94\u7528\u5728\u5404\u7c7b\u7b97\u6cd5\u4e4b\u4e2d\uff0c\u4e5f\u53ef\u7528\u4e8e\u5b9e\u73b0\u5404\u79cd\u590d\u6742\u6570\u636e\u7ed3\u6784\u3002
- \u968f\u673a\u8bbf\u95ee\uff1a\u5982\u679c\u6211\u4eec\u60f3\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u968f\u673a\u62bd\u6837\u3002
- \u6392\u5e8f\u548c\u641c\u7d22\uff1a\u6570\u7ec4\u662f\u6392\u5e8f\u548c\u641c\u7d22\u7b97\u6cd5\u6700\u5e38\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u4e8c\u5206\u67e5\u627e\u7b49\u90fd\u4e3b\u8981\u5728\u6570\u7ec4\u4e0a\u8fdb\u884c\u3002
- \u67e5\u627e\u8868\uff1a\u5f53\u9700\u8981\u5feb\u901f\u67e5\u627e\u4e00\u4e2a\u5143\u7d20\u6216\u5176\u5bf9\u5e94\u5173\u7cfb\u65f6\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u4f5c\u4e3a\u67e5\u627e\u8868\u3002\u5047\u5982\u6211\u4eec\u60f3\u5b9e\u73b0\u5b57\u7b26\u5230 ASCII \u7801\u7684\u6620\u5c04\uff0c\u5219\u53ef\u4ee5\u5c06\u5b57\u7b26\u7684 ASCII \u7801\u503c\u4f5c\u4e3a\u7d22\u5f15\uff0c\u5bf9\u5e94\u7684\u5143\u7d20\u5b58\u653e\u5728\u6570\u7ec4\u4e2d\u7684\u5bf9\u5e94\u4f4d\u7f6e\u3002
- \u673a\u5668\u5b66\u4e60\uff1a\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002
- \u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff1a\u6570\u7ec4\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002\u4f8b\u5982\uff0c\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\u3002
\u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u5728\u4e00\u4e2a\u590d\u6742\u7684\u7cfb\u7edf\u8fd0\u884c\u73af\u5883\u4e0b\uff0c\u7a7a\u95f2\u7684\u5185\u5b58\u7a7a\u95f4\u53ef\u80fd\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5b58\u50a8\u6570\u7ec4\u7684\u5185\u5b58\u7a7a\u95f4\u5fc5\u987b\u662f\u8fde\u7eed\u7684\uff0c\u800c\u5f53\u6570\u7ec4\u975e\u5e38\u5927\u65f6\uff0c\u5185\u5b58\u53ef\u80fd\u65e0\u6cd5\u63d0\u4f9b\u5982\u6b64\u5927\u7684\u8fde\u7eed\u7a7a\u95f4\u3002\u6b64\u65f6\u94fe\u8868\u7684\u7075\u6d3b\u6027\u4f18\u52bf\u5c31\u4f53\u73b0\u51fa\u6765\u4e86\u3002
\u300c\u94fe\u8868 linked list\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u4e00\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u5404\u4e2a\u8282\u70b9\u901a\u8fc7\u201c\u5f15\u7528\u201d\u76f8\u8fde\u63a5\u3002\u5f15\u7528\u8bb0\u5f55\u4e86\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684\u5185\u5b58\u5730\u5740\uff0c\u901a\u8fc7\u5b83\u53ef\u4ee5\u4ece\u5f53\u524d\u8282\u70b9\u8bbf\u95ee\u5230\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002
\u94fe\u8868\u7684\u8bbe\u8ba1\u4f7f\u5f97\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\uff0c\u5b83\u4eec\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002
\u56fe 4-5 \u00a0 \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f
\u89c2\u5bdf\u56fe 4-5 \uff0c\u94fe\u8868\u7684\u7ec4\u6210\u5355\u4f4d\u662f\u300c\u8282\u70b9 node\u300d\u5bf9\u8c61\u3002\u6bcf\u4e2a\u8282\u70b9\u90fd\u5305\u542b\u4e24\u9879\u6570\u636e\uff1a\u8282\u70b9\u7684\u201c\u503c\u201d\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u201c\u5f15\u7528\u201d\u3002
- \u94fe\u8868\u7684\u9996\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5934\u8282\u70b9\u201d\uff0c\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5c3e\u8282\u70b9\u201d\u3002
- \u5c3e\u8282\u70b9\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5b83\u5728 Java\u3001C++ \u548c Python \u4e2d\u5206\u522b\u88ab\u8bb0\u4e3a \\(\\text{null}\\)\u3001\\(\\text{nullptr}\\) \u548c \\(\\text{None}\\) \u3002
- \u5728 C\u3001C++\u3001Go \u548c Rust \u7b49\u652f\u6301\u6307\u9488\u7684\u8bed\u8a00\u4e2d\uff0c\u4e0a\u8ff0\u7684\u201c\u5f15\u7528\u201d\u5e94\u88ab\u66ff\u6362\u4e3a\u201c\u6307\u9488\u201d\u3002
\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u94fe\u8868\u8282\u70b9
PythonC++JavaC#GoSwiftJSTSDartRustCZigListNode\u9664\u4e86\u5305\u542b\u503c\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u4e00\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u56e0\u6b64\u5728\u76f8\u540c\u6570\u636e\u91cf\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002class ListNode:\n \"\"\"\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n def __init__(self, val: int):\n self.val: int = val # \u8282\u70b9\u503c\n self.next: ListNode | None = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n ListNode(int x) : val(x), next(nullptr) {} // \u6784\u9020\u51fd\u6570\n};\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n ListNode(int x) { val = x; } // \u6784\u9020\u51fd\u6570\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) { //\u6784\u9020\u51fd\u6570\n int val = x; // \u8282\u70b9\u503c\n ListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\n Val int // \u8282\u70b9\u503c\n Next *ListNode // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\n return &ListNode{\n Val: val,\n Next: nil,\n }\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n var val: Int // \u8282\u70b9\u503c\n var next: ListNode? // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\n init(x: Int) { // \u6784\u9020\u51fd\u6570\n val = x\n }\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n constructor(val, next) {\n this.val = (val === undefined ? 0 : val); // \u8282\u70b9\u503c\n this.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n val: number;\n next: ListNode | null;\n constructor(val?: number, next?: ListNode | null) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = next === undefined ? null : next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n ListNode(this.val, [this.next]); // \u6784\u9020\u51fd\u6570\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\n#[derive(Debug)]\nstruct ListNode {\n val: i32, // \u8282\u70b9\u503c\n next: Option<Rc<RefCell<ListNode>>>, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n int val; // \u8282\u70b9\u503c\n struct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n ListNode *node;\n node = (ListNode *) malloc(sizeof(ListNode));\n node->val = val;\n node->next = NULL;\n return node;\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1 \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u94fe\u8868","text":"// \u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n return struct {\n const Self = @This();\n\n val: T = 0, // \u8282\u70b9\u503c\n next: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n\n // \u6784\u9020\u51fd\u6570\n pub fn init(self: *Self, x: i32) void {\n self.val = x;\n self.next = null;\n }\n };\n}\n\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u5173\u7cfb\u3002\u521d\u59cb\u5316\u5b8c\u6210\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u901a\u8fc7\u5f15\u7528\u6307\u5411
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.pynext\u4f9d\u6b21\u8bbf\u95ee\u6240\u6709\u8282\u70b9\u3002
linked_list.cpp# \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
linked_list.java/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
linked_list.cs/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.go/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.swift/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n
linked_list.js/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
linked_list.ts/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.dart/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.rs/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\\\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.c/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n
linked_list.zig/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n// \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = &n1;\nn1.next = &n2;\nn2.next = &n3;\nn3.next = &n4;\n\u6570\u7ec4\u6574\u4f53\u662f\u4e00\u4e2a\u53d8\u91cf\uff0c\u6bd4\u5982\u6570\u7ec4
"},{"location":"chapter_array_and_linkedlist/linked_list/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"nums\u5305\u542b\u5143\u7d20nums[0]\u548cnums[1]\u7b49\uff0c\u800c\u94fe\u8868\u662f\u7531\u591a\u4e2a\u72ec\u7acb\u7684\u8282\u70b9\u5bf9\u8c61\u7ec4\u6210\u7684\u3002\u6211\u4eec\u901a\u5e38\u5c06\u5934\u8282\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u6bd4\u5982\u4ee5\u4e0a\u4ee3\u7801\u4e2d\u7684\u94fe\u8868\u53ef\u8bb0\u4f5c\u94fe\u8868n0\u3002\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u8282\u70b9\u975e\u5e38\u5bb9\u6613\u3002\u5982\u56fe 4-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u5728\u76f8\u90bb\u7684\u4e24\u4e2a\u8282\u70b9
n0\u548cn1\u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u8282\u70b9P\uff0c\u5219\u53ea\u9700\u6539\u53d8\u4e24\u4e2a\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5728\u6570\u7ec4\u4e2d\u63d2\u5165\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u6548\u7387\u8f83\u4f4e\u3002
\u56fe 4-6 \u00a0 \u94fe\u8868\u63d2\u5165\u8282\u70b9\u793a\u4f8b
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py
linked_list.cppdef insert(n0: ListNode, P: ListNode):\n \"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\n n1 = n0.next\n P.next = n1\n n0.next = P\n
linked_list.java/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n ListNode *n1 = n0->next;\n P->next = n1;\n n0->next = P;\n}\n
linked_list.cs/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n ListNode n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.go/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid Insert(ListNode n0, ListNode P) {\n ListNode? n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.swift/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n n1 := n0.Next\n P.Next = n1\n n0.Next = P\n}\n
linked_list.js/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\n let n1 = n0.next\n P.next = n1\n n0.next = P\n}\n
linked_list.ts/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\n const n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.dart/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\n const n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.rs/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n ListNode? n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.c/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\n#[allow(non_snake_case)]\npub fn insert<T>(n0: &Rc<RefCell<ListNode<T>>>, P: Rc<RefCell<ListNode<T>>>) {\n let n1 = n0.borrow_mut().next.take();\n P.borrow_mut().next = n1;\n n0.borrow_mut().next = Some(P);\n}\n
linked_list.zig/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n ListNode *n1 = n0->next;\n P->next = n1;\n n0->next = P;\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"// \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\n var n1 = n0.?.next;\n P.?.next = n1;\n n0.?.next = P;\n}\n\u5982\u56fe 4-7 \u6240\u793a\uff0c\u5728\u94fe\u8868\u4e2d\u5220\u9664\u8282\u70b9\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u53d8\u4e00\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u3002
\u8bf7\u6ce8\u610f\uff0c\u5c3d\u7ba1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\u8282\u70b9
P\u4ecd\u7136\u6307\u5411n1\uff0c\u4f46\u5b9e\u9645\u4e0a\u904d\u5386\u6b64\u94fe\u8868\u5df2\u7ecf\u65e0\u6cd5\u8bbf\u95ee\u5230P\uff0c\u8fd9\u610f\u5473\u7740P\u5df2\u7ecf\u4e0d\u518d\u5c5e\u4e8e\u8be5\u94fe\u8868\u4e86\u3002\u56fe 4-7 \u00a0 \u94fe\u8868\u5220\u9664\u8282\u70b9
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py
linked_list.cppdef remove(n0: ListNode):\n \"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n if not n0.next:\n return\n # n0 -> P -> n1\n P = n0.next\n n1 = P.next\n n0.next = n1\n
linked_list.java/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\n if (n0->next == nullptr)\n return;\n // n0 -> P -> n1\n ListNode *P = n0->next;\n ListNode *n1 = P->next;\n n0->next = n1;\n // \u91ca\u653e\u5185\u5b58\n delete P;\n}\n
linked_list.cs/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n if (n0.next == null)\n return;\n // n0 -> P -> n1\n ListNode P = n0.next;\n ListNode n1 = P.next;\n n0.next = n1;\n}\n
linked_list.go/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid Remove(ListNode n0) {\n if (n0.next == null)\n return;\n // n0 -> P -> n1\n ListNode P = n0.next;\n ListNode? n1 = P.next;\n n0.next = n1;\n}\n
linked_list.swift/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeItem(n0 *ListNode) {\n if n0.Next == nil {\n return\n }\n // n0 -> P -> n1\n P := n0.Next\n n1 := P.Next\n n0.Next = n1\n}\n
linked_list.js/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\n if n0.next == nil {\n return\n }\n // n0 -> P -> n1\n let P = n0.next\n let n1 = P?.next\n n0.next = n1\n P?.next = nil\n}\n
linked_list.ts/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\n if (!n0.next) return;\n // n0 -> P -> n1\n const P = n0.next;\n const n1 = P.next;\n n0.next = n1;\n}\n
linked_list.dart/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\n if (!n0.next) {\n return;\n }\n // n0 -> P -> n1\n const P = n0.next;\n const n1 = P.next;\n n0.next = n1;\n}\n
linked_list.rs/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n if (n0.next == null) return;\n // n0 -> P -> n1\n ListNode P = n0.next!;\n ListNode? n1 = P.next;\n n0.next = n1;\n}\n
linked_list.c/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n#[allow(non_snake_case)]\npub fn remove<T>(n0: &Rc<RefCell<ListNode<T>>>) {\n if n0.borrow().next.is_none() {return};\n // n0 -> P -> n1\n let P = n0.borrow_mut().next.take();\n if let Some(node) = P {\n let n1 = node.borrow_mut().next.take();\n n0.borrow_mut().next = n1;\n }\n}\n
linked_list.zig/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(ListNode *n0) {\n if (!n0->next)\n return;\n // n0 -> P -> n1\n ListNode *P = n0->next;\n ListNode *n1 = P->next;\n n0->next = n1;\n // \u91ca\u653e\u5185\u5b58\n free(P);\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#4","title":"4. \u00a0 \u8bbf\u95ee\u8282\u70b9","text":"// \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\n if (n0.?.next == null) return;\n // n0 -> P -> n1\n var P = n0.?.next;\n var n1 = P.?.next;\n n0.?.next = n1;\n}\n\u5728\u94fe\u8868\u4e2d\u8bbf\u95ee\u8282\u70b9\u7684\u6548\u7387\u8f83\u4f4e\u3002\u5982\u4e0a\u4e00\u8282\u6240\u8ff0\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u5143\u7d20\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u7a0b\u5e8f\u9700\u8981\u4ece\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u9010\u4e2a\u5411\u540e\u904d\u5386\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u8282\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8bbf\u95ee\u94fe\u8868\u7684\u7b2c \\(i\\) \u4e2a\u8282\u70b9\u9700\u8981\u5faa\u73af \\(i - 1\\) \u8f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py
linked_list.cppdef access(head: ListNode, index: int) -> ListNode | None:\n \"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\n for _ in range(index):\n if not head:\n return None\n head = head.next\n return head\n
linked_list.java/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n for (int i = 0; i < index; i++) {\n if (head == nullptr)\n return nullptr;\n head = head->next;\n }\n return head;\n}\n
linked_list.cs/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\n for (int i = 0; i < index; i++) {\n if (head == null)\n return null;\n head = head.next;\n }\n return head;\n}\n
linked_list.go/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? Access(ListNode? head, int index) {\n for (int i = 0; i < index; i++) {\n if (head == null)\n return null;\n head = head.next;\n }\n return head;\n}\n
linked_list.swift/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\n for i := 0; i < index; i++ {\n if head == nil {\n return nil\n }\n head = head.Next\n }\n return head\n}\n
linked_list.js/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -> ListNode? {\n var head: ListNode? = head\n for _ in 0 ..< index {\n if head == nil {\n return nil\n }\n head = head?.next\n }\n return head\n}\n
linked_list.ts/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\n for (let i = 0; i < index; i++) {\n if (!head) {\n return null;\n }\n head = head.next;\n }\n return head;\n}\n
linked_list.dart/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n for (let i = 0; i < index; i++) {\n if (!head) {\n return null;\n }\n head = head.next;\n }\n return head;\n}\n
linked_list.rs/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\n for (var i = 0; i < index; i++) {\n if (head == null) return null;\n head = head.next;\n }\n return head;\n}\n
linked_list.c/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\npub fn access<T>(head: Rc<RefCell<ListNode<T>>>, index: i32) -> Rc<RefCell<ListNode<T>>> {\n if index <= 0 {return head};\n if let Some(node) = &head.borrow_mut().next {\n return access(node.clone(), index - 1);\n }\n return head;\n}\n
linked_list.zig/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n for (int i = 0; i < index; i++) {\n if (head == NULL)\n return NULL;\n head = head->next;\n }\n return head;\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#5","title":"5. \u00a0 \u67e5\u627e\u8282\u70b9","text":"// \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\n var head = node;\n var i: i32 = 0;\n while (i < index) : (i += 1) {\n head = head.?.next;\n if (head == null) return null;\n }\n return head;\n}\n\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u5176\u4e2d\u503c\u4e3a
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.pytarget\u7684\u8282\u70b9\uff0c\u8f93\u51fa\u8be5\u8282\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002\u6b64\u8fc7\u7a0b\u4e5f\u5c5e\u4e8e\u7ebf\u6027\u67e5\u627e\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
linked_list.cppdef find(head: ListNode, target: int) -> int:\n \"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n index = 0\n while head:\n if head.val == target:\n return index\n head = head.next\n index += 1\n return -1\n
linked_list.java/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n int index = 0;\n while (head != nullptr) {\n if (head->val == target)\n return index;\n head = head->next;\n index++;\n }\n return -1;\n}\n
linked_list.cs/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\n int index = 0;\n while (head != null) {\n if (head.val == target)\n return index;\n head = head.next;\n index++;\n }\n return -1;\n}\n
linked_list.go/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint Find(ListNode? head, int target) {\n int index = 0;\n while (head != null) {\n if (head.val == target)\n return index;\n head = head.next;\n index++;\n }\n return -1;\n}\n
linked_list.swift/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\n index := 0\n for head != nil {\n if head.Val == target {\n return index\n }\n head = head.Next\n index++\n }\n return -1\n}\n
linked_list.js/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -> Int {\n var head: ListNode? = head\n var index = 0\n while head != nil {\n if head?.val == target {\n return index\n }\n head = head?.next\n index += 1\n }\n return -1\n}\n
linked_list.ts/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\n let index = 0;\n while (head !== null) {\n if (head.val === target) {\n return index;\n }\n head = head.next;\n index += 1;\n }\n return -1;\n}\n
linked_list.dart/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\n let index = 0;\n while (head !== null) {\n if (head.val === target) {\n return index;\n }\n head = head.next;\n index += 1;\n }\n return -1;\n}\n
linked_list.rs/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\n int index = 0;\n while (head != null) {\n if (head.val == target) {\n return index;\n }\n head = head.next;\n index++;\n }\n return -1;\n}\n
linked_list.c/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\npub fn find<T: PartialEq>(head: Rc<RefCell<ListNode<T>>>, target: T, index: i32) -> i32 {\n if head.borrow().val == target {return index};\n if let Some(node) = &head.borrow_mut().next {\n return find(node.clone(), target, index + 1);\n }\n return -1;\n}\n
linked_list.zig/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n int index = 0;\n while (head) {\n if (head->val == target)\n return index;\n head = head->next;\n index++;\n }\n return -1;\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","title":"4.2.2 \u00a0 \u6570\u7ec4 vs. \u94fe\u8868","text":"// \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\n var head = node;\n var index: i32 = 0;\n while (head != null) {\n if (head.?.val == target) return index;\n head = head.?.next;\n index += 1;\n }\n return -1;\n}\n\u8868 4-1 \u603b\u7ed3\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u7684\u5404\u9879\u7279\u70b9\u5e76\u5bf9\u6bd4\u4e86\u64cd\u4f5c\u6548\u7387\u3002\u7531\u4e8e\u5b83\u4eec\u91c7\u7528\u4e24\u79cd\u76f8\u53cd\u7684\u5b58\u50a8\u7b56\u7565\uff0c\u56e0\u6b64\u5404\u79cd\u6027\u8d28\u548c\u64cd\u4f5c\u6548\u7387\u4e5f\u5448\u73b0\u5bf9\u7acb\u7684\u7279\u70b9\u3002
\u8868 4-1 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u6548\u7387\u5bf9\u6bd4
\u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u5206\u6563\u5185\u5b58\u7a7a\u95f4 \u5bb9\u91cf\u6269\u5c55 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u53ef\u7075\u6d3b\u6269\u5c55 \u5185\u5b58\u6548\u7387 \u5143\u7d20\u5360\u7528\u5185\u5b58\u5c11\u3001\u4f46\u53ef\u80fd\u6d6a\u8d39\u7a7a\u95f4 \u5143\u7d20\u5360\u7528\u5185\u5b58\u591a \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3 \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"\u5982\u56fe 4-8 \u6240\u793a\uff0c\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u4e09\u79cd\u3002
- \u5355\u5411\u94fe\u8868\uff1a\u5373\u524d\u9762\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u8282\u70b9\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u8282\u70b9\u79f0\u4e3a\u5934\u8282\u70b9\uff0c\u5c06\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u79f0\u4e3a\u5c3e\u8282\u70b9\uff0c\u5c3e\u8282\u70b9\u6307\u5411\u7a7a \\(\\text{None}\\) \u3002
- \u73af\u5f62\u94fe\u8868\uff1a\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u8282\u70b9\u6307\u5411\u5934\u8282\u70b9\uff08\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u4efb\u610f\u8282\u70b9\u90fd\u53ef\u4ee5\u89c6\u4f5c\u5934\u8282\u70b9\u3002
- \u53cc\u5411\u94fe\u8868\uff1a\u4e0e\u5355\u5411\u94fe\u8868\u76f8\u6bd4\uff0c\u53cc\u5411\u94fe\u8868\u8bb0\u5f55\u4e86\u4e24\u4e2a\u65b9\u5411\u7684\u5f15\u7528\u3002\u53cc\u5411\u94fe\u8868\u7684\u8282\u70b9\u5b9a\u4e49\u540c\u65f6\u5305\u542b\u6307\u5411\u540e\u7ee7\u8282\u70b9\uff08\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\u548c\u524d\u9a71\u8282\u70b9\uff08\u4e0a\u4e00\u4e2a\u8282\u70b9\uff09\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u76f8\u8f83\u4e8e\u5355\u5411\u94fe\u8868\uff0c\u53cc\u5411\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u76f8\u5e94\u5730\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002
class ListNode:\n \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n def __init__(self, val: int):\n self.val: int = val # \u8282\u70b9\u503c\n self.next: ListNode | None = None # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n self.prev: ListNode | None = None # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n ListNode(int x) : val(x), next(nullptr), prev(nullptr) {} // \u6784\u9020\u51fd\u6570\n};\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n ListNode prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n ListNode(int x) { val = x; } // \u6784\u9020\u51fd\u6570\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) { // \u6784\u9020\u51fd\u6570\n int val = x; // \u8282\u70b9\u503c\n ListNode next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n ListNode prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\n Val int // \u8282\u70b9\u503c\n Next *DoublyListNode // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n Prev *DoublyListNode // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\n return &DoublyListNode{\n Val: val,\n Next: nil,\n Prev: nil,\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n var val: Int // \u8282\u70b9\u503c\n var next: ListNode? // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n var prev: ListNode? // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n\n init(x: Int) { // \u6784\u9020\u51fd\u6570\n val = x\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n constructor(val, next, prev) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = next === undefined ? null : next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n this.prev = prev === undefined ? null : prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n val: number;\n next: ListNode | null;\n prev: ListNode | null;\n constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = next === undefined ? null : next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n this.prev = prev === undefined ? null : prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n ListNode prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n ListNode(this.val, [this.next, this.prev]); // \u6784\u9020\u51fd\u6570\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\u578b */\n#[derive(Debug)]\nstruct ListNode {\n val: i32, // \u8282\u70b9\u503c\n next: Option<Rc<RefCell<ListNode>>>, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n prev: Option<Rc<RefCell<ListNode>>>, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nimpl ListNode {\n fn new(val: i32) -> Self {\n ListNode {\n val,\n next: None,\n prev: None,\n }\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n int val; // \u8282\u70b9\u503c\n struct ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n struct ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n ListNode *node;\n node = (ListNode *) malloc(sizeof(ListNode));\n node->val = val;\n node->next = NULL;\n node->prev = NULL;\n return node;\n}\n// \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n return struct {\n const Self = @This();\n\n val: T = 0, // \u8282\u70b9\u503c\n next: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n prev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n\n // \u6784\u9020\u51fd\u6570\n pub fn init(self: *Self, x: i32) void {\n self.val = x;\n self.next = null;\n self.prev = null;\n }\n };\n}\n\u56fe 4-8 \u00a0 \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b
"},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4 \u00a0 \u94fe\u8868\u5178\u578b\u5e94\u7528","text":"\u5355\u5411\u94fe\u8868\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u548c\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002
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- LRU \u7b97\u6cd5\uff1a\u5728\u7f13\u5b58\u6dd8\u6c70\uff08LRU\uff09\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5feb\u901f\u627e\u5230\u6700\u8fd1\u6700\u5c11\u4f7f\u7528\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u652f\u6301\u5feb\u901f\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u3002\u8fd9\u65f6\u5019\u4f7f\u7528\u53cc\u5411\u94fe\u8868\u5c31\u975e\u5e38\u5408\u9002\u3002
\u73af\u5f62\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5468\u671f\u6027\u64cd\u4f5c\u7684\u573a\u666f\uff0c\u6bd4\u5982\u64cd\u4f5c\u7cfb\u7edf\u7684\u8d44\u6e90\u8c03\u5ea6\u3002
- \u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\uff1a\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\u662f\u4e00\u79cd\u5e38\u89c1\u7684 CPU \u8c03\u5ea6\u7b97\u6cd5\uff0c\u5b83\u9700\u8981\u5bf9\u4e00\u7ec4\u8fdb\u7a0b\u8fdb\u884c\u5faa\u73af\u3002\u6bcf\u4e2a\u8fdb\u7a0b\u88ab\u8d4b\u4e88\u4e00\u4e2a\u65f6\u95f4\u7247\uff0c\u5f53\u65f6\u95f4\u7247\u7528\u5b8c\u65f6\uff0cCPU \u5c06\u5207\u6362\u5230\u4e0b\u4e00\u4e2a\u8fdb\u7a0b\u3002\u8fd9\u79cd\u5faa\u73af\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u73af\u5f62\u94fe\u8868\u6765\u5b9e\u73b0\u3002
- \u6570\u636e\u7f13\u51b2\u533a\uff1a\u5728\u67d0\u4e9b\u6570\u636e\u7f13\u51b2\u533a\u7684\u5b9e\u73b0\u4e2d\uff0c\u4e5f\u53ef\u80fd\u4f1a\u4f7f\u7528\u73af\u5f62\u94fe\u8868\u3002\u6bd4\u5982\u5728\u97f3\u9891\u3001\u89c6\u9891\u64ad\u653e\u5668\u4e2d\uff0c\u6570\u636e\u6d41\u53ef\u80fd\u4f1a\u88ab\u5206\u6210\u591a\u4e2a\u7f13\u51b2\u5757\u5e76\u653e\u5165\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\uff0c\u4ee5\u4fbf\u5b9e\u73b0\u65e0\u7f1d\u64ad\u653e\u3002
\u300c\u5217\u8868 list\u300d\u662f\u4e00\u4e2a\u62bd\u8c61\u7684\u6570\u636e\u7ed3\u6784\u6982\u5ff5\uff0c\u5b83\u8868\u793a\u5143\u7d20\u7684\u6709\u5e8f\u96c6\u5408\uff0c\u652f\u6301\u5143\u7d20\u8bbf\u95ee\u3001\u4fee\u6539\u3001\u6dfb\u52a0\u3001\u5220\u9664\u548c\u904d\u5386\u7b49\u64cd\u4f5c\uff0c\u65e0\u987b\u4f7f\u7528\u8005\u8003\u8651\u5bb9\u91cf\u9650\u5236\u7684\u95ee\u9898\u3002\u5217\u8868\u53ef\u4ee5\u57fa\u4e8e\u94fe\u8868\u6216\u6570\u7ec4\u5b9e\u73b0\u3002
- \u94fe\u8868\u5929\u7136\u53ef\u4ee5\u770b\u4f5c\u4e00\u4e2a\u5217\u8868\uff0c\u5176\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\u64cd\u4f5c\uff0c\u5e76\u4e14\u53ef\u4ee5\u7075\u6d3b\u52a8\u6001\u6269\u5bb9\u3002
- \u6570\u7ec4\u4e5f\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\uff0c\u4f46\u7531\u4e8e\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u53ea\u80fd\u770b\u4f5c\u4e00\u4e2a\u5177\u6709\u957f\u5ea6\u9650\u5236\u7684\u5217\u8868\u3002
\u5f53\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u5217\u8868\u65f6\uff0c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6027\u8d28\u4f1a\u5bfc\u81f4\u5217\u8868\u7684\u5b9e\u7528\u6027\u964d\u4f4e\u3002\u8fd9\u662f\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u65e0\u6cd5\u4e8b\u5148\u786e\u5b9a\u9700\u8981\u5b58\u50a8\u591a\u5c11\u6570\u636e\uff0c\u4ece\u800c\u96be\u4ee5\u9009\u62e9\u5408\u9002\u7684\u5217\u8868\u957f\u5ea6\u3002\u82e5\u957f\u5ea6\u8fc7\u5c0f\uff0c\u5219\u5f88\u53ef\u80fd\u65e0\u6cd5\u6ee1\u8db3\u4f7f\u7528\u9700\u6c42\uff1b\u82e5\u957f\u5ea6\u8fc7\u5927\uff0c\u5219\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u300c\u52a8\u6001\u6570\u7ec4 dynamic array\u300d\u6765\u5b9e\u73b0\u5217\u8868\u3002\u5b83\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u5404\u9879\u4f18\u70b9\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u8fdb\u884c\u52a8\u6001\u6269\u5bb9\u3002
\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u6807\u51c6\u5e93\u63d0\u4f9b\u7684\u5217\u8868\u662f\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982 Python \u4e2d\u7684
"},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1 \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u5217\u8868","text":"list\u3001Java \u4e2d\u7684ArrayList\u3001C++ \u4e2d\u7684vector\u548c C# \u4e2d\u7684List\u7b49\u3002\u5728\u63a5\u4e0b\u6765\u7684\u8ba8\u8bba\u4e2d\uff0c\u6211\u4eec\u5c06\u628a\u201c\u5217\u8868\u201d\u548c\u201c\u52a8\u6001\u6570\u7ec4\u201d\u89c6\u4e3a\u7b49\u540c\u7684\u6982\u5ff5\u3002\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u8fd9\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nnums1: list[int] = []\n# \u6709\u521d\u59cb\u503c\nnums: list[int] = [1, 3, 2, 5, 4]\n
list.java/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 nums\n// \u65e0\u521d\u59cb\u503c\nvector<int> nums1;\n// \u6709\u521d\u59cb\u503c\nvector<int> nums = { 1, 3, 2, 5, 4 };\n
list.cs/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<Integer> nums1 = new ArrayList<>();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList<Integer> nums = new ArrayList<>(Arrays.asList(numbers));\n
list_test.go/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> nums1 = [];\n// \u6709\u521d\u59cb\u503c\nint[] numbers = [1, 3, 2, 5, 4];\nList<int> nums = [.. numbers];\n
list.swift/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nnums1 := []int{}\n// \u6709\u521d\u59cb\u503c\nnums := []int{1, 3, 2, 5, 4}\n
list.js/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar nums = [1, 3, 2, 5, 4]\n
list.ts/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1 = [];\n// \u6709\u521d\u59cb\u503c\nconst nums = [1, 3, 2, 5, 4];\n
list.dart/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst nums: number[] = [1, 3, 2, 5, 4];\n
list.rs/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> nums1 = [];\n// \u6709\u521d\u59cb\u503c\nList<int> nums = [1, 3, 2, 5, 4];\n
list.c/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: Vec<i32> = Vec::new();\n// \u6709\u521d\u59cb\u503c\nlet nums: Vec<i32> = vec![1, 3, 2, 5, 4];\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"// \u521d\u59cb\u5316\u5217\u8868\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 });\n\u5217\u8868\u672c\u8d28\u4e0a\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u8bbf\u95ee\u5143\u7d20\nnum: int = nums[1] # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0 # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.java/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.cs/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums.get(1); // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums.set(1, 0); // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list_test.go/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.swift/* \u8bbf\u95ee\u5143\u7d20 */\nnum := nums[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.js/* \u8bbf\u95ee\u5143\u7d20 */\nlet num = nums[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.ts/* \u8bbf\u95ee\u5143\u7d20 */\nconst num = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.dart/* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.rs/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.c/* \u8bbf\u95ee\u5143\u7d20 */\nlet num: i32 = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#3","title":"3. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20","text":"// \u8bbf\u95ee\u5143\u7d20\nvar num = nums.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n// \u66f4\u65b0\u5143\u7d20\nnums.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0 \n\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u76f8\u540c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u6e05\u7a7a\u5217\u8868\nnums.clear()\n\n# \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6) # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n# \u5220\u9664\u5143\u7d20\nnums.pop(3) # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.java/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(nums.begin() + 3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.erase(nums.begin() + 3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.cs/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list_test.go/* \u6e05\u7a7a\u5217\u8868 */\nnums.Clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.Insert(3, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.RemoveAt(3);\n
list.swift/* \u6e05\u7a7a\u5217\u8868 */\nnums = nil\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums = append(nums[:3], nums[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.js/* \u6e05\u7a7a\u5217\u8868 */\nnums.removeAll()\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.ts/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n
list.dart/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n
list.rs/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.removeAt(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.c/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#4","title":"4. \u00a0 \u904d\u5386\u5217\u8868","text":"// \u6e05\u7a7a\u5217\u8868\nnums.clearRetainingCapacity();\n\n// \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n\n// \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry nums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n// \u5220\u9664\u5143\u7d20\n_ = nums.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u6839\u636e\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u5404\u5143\u7d20\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in range(len(nums)):\n count += nums[i]\n\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\nfor num in nums:\n count += num\n
list.java/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < nums.size(); i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int num : nums) {\n count += num;\n}\n
list.cs/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < nums.size(); i++) {\n count += nums.get(i);\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\nfor (int num : nums) {\n count += num;\n}\n
list_test.go/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < nums.Count; i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int num in nums) {\n count += num;\n}\n
list.swift/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i < len(nums); i++ {\n count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _, num := range nums {\n count += num\n}\n
list.js/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor i in nums.indices {\n count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor num in nums {\n count += num\n}\n
list.ts/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < nums.length; i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n count += num;\n}\n
list.dart/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < nums.length; i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n count += num;\n}\n
list.rs/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (var i = 0; i < nums.length; i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (var num in nums) {\n count += num;\n}\n
list.c// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nlet mut _count = 0;\nfor i in 0..nums.len() {\n _count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\n_count = 0;\nfor num in &nums {\n _count += num;\n}\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#5","title":"5. \u00a0 \u62fc\u63a5\u5217\u8868","text":"// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < nums.items.len) : (i += 1) {\n count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (nums.items) |num| {\n count += num;\n}\n\u7ed9\u5b9a\u4e00\u4e2a\u65b0\u5217\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.pynums1\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u62fc\u63a5\u5230\u539f\u5217\u8868\u7684\u5c3e\u90e8\u3002
list.cpp# \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1 # \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.java/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector<int> nums1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n
list.cs/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<Integer> nums1 = new ArrayList<>(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list_test.go/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.swift/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...) // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.js/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.ts/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.dart/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.rs/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.c/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1: Vec<i32> = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#6","title":"6. \u00a0 \u6392\u5e8f\u5217\u8868","text":"// \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u67e5\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u548c\u201c\u53cc\u6307\u9488\u201d\u7b97\u6cd5\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u6392\u5e8f\u5217\u8868\nnums.sort() # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.java/* \u6392\u5e8f\u5217\u8868 */\nsort(nums.begin(), nums.end()); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.cs/* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(nums); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list_test.go/* \u6392\u5e8f\u5217\u8868 */\nnums.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.swift/* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(nums) // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.js/* \u6392\u5e8f\u5217\u8868 */\nnums.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.ts/* \u6392\u5e8f\u5217\u8868 */ \nnums.sort((a, b) => a - b); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.dart/* \u6392\u5e8f\u5217\u8868 */\nnums.sort((a, b) => a - b); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.rs/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.c/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2 \u00a0 \u5217\u8868\u5b9e\u73b0","text":"// \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u4e86\u5217\u8868\uff0c\u4f8b\u5982 Java\u3001C++\u3001Python \u7b49\u3002\u5b83\u4eec\u7684\u5b9e\u73b0\u6bd4\u8f83\u590d\u6742\uff0c\u5404\u4e2a\u53c2\u6570\u7684\u8bbe\u5b9a\u4e5f\u975e\u5e38\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u6269\u5bb9\u500d\u6570\u7b49\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u8fdb\u884c\u5b66\u4e60\u3002
\u4e3a\u4e86\u52a0\u6df1\u5bf9\u5217\u8868\u5de5\u4f5c\u539f\u7406\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5c1d\u8bd5\u5b9e\u73b0\u4e00\u4e2a\u7b80\u6613\u7248\u5217\u8868\uff0c\u5305\u62ec\u4ee5\u4e0b\u4e09\u4e2a\u91cd\u70b9\u8bbe\u8ba1\u3002
- \u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002
- \u6570\u91cf\u8bb0\u5f55\uff1a\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf
size\uff0c\u7528\u4e8e\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u548c\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002 - \u6269\u5bb9\u673a\u5236\uff1a\u82e5\u63d2\u5165\u5143\u7d20\u65f6\u5217\u8868\u5bb9\u91cf\u5df2\u6ee1\uff0c\u5219\u9700\u8981\u8fdb\u884c\u6269\u5bb9\u3002\u5148\u6839\u636e\u6269\u5bb9\u500d\u6570\u521b\u5efa\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u518d\u5c06\u5f53\u524d\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52a8\u81f3\u65b0\u6570\u7ec4\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002
my_list.cppclass MyList:\n \"\"\"\u5217\u8868\u7c7b\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._capacity: int = 10 # \u5217\u8868\u5bb9\u91cf\n self._arr: list[int] = [0] * self._capacity # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n self._size: int = 0 # \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n self._extend_ratio: int = 2 # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\n return self._size\n\n def capacity(self) -> int:\n \"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\n return self._capacity\n\n def get(self, index: int) -> int:\n \"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if index < 0 or index >= self._size:\n raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n return self._arr[index]\n\n def set(self, num: int, index: int):\n \"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\n if index < 0 or index >= self._size:\n raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n self._arr[index] = num\n\n def add(self, num: int):\n \"\"\"\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if self.size() == self.capacity():\n self.extend_capacity()\n self._arr[self._size] = num\n self._size += 1\n\n def insert(self, num: int, index: int):\n \"\"\"\u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\n if index < 0 or index >= self._size:\n raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if self._size == self.capacity():\n self.extend_capacity()\n # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for j in range(self._size - 1, index - 1, -1):\n self._arr[j + 1] = self._arr[j]\n self._arr[index] = num\n # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self._size += 1\n\n def remove(self, index: int) -> int:\n \"\"\"\u5220\u9664\u5143\u7d20\"\"\"\n if index < 0 or index >= self._size:\n raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n num = self._arr[index]\n # \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for j in range(index, self._size - 1):\n self._arr[j] = self._arr[j + 1]\n # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self._size -= 1\n # \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num\n\n def extend_capacity(self):\n \"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 __extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n self._capacity = len(self._arr)\n\n def to_array(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\n return self._arr[: self._size]\n
my_list.java/* \u5217\u8868\u7c7b */\nclass MyList {\n private:\n int *arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n int arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\n int arrSize = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n MyList() {\n arr = new int[arrCapacity];\n }\n\n /* \u6790\u6784\u65b9\u6cd5 */\n ~MyList() {\n delete[] arr;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n int size() {\n return arrSize;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n int capacity() {\n return arrCapacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n int get(int index) {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= size())\n throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n return arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n void set(int index, int num) {\n if (index < 0 || index >= size())\n throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n void add(int num) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size() == capacity())\n extendCapacity();\n arr[size()] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n void insert(int index, int num) {\n if (index < 0 || index >= size())\n throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size() == capacity())\n extendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int j = size() - 1; j >= index; j--) {\n arr[j + 1] = arr[j];\n }\n arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n int remove(int index) {\n if (index < 0 || index >= size())\n throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n int num = arr[index];\n // \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int j = index; j < size() - 1; j++) {\n arr[j] = arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n void extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n int newCapacity = capacity() * extendRatio;\n int *tmp = arr;\n arr = new int[newCapacity];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < size(); i++) {\n arr[i] = tmp[i];\n }\n // \u91ca\u653e\u5185\u5b58\n delete[] tmp;\n arrCapacity = newCapacity;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\n vector<int> toVector() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n vector<int> vec(size());\n for (int i = 0; i < size(); i++) {\n vec[i] = arr[i];\n }\n return vec;\n }\n};\n
my_list.cs/* \u5217\u8868\u7c7b */\nclass MyList {\n private int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n private int capacity = 10; // \u5217\u8868\u5bb9\u91cf\n private int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n private int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public MyList() {\n arr = new int[capacity];\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n public int size() {\n return size;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n public int capacity() {\n return capacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n public int get(int index) {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= size)\n throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n return arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n public void set(int index, int num) {\n if (index < 0 || index >= size)\n throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n public void add(int num) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size == capacity())\n extendCapacity();\n arr[size] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n size++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n public void insert(int index, int num) {\n if (index < 0 || index >= size)\n throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size == capacity())\n extendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int j = size - 1; j >= index; j--) {\n arr[j + 1] = arr[j];\n }\n arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n size++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n public int remove(int index) {\n if (index < 0 || index >= size)\n throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n int num = arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int j = index; j < size - 1; j++) {\n arr[j] = arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n size--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n public void extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n arr = Arrays.copyOf(arr, capacity() * extendRatio);\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n capacity = arr.length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n public int[] toArray() {\n int size = size();\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] arr = new int[size];\n for (int i = 0; i < size; i++) {\n arr[i] = get(i);\n }\n return arr;\n }\n}\n
my_list.go/* \u5217\u8868\u7c7b */\nclass MyList {\n private int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n private int arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\n private int arrSize = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n private readonly int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public MyList() {\n arr = new int[arrCapacity];\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n public int Size() {\n return arrSize;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n public int Capacity() {\n return arrCapacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n public int Get(int index) {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= arrSize)\n throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n return arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n public void Set(int index, int num) {\n if (index < 0 || index >= arrSize)\n throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n public void Add(int num) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (arrSize == arrCapacity)\n ExtendCapacity();\n arr[arrSize] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n public void Insert(int index, int num) {\n if (index < 0 || index >= arrSize)\n throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (arrSize == arrCapacity)\n ExtendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int j = arrSize - 1; j >= index; j--) {\n arr[j + 1] = arr[j];\n }\n arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n public int Remove(int index) {\n if (index < 0 || index >= arrSize)\n throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n int num = arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int j = index; j < arrSize - 1; j++) {\n arr[j] = arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n public void ExtendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a arrCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n Array.Resize(ref arr, arrCapacity * extendRatio);\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n arrCapacity = arr.Length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n public int[] ToArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] arr = new int[arrSize];\n for (int i = 0; i < arrSize; i++) {\n arr[i] = Get(i);\n }\n return arr;\n }\n}\n
my_list.swift/* \u5217\u8868\u7c7b */\ntype myList struct {\n arrCapacity int\n arr []int\n arrSize int\n extendRatio int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\n return &myList{\n arrCapacity: 10, // \u5217\u8868\u5bb9\u91cf\n arr: make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n arrSize: 0, // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n extendRatio: 2, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n }\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\n return l.arrSize\n}\n\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\n return l.arrCapacity\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if index < 0 || index >= l.arrSize {\n panic(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n return l.arr[index]\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\n if index < 0 || index >= l.arrSize {\n panic(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n l.arr[index] = num\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if l.arrSize == l.arrCapacity {\n l.extendCapacity()\n }\n l.arr[l.arrSize] = num\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n l.arrSize++\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\n if index < 0 || index >= l.arrSize {\n panic(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if l.arrSize == l.arrCapacity {\n l.extendCapacity()\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for j := l.arrSize - 1; j >= index; j-- {\n l.arr[j+1] = l.arr[j]\n }\n l.arr[index] = num\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n l.arrSize++\n}\n\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\n if index < 0 || index >= l.arrSize {\n panic(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n num := l.arr[index]\n // \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for j := index; j < l.arrSize-1; j++ {\n l.arr[j] = l.arr[j+1]\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n l.arrSize--\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n l.arrCapacity = len(l.arr)\n}\n\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n return l.arr[:l.arrSize]\n}\n
my_list.js/* \u5217\u8868\u7c7b */\nclass MyList {\n private var arr: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n private var _capacity = 10 // \u5217\u8868\u5bb9\u91cf\n private var _size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n private let extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init() {\n arr = Array(repeating: 0, count: _capacity)\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n func size() -> Int {\n _size\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n func capacity() -> Int {\n _capacity\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n func get(index: Int) -> Int {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\n if index < 0 || index >= _size {\n fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n return arr[index]\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n func set(index: Int, num: Int) {\n if index < 0 || index >= _size {\n fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n arr[index] = num\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n func add(num: Int) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if _size == _capacity {\n extendCapacity()\n }\n arr[_size] = num\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size += 1\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n func insert(index: Int, num: Int) {\n if index < 0 || index >= _size {\n fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if _size == _capacity {\n extendCapacity()\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for j in sequence(first: _size - 1, next: { $0 >= index + 1 ? $0 - 1 : nil }) {\n arr[j + 1] = arr[j]\n }\n arr[index] = num\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size += 1\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n @discardableResult\n func remove(index: Int) -> Int {\n if index < 0 || index >= _size {\n fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n let num = arr[index]\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for j in index ..< (_size - 1) {\n arr[j] = arr[j + 1]\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size -= 1\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n func extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n arr = arr + Array(repeating: 0, count: _capacity * (extendRatio - 1))\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n _capacity = arr.count\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n func toArray() -> [Int] {\n var arr = Array(repeating: 0, count: _size)\n for i in 0 ..< _size {\n arr[i] = get(index: i)\n }\n return arr\n }\n}\n
my_list.ts/* \u5217\u8868\u7c7b */\nclass MyList {\n #arr = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n #capacity = 10; // \u5217\u8868\u5bb9\u91cf\n #size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n #extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.#arr = new Array(this.#capacity);\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n size() {\n return this.#size;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n capacity() {\n return this.#capacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n get(index) {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n return this.#arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n set(index, num) {\n if (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n this.#arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n add(num) {\n // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n if (this.#size === this.#capacity) {\n this.extendCapacity();\n }\n // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n this.#arr[this.#size] = num;\n this.#size++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n insert(index, num) {\n if (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (this.#size === this.#capacity) {\n this.extendCapacity();\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (let j = this.#size - 1; j >= index; j--) {\n this.#arr[j + 1] = this.#arr[j];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n this.#arr[index] = num;\n this.#size++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n remove(index) {\n if (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n let num = this.#arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (let j = index; j < this.#size - 1; j++) {\n this.#arr[j] = this.#arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n this.#size--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n this.#arr = this.#arr.concat(\n new Array(this.capacity() * (this.#extendRatio - 1))\n );\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n this.#capacity = this.#arr.length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n toArray() {\n let size = this.size();\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const arr = new Array(size);\n for (let i = 0; i < size; i++) {\n arr[i] = this.get(i);\n }\n return arr;\n }\n}\n
my_list.dart/* \u5217\u8868\u7c7b */\nclass MyList {\n private arr: Array<number>; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n private _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\n private _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n private extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.arr = new Array(this._capacity);\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n public size(): number {\n return this._size;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n public capacity(): number {\n return this._capacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n public get(index: number): number {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n return this.arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n public set(index: number, num: number): void {\n if (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n this.arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n public add(num: number): void {\n // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n if (this._size === this._capacity) this.extendCapacity();\n // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n this.arr[this._size] = num;\n this._size++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n public insert(index: number, num: number): void {\n if (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (this._size === this._capacity) {\n this.extendCapacity();\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (let j = this._size - 1; j >= index; j--) {\n this.arr[j + 1] = this.arr[j];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n this.arr[index] = num;\n this._size++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n public remove(index: number): number {\n if (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n let num = this.arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (let j = index; j < this._size - 1; j++) {\n this.arr[j] = this.arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n this._size--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n public extendCapacity(): void {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n this.arr = this.arr.concat(\n new Array(this.capacity() * (this.extendRatio - 1))\n );\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n this._capacity = this.arr.length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n public toArray(): number[] {\n let size = this.size();\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const arr = new Array(size);\n for (let i = 0; i < size; i++) {\n arr[i] = this.get(i);\n }\n return arr;\n }\n}\n
my_list.rs/* \u5217\u8868\u7c7b */\nclass MyList {\n late List<int> _arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n int _capacity = 10; // \u5217\u8868\u5bb9\u91cf\n int _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n int _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n MyList() {\n _arr = List.filled(_capacity, 0);\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n int size() => _size;\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n int capacity() => _capacity;\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n int get(int index) {\n if (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n return _arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n void set(int index, int _num) {\n if (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n _arr[index] = _num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n void add(int _num) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (_size == _capacity) extendCapacity();\n _arr[_size] = _num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n void insert(int index, int _num) {\n if (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (_size == _capacity) extendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (var j = _size - 1; j >= index; j--) {\n _arr[j + 1] = _arr[j];\n }\n _arr[index] = _num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n int remove(int index) {\n if (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n int _num = _arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (var j = index; j < _size - 1; j++) {\n _arr[j] = _arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return _num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n void extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n final _newNums = List.filled(_capacity * _extendRatio, 0);\n // \u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n List.copyRange(_newNums, 0, _arr);\n // \u66f4\u65b0 _arr \u7684\u5f15\u7528\n _arr = _newNums;\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n _capacity = _arr.length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n List<int> toArray() {\n List<int> arr = [];\n for (var i = 0; i < _size; i++) {\n arr.add(get(i));\n }\n return arr;\n }\n}\n
my_list.c/* \u5217\u8868\u7c7b */\n#[allow(dead_code)]\nstruct MyList {\n arr: Vec<i32>, // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n capacity: usize, // \u5217\u8868\u5bb9\u91cf\n size: usize, // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n extend_ratio: usize, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n\n#[allow(unused,unused_comparisons)]\nimpl MyList {\n /* \u6784\u9020\u65b9\u6cd5 */\n pub fn new(capacity: usize) -> Self {\n let mut vec = Vec::new(); \n vec.resize(capacity, 0);\n Self {\n arr: vec,\n capacity,\n size: 0,\n extend_ratio: 2,\n }\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n pub fn size(&self) -> usize {\n return self.size;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n pub fn capacity(&self) -> usize {\n return self.capacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n pub fn get(&self, index: usize) -> i32 {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if index >= self.size {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n return self.arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n pub fn set(&mut self, index: usize, num: i32) {\n if index >= self.size {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n self.arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n pub fn add(&mut self, num: i32) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if self.size == self.capacity() {\n self.extend_capacity();\n }\n self.arr[self.size] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.size += 1;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n pub fn insert(&mut self, index: usize, num: i32) {\n if index >= self.size() {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if self.size == self.capacity() {\n self.extend_capacity();\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for j in (index..self.size).rev() {\n self.arr[j + 1] = self.arr[j];\n }\n self.arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.size += 1;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n pub fn remove(&mut self, index: usize) -> i32 {\n if index >= self.size() {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n let num = self.arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for j in (index..self.size - 1) {\n self.arr[j] = self.arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.size -= 1;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n pub fn extend_capacity(&mut self) {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n let new_capacity = self.capacity * self.extend_ratio;\n self.arr.resize(new_capacity, 0);\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n self.capacity = new_capacity;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n pub fn to_array(&mut self) -> Vec<i32> {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n let mut arr = Vec::new();\n for i in 0..self.size {\n arr.push(self.get(i));\n }\n arr\n }\n}\n
my_list.zig/* \u5217\u8868\u7c7b */\ntypedef struct {\n int *arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n int capacity; // \u5217\u8868\u5bb9\u91cf\n int size; // \u5217\u8868\u5927\u5c0f\n int extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n} MyList;\n\n/* \u6784\u9020\u51fd\u6570 */\nMyList *newMyList() {\n MyList *nums = malloc(sizeof(MyList));\n nums->capacity = 10;\n nums->arr = malloc(sizeof(int) * nums->capacity);\n nums->size = 0;\n nums->extendRatio = 2;\n return nums;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(MyList *nums) {\n free(nums->arr);\n free(nums);\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(MyList *nums) {\n return nums->size;\n}\n\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(MyList *nums) {\n return nums->capacity;\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(MyList *nums, int index) {\n assert(index >= 0 && index < nums->size);\n return nums->arr[index];\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\n assert(index >= 0 && index < nums->size);\n nums->arr[index] = num;\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\n if (size(nums) == capacity(nums)) {\n extendCapacity(nums); // \u6269\u5bb9\n }\n nums->arr[size(nums)] = num;\n nums->size++;\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\n assert(index >= 0 && index < size(nums));\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size(nums) == capacity(nums)) {\n extendCapacity(nums); // \u6269\u5bb9\n }\n for (int i = size(nums); i > index; --i) {\n nums->arr[i] = nums->arr[i - 1];\n }\n nums->arr[index] = num;\n nums->size++;\n}\n\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeItem(MyList *nums, int index) {\n assert(index >= 0 && index < size(nums));\n int num = nums->arr[index];\n for (int i = index; i < size(nums) - 1; i++) {\n nums->arr[i] = nums->arr[i + 1];\n }\n nums->size--;\n return num;\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n // \u5148\u5206\u914d\u7a7a\u95f4\n int newCapacity = capacity(nums) * nums->extendRatio;\n int *extend = (int *)malloc(sizeof(int) * newCapacity);\n int *temp = nums->arr;\n\n // \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\n for (int i = 0; i < size(nums); i++)\n extend[i] = nums->arr[i];\n\n // \u91ca\u653e\u65e7\u6570\u636e\n free(temp);\n\n // \u66f4\u65b0\u65b0\u6570\u636e\n nums->arr = extend;\n nums->capacity = newCapacity;\n}\n\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(MyList *nums) {\n return nums->arr;\n}\n
"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *","text":"// \u5217\u8868\u7c7b\nfn MyList(comptime T: type) type {\n return struct {\n const Self = @This();\n\n arr: []T = undefined, // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n arrCapacity: usize = 10, // \u5217\u8868\u5bb9\u91cf\n numSize: usize = 0, // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n extendRatio: usize = 2, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n @memset(self.arr, @as(T, 0));\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n pub fn size(self: *Self) usize {\n return self.numSize;\n }\n\n // \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n pub fn capacity(self: *Self) usize {\n return self.arrCapacity;\n }\n\n // \u8bbf\u95ee\u5143\u7d20\n pub fn get(self: *Self, index: usize) T {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n return self.arr[index];\n } \n\n // \u66f4\u65b0\u5143\u7d20\n pub fn set(self: *Self, index: usize, num: T) void {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n self.arr[index] = num;\n } \n\n // \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\n pub fn add(self: *Self, num: T) !void {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (self.size() == self.capacity()) try self.extendCapacity();\n self.arr[self.size()] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.numSize += 1;\n } \n\n // \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\n pub fn insert(self: *Self, index: usize, num: T) !void {\n if (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (self.size() == self.capacity()) try self.extendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n var j = self.size() - 1;\n while (j >= index) : (j -= 1) {\n self.arr[j + 1] = self.arr[j];\n }\n self.arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.numSize += 1;\n }\n\n // \u5220\u9664\u5143\u7d20\n pub fn remove(self: *Self, index: usize) T {\n if (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n var num = self.arr[index];\n // \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n var j = index;\n while (j < self.size() - 1) : (j += 1) {\n self.arr[j] = self.arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.numSize -= 1;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n // \u5217\u8868\u6269\u5bb9\n pub fn extendCapacity(self: *Self) !void {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n var newCapacity = self.capacity() * self.extendRatio;\n var extend = try self.mem_allocator.alloc(T, newCapacity);\n @memset(extend, @as(T, 0));\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n std.mem.copy(T, extend, self.arr);\n self.arr = extend;\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n self.arrCapacity = newCapacity;\n }\n\n // \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n pub fn toArray(self: *Self) ![]T {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n var arr = try self.mem_allocator.alloc(T, self.size());\n @memset(arr, @as(T, 0));\n for (arr, 0..) |*num, i| {\n num.* = self.get(i);\n }\n return arr;\n }\n };\n}\n\u5728\u672c\u7ae0\u7684\u524d\u4e24\u8282\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u8fd9\u4e24\u79cd\u57fa\u7840\u4e14\u91cd\u8981\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5b83\u4eec\u5206\u522b\u4ee3\u8868\u4e86\u201c\u8fde\u7eed\u5b58\u50a8\u201d\u548c\u201c\u5206\u6563\u5b58\u50a8\u201d\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u3002
\u5b9e\u9645\u4e0a\uff0c\u7269\u7406\u7ed3\u6784\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u51b3\u5b9a\u4e86\u7a0b\u5e8f\u5bf9\u5185\u5b58\u548c\u7f13\u5b58\u7684\u4f7f\u7528\u6548\u7387\uff0c\u8fdb\u800c\u5f71\u54cd\u7b97\u6cd5\u7a0b\u5e8f\u7684\u6574\u4f53\u6027\u80fd\u3002
"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","title":"4.4.1 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u8bbe\u5907","text":"\u8ba1\u7b97\u673a\u4e2d\u5305\u62ec\u4e09\u79cd\u7c7b\u578b\u7684\u5b58\u50a8\u8bbe\u5907\uff1a\u300c\u786c\u76d8 hard disk\u300d\u3001\u300c\u5185\u5b58 random-access memory, RAM\u300d\u3001\u300c\u7f13\u5b58 cache memory\u300d\u3002\u8868 4-2 \u5c55\u793a\u4e86\u5b83\u4eec\u5728\u8ba1\u7b97\u673a\u7cfb\u7edf\u4e2d\u7684\u4e0d\u540c\u89d2\u8272\u548c\u6027\u80fd\u7279\u70b9\u3002
\u8868 4-2 \u00a0 \u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u8bbe\u5907
\u786c\u76d8 \u5185\u5b58 \u7f13\u5b58 \u7528\u9014 \u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5305\u62ec\u64cd\u4f5c\u7cfb\u7edf\u3001\u7a0b\u5e8f\u3001\u6587\u4ef6\u7b49 \u4e34\u65f6\u5b58\u50a8\u5f53\u524d\u8fd0\u884c\u7684\u7a0b\u5e8f\u548c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e \u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u51cf\u5c11 CPU \u8bbf\u95ee\u5185\u5b58\u7684\u6b21\u6570 \u6613\u5931\u6027 \u65ad\u7535\u540e\u6570\u636e\u4e0d\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u5bb9\u91cf \u8f83\u5927\uff0cTB \u7ea7\u522b \u8f83\u5c0f\uff0cGB \u7ea7\u522b \u975e\u5e38\u5c0f\uff0cMB \u7ea7\u522b \u901f\u5ea6 \u8f83\u6162\uff0c\u51e0\u767e\u5230\u51e0\u5343 MB/s \u8f83\u5feb\uff0c\u51e0\u5341 GB/s \u975e\u5e38\u5feb\uff0c\u51e0\u5341\u5230\u51e0\u767e GB/s \u4ef7\u683c \u8f83\u4fbf\u5b9c\uff0c\u51e0\u6bdb\u5230\u51e0\u5143 / GB \u8f83\u8d35\uff0c\u51e0\u5341\u5230\u51e0\u767e\u5143 / GB \u975e\u5e38\u8d35\uff0c\u968f CPU \u6253\u5305\u8ba1\u4ef7\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf\u60f3\u8c61\u4e3a\u56fe 4-9 \u6240\u793a\u7684\u91d1\u5b57\u5854\u7ed3\u6784\u3002\u8d8a\u9760\u8fd1\u91d1\u5b57\u5854\u9876\u7aef\u7684\u5b58\u50a8\u8bbe\u5907\u7684\u901f\u5ea6\u8d8a\u5feb\u3001\u5bb9\u91cf\u8d8a\u5c0f\u3001\u6210\u672c\u8d8a\u9ad8\u3002\u8fd9\u79cd\u591a\u5c42\u7ea7\u7684\u8bbe\u8ba1\u5e76\u975e\u5076\u7136\uff0c\u800c\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u548c\u5de5\u7a0b\u5e08\u4eec\u7ecf\u8fc7\u6df1\u601d\u719f\u8651\u7684\u7ed3\u679c\u3002
- \u786c\u76d8\u96be\u4ee5\u88ab\u5185\u5b58\u53d6\u4ee3\u3002\u9996\u5148\uff0c\u5185\u5b58\u4e2d\u7684\u6570\u636e\u5728\u65ad\u7535\u540e\u4f1a\u4e22\u5931\uff0c\u56e0\u6b64\u5b83\u4e0d\u9002\u5408\u957f\u671f\u5b58\u50a8\u6570\u636e\uff1b\u5176\u6b21\uff0c\u5185\u5b58\u7684\u6210\u672c\u662f\u786c\u76d8\u7684\u51e0\u5341\u500d\uff0c\u8fd9\u4f7f\u5f97\u5b83\u96be\u4ee5\u5728\u6d88\u8d39\u8005\u5e02\u573a\u666e\u53ca\u3002
- \u7f13\u5b58\u7684\u5927\u5bb9\u91cf\u548c\u9ad8\u901f\u5ea6\u96be\u4ee5\u517c\u5f97\u3002\u968f\u7740 L1\u3001L2\u3001L3 \u7f13\u5b58\u7684\u5bb9\u91cf\u9010\u6b65\u589e\u5927\uff0c\u5176\u7269\u7406\u5c3a\u5bf8\u4f1a\u53d8\u5927\uff0c\u4e0e CPU \u6838\u5fc3\u4e4b\u95f4\u7684\u7269\u7406\u8ddd\u79bb\u4f1a\u53d8\u8fdc\uff0c\u4ece\u800c\u5bfc\u81f4\u6570\u636e\u4f20\u8f93\u65f6\u95f4\u589e\u52a0\uff0c\u5143\u7d20\u8bbf\u95ee\u5ef6\u8fdf\u53d8\u9ad8\u3002\u5728\u5f53\u524d\u6280\u672f\u4e0b\uff0c\u591a\u5c42\u7ea7\u7684\u7f13\u5b58\u7ed3\u6784\u662f\u5bb9\u91cf\u3001\u901f\u5ea6\u548c\u6210\u672c\u4e4b\u95f4\u7684\u6700\u4f73\u5e73\u8861\u70b9\u3002
\u56fe 4-9 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf
Note
\u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u5c42\u6b21\u7ed3\u6784\u4f53\u73b0\u4e86\u901f\u5ea6\u3001\u5bb9\u91cf\u548c\u6210\u672c\u4e09\u8005\u4e4b\u95f4\u7684\u7cbe\u5999\u5e73\u8861\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u79cd\u6743\u8861\u666e\u904d\u5b58\u5728\u4e8e\u6240\u6709\u5de5\u4e1a\u9886\u57df\uff0c\u5b83\u8981\u6c42\u6211\u4eec\u5728\u4e0d\u540c\u7684\u4f18\u52bf\u548c\u9650\u5236\u4e4b\u95f4\u627e\u5230\u6700\u4f73\u5e73\u8861\u70b9\u3002
\u603b\u7684\u6765\u8bf4\uff0c\u786c\u76d8\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u5927\u91cf\u6570\u636e\uff0c\u5185\u5b58\u7528\u4e8e\u4e34\u65f6\u5b58\u50a8\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\uff0c\u800c\u7f13\u5b58\u5219\u7528\u4e8e\u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u4ee5\u63d0\u9ad8\u7a0b\u5e8f\u8fd0\u884c\u6548\u7387\u3002\u4e09\u8005\u5171\u540c\u534f\u4f5c\uff0c\u786e\u4fdd\u8ba1\u7b97\u673a\u7cfb\u7edf\u9ad8\u6548\u8fd0\u884c\u3002
\u5982\u56fe 4-10 \u6240\u793a\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4f1a\u4ece\u786c\u76d8\u4e2d\u88ab\u8bfb\u53d6\u5230\u5185\u5b58\u4e2d\uff0c\u4f9b CPU \u8ba1\u7b97\u4f7f\u7528\u3002\u7f13\u5b58\u53ef\u4ee5\u770b\u4f5c CPU \u7684\u4e00\u90e8\u5206\uff0c\u5b83\u901a\u8fc7\u667a\u80fd\u5730\u4ece\u5185\u5b58\u52a0\u8f7d\u6570\u636e\uff0c\u7ed9 CPU \u63d0\u4f9b\u9ad8\u901f\u7684\u6570\u636e\u8bfb\u53d6\uff0c\u4ece\u800c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\uff0c\u51cf\u5c11\u5bf9\u8f83\u6162\u7684\u5185\u5b58\u7684\u4f9d\u8d56\u3002
\u56fe 4-10 \u00a0 \u786c\u76d8\u3001\u5185\u5b58\u548c\u7f13\u5b58\u4e4b\u95f4\u7684\u6570\u636e\u6d41\u901a
"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","title":"4.4.2 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u5185\u5b58\u6548\u7387","text":"\u5728\u5185\u5b58\u7a7a\u95f4\u5229\u7528\u65b9\u9762\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5404\u81ea\u5177\u6709\u4f18\u52bf\u548c\u5c40\u9650\u6027\u3002
\u4e00\u65b9\u9762\uff0c\u5185\u5b58\u662f\u6709\u9650\u7684\uff0c\u4e14\u540c\u4e00\u5757\u5185\u5b58\u4e0d\u80fd\u88ab\u591a\u4e2a\u7a0b\u5e8f\u5171\u4eab\uff0c\u56e0\u6b64\u6211\u4eec\u5e0c\u671b\u6570\u636e\u7ed3\u6784\u80fd\u591f\u5c3d\u53ef\u80fd\u9ad8\u6548\u5730\u5229\u7528\u7a7a\u95f4\u3002\u6570\u7ec4\u7684\u5143\u7d20\u7d27\u5bc6\u6392\u5217\uff0c\u4e0d\u9700\u8981\u989d\u5916\u7684\u7a7a\u95f4\u6765\u5b58\u50a8\u94fe\u8868\u8282\u70b9\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u56e0\u6b64\u7a7a\u95f4\u6548\u7387\u66f4\u9ad8\u3002\u7136\u800c\uff0c\u6570\u7ec4\u9700\u8981\u4e00\u6b21\u6027\u5206\u914d\u8db3\u591f\u7684\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u8fd9\u53ef\u80fd\u5bfc\u81f4\u5185\u5b58\u6d6a\u8d39\uff0c\u6570\u7ec4\u6269\u5bb9\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6210\u672c\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u94fe\u8868\u4ee5\u201c\u8282\u70b9\u201d\u4e3a\u5355\u4f4d\u8fdb\u884c\u52a8\u6001\u5185\u5b58\u5206\u914d\u548c\u56de\u6536\uff0c\u63d0\u4f9b\u4e86\u66f4\u5927\u7684\u7075\u6d3b\u6027\u3002
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"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","title":"4.4.3 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u7f13\u5b58\u6548\u7387","text":"\u7f13\u5b58\u867d\u7136\u5728\u7a7a\u95f4\u5bb9\u91cf\u4e0a\u8fdc\u5c0f\u4e8e\u5185\u5b58\uff0c\u4f46\u5b83\u6bd4\u5185\u5b58\u5feb\u5f97\u591a\uff0c\u5728\u7a0b\u5e8f\u6267\u884c\u901f\u5ea6\u4e0a\u8d77\u7740\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\u3002\u7531\u4e8e\u7f13\u5b58\u7684\u5bb9\u91cf\u6709\u9650\uff0c\u53ea\u80fd\u5b58\u50a8\u4e00\u5c0f\u90e8\u5206\u9891\u7e41\u8bbf\u95ee\u7684\u6570\u636e\uff0c\u56e0\u6b64\u5f53 CPU \u5c1d\u8bd5\u8bbf\u95ee\u7684\u6570\u636e\u4e0d\u5728\u7f13\u5b58\u4e2d\u65f6\uff0c\u5c31\u4f1a\u53d1\u751f\u300c\u7f13\u5b58\u672a\u547d\u4e2d cache miss\u300d\uff0c\u6b64\u65f6 CPU \u4e0d\u5f97\u4e0d\u4ece\u901f\u5ea6\u8f83\u6162\u7684\u5185\u5b58\u4e2d\u52a0\u8f7d\u6240\u9700\u6570\u636e\u3002
\u663e\u7136\uff0c\u201c\u7f13\u5b58\u672a\u547d\u4e2d\u201d\u8d8a\u5c11\uff0cCPU \u8bfb\u5199\u6570\u636e\u7684\u6548\u7387\u5c31\u8d8a\u9ad8\uff0c\u7a0b\u5e8f\u6027\u80fd\u4e5f\u5c31\u8d8a\u597d\u3002\u6211\u4eec\u5c06 CPU \u4ece\u7f13\u5b58\u4e2d\u6210\u529f\u83b7\u53d6\u6570\u636e\u7684\u6bd4\u4f8b\u79f0\u4e3a\u300c\u7f13\u5b58\u547d\u4e2d\u7387 cache hit rate\u300d\uff0c\u8fd9\u4e2a\u6307\u6807\u901a\u5e38\u7528\u6765\u8861\u91cf\u7f13\u5b58\u6548\u7387\u3002
\u4e3a\u4e86\u5c3d\u53ef\u80fd\u8fbe\u5230\u66f4\u9ad8\u7684\u6548\u7387\uff0c\u7f13\u5b58\u4f1a\u91c7\u53d6\u4ee5\u4e0b\u6570\u636e\u52a0\u8f7d\u673a\u5236\u3002
- \u7f13\u5b58\u884c\uff1a\u7f13\u5b58\u4e0d\u662f\u5355\u4e2a\u5b57\u8282\u5730\u5b58\u50a8\u4e0e\u52a0\u8f7d\u6570\u636e\uff0c\u800c\u662f\u4ee5\u7f13\u5b58\u884c\u4e3a\u5355\u4f4d\u3002\u76f8\u6bd4\u4e8e\u5355\u4e2a\u5b57\u8282\u7684\u4f20\u8f93\uff0c\u7f13\u5b58\u884c\u7684\u4f20\u8f93\u5f62\u5f0f\u66f4\u52a0\u9ad8\u6548\u3002
- \u9884\u53d6\u673a\u5236\uff1a\u5904\u7406\u5668\u4f1a\u5c1d\u8bd5\u9884\u6d4b\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\uff08\u4f8b\u5982\u987a\u5e8f\u8bbf\u95ee\u3001\u56fa\u5b9a\u6b65\u957f\u8df3\u8dc3\u8bbf\u95ee\u7b49\uff09\uff0c\u5e76\u6839\u636e\u7279\u5b9a\u6a21\u5f0f\u5c06\u6570\u636e\u52a0\u8f7d\u81f3\u7f13\u5b58\u4e4b\u4e2d\uff0c\u4ece\u800c\u63d0\u5347\u547d\u4e2d\u7387\u3002
- \u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u9644\u8fd1\u7684\u6570\u636e\u53ef\u80fd\u8fd1\u671f\u4e5f\u4f1a\u88ab\u8bbf\u95ee\u3002\u56e0\u6b64\uff0c\u7f13\u5b58\u5728\u52a0\u8f7d\u67d0\u4e00\u6570\u636e\u65f6\uff0c\u4e5f\u4f1a\u52a0\u8f7d\u5176\u9644\u8fd1\u7684\u6570\u636e\uff0c\u4ee5\u63d0\u9ad8\u547d\u4e2d\u7387\u3002
- \u65f6\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u5728\u4e0d\u4e45\u7684\u5c06\u6765\u5f88\u53ef\u80fd\u518d\u6b21\u88ab\u8bbf\u95ee\u3002\u7f13\u5b58\u5229\u7528\u8fd9\u4e00\u539f\u7406\uff0c\u901a\u8fc7\u4fdd\u7559\u6700\u8fd1\u8bbf\u95ee\u8fc7\u7684\u6570\u636e\u6765\u63d0\u9ad8\u547d\u4e2d\u7387\u3002
\u5b9e\u9645\u4e0a\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5bf9\u7f13\u5b58\u7684\u5229\u7528\u6548\u7387\u662f\u4e0d\u540c\u7684\uff0c\u4e3b\u8981\u4f53\u73b0\u5728\u4ee5\u4e0b\u51e0\u4e2a\u65b9\u9762\u3002
- \u5360\u7528\u7a7a\u95f4\uff1a\u94fe\u8868\u5143\u7d20\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u7a7a\u95f4\u66f4\u591a\uff0c\u5bfc\u81f4\u7f13\u5b58\u4e2d\u5bb9\u7eb3\u7684\u6709\u6548\u6570\u636e\u91cf\u66f4\u5c11\u3002
- \u7f13\u5b58\u884c\uff1a\u94fe\u8868\u6570\u636e\u5206\u6563\u5728\u5185\u5b58\u5404\u5904\uff0c\u800c\u7f13\u5b58\u662f\u201c\u6309\u884c\u52a0\u8f7d\u201d\u7684\uff0c\u56e0\u6b64\u52a0\u8f7d\u5230\u65e0\u6548\u6570\u636e\u7684\u6bd4\u4f8b\u66f4\u9ad8\u3002
- \u9884\u53d6\u673a\u5236\uff1a\u6570\u7ec4\u6bd4\u94fe\u8868\u7684\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\u66f4\u5177\u201c\u53ef\u9884\u6d4b\u6027\u201d\uff0c\u5373\u7cfb\u7edf\u66f4\u5bb9\u6613\u731c\u51fa\u5373\u5c06\u88ab\u52a0\u8f7d\u7684\u6570\u636e\u3002
- \u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u6570\u7ec4\u88ab\u5b58\u50a8\u5728\u96c6\u4e2d\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u56e0\u6b64\u88ab\u52a0\u8f7d\u6570\u636e\u9644\u8fd1\u7684\u6570\u636e\u66f4\u6709\u53ef\u80fd\u5373\u5c06\u88ab\u8bbf\u95ee\u3002
\u603b\u4f53\u800c\u8a00\uff0c\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u5728\u64cd\u4f5c\u6548\u7387\u4e0a\u901a\u5e38\u4f18\u4e8e\u94fe\u8868\u3002\u8fd9\u4f7f\u5f97\u5728\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u5f80\u5f80\u66f4\u53d7\u6b22\u8fce\u3002
\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9ad8\u7f13\u5b58\u6548\u7387\u5e76\u4e0d\u610f\u5473\u7740\u6570\u7ec4\u5728\u6240\u6709\u60c5\u51b5\u4e0b\u90fd\u4f18\u4e8e\u94fe\u8868\u3002\u5b9e\u9645\u5e94\u7528\u4e2d\u9009\u62e9\u54ea\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u6765\u51b3\u5b9a\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5b9e\u73b0\u201c\u6808\u201d\u6570\u636e\u7ed3\u6784\uff08\u4e0b\u4e00\u7ae0\u4f1a\u8be6\u7ec6\u4ecb\u7ecd\uff09\uff0c\u4f46\u5b83\u4eec\u9002\u7528\u4e8e\u4e0d\u540c\u573a\u666f\u3002
- \u5728\u505a\u7b97\u6cd5\u9898\u65f6\uff0c\u6211\u4eec\u4f1a\u503e\u5411\u4e8e\u9009\u62e9\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\uff0c\u56e0\u4e3a\u5b83\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u64cd\u4f5c\u6548\u7387\u548c\u968f\u673a\u8bbf\u95ee\u7684\u80fd\u529b\uff0c\u4ee3\u4ef7\u4ec5\u662f\u9700\u8981\u9884\u5148\u4e3a\u6570\u7ec4\u5206\u914d\u4e00\u5b9a\u7684\u5185\u5b58\u7a7a\u95f4\u3002
- \u5982\u679c\u6570\u636e\u91cf\u975e\u5e38\u5927\u3001\u52a8\u6001\u6027\u5f88\u9ad8\u3001\u6808\u7684\u9884\u671f\u5927\u5c0f\u96be\u4ee5\u4f30\u8ba1\uff0c\u90a3\u4e48\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u66f4\u52a0\u5408\u9002\u3002\u94fe\u8868\u80fd\u591f\u5c06\u5927\u91cf\u6570\u636e\u5206\u6563\u5b58\u50a8\u4e8e\u5185\u5b58\u7684\u4e0d\u540c\u90e8\u5206\uff0c\u5e76\u4e14\u907f\u514d\u4e86\u6570\u7ec4\u6269\u5bb9\u4ea7\u751f\u7684\u989d\u5916\u5f00\u9500\u3002
- \u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5206\u522b\u4ee3\u8868\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u4e24\u79cd\u5b58\u50a8\u65b9\u5f0f\uff1a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u4e24\u8005\u7684\u7279\u70b9\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u6027\u3002
- \u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5360\u7528\u5185\u5b58\u8f83\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002
- \u94fe\u8868\u901a\u8fc7\u66f4\u6539\u5f15\u7528\uff08\u6307\u9488\uff09\u5b9e\u73b0\u9ad8\u6548\u7684\u8282\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\uff1b\u4f46\u8282\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u8f83\u591a\u3002
- \u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u5355\u5411\u94fe\u8868\u3001\u73af\u5f62\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\uff0c\u5b83\u4eec\u5206\u522b\u5177\u6709\u5404\u81ea\u7684\u5e94\u7528\u573a\u666f\u3002
- \u5217\u8868\u662f\u4e00\u79cd\u652f\u6301\u589e\u5220\u67e5\u6539\u7684\u5143\u7d20\u6709\u5e8f\u96c6\u5408\uff0c\u901a\u5e38\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\uff0c\u5176\u4fdd\u7559\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u540c\u65f6\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\u3002
- \u5217\u8868\u7684\u51fa\u73b0\u5927\u5e45\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
- \u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u6570\u7ec4\u53ef\u63d0\u4f9b\u66f4\u9ad8\u7684\u5185\u5b58\u7a7a\u95f4\u6548\u7387\uff0c\u800c\u94fe\u8868\u5219\u5728\u5185\u5b58\u4f7f\u7528\u4e0a\u66f4\u52a0\u7075\u6d3b\u3002
- \u7f13\u5b58\u901a\u8fc7\u7f13\u5b58\u884c\u3001\u9884\u53d6\u673a\u5236\u4ee5\u53ca\u7a7a\u95f4\u5c40\u90e8\u6027\u548c\u65f6\u95f4\u5c40\u90e8\u6027\u7b49\u6570\u636e\u52a0\u8f7d\u673a\u5236\uff0c\u4e3a CPU \u63d0\u4f9b\u5feb\u901f\u6570\u636e\u8bbf\u95ee\uff0c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\u3002
- \u7531\u4e8e\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u901a\u5e38\u6bd4\u94fe\u8868\u66f4\u9ad8\u6548\u3002\u5728\u9009\u62e9\u6570\u636e\u7ed3\u6784\u65f6\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u548c\u573a\u666f\u505a\u51fa\u6070\u5f53\u9009\u62e9\u3002
\u6570\u7ec4\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5b58\u50a8\u5728\u5806\u4e0a\uff0c\u5bf9\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u5426\u6709\u5f71\u54cd\uff1f
\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5806\u4e0a\u7684\u6570\u7ec4\u90fd\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u6570\u636e\u64cd\u4f5c\u6548\u7387\u57fa\u672c\u4e00\u81f4\u3002\u7136\u800c\uff0c\u6808\u548c\u5806\u5177\u6709\u5404\u81ea\u7684\u7279\u70b9\uff0c\u4ece\u800c\u5bfc\u81f4\u4ee5\u4e0b\u4e0d\u540c\u70b9\u3002
- \u5206\u914d\u548c\u91ca\u653e\u6548\u7387\uff1a\u6808\u662f\u4e00\u5757\u8f83\u5c0f\u7684\u5185\u5b58\uff0c\u5206\u914d\u7531\u7f16\u8bd1\u5668\u81ea\u52a8\u5b8c\u6210\uff1b\u800c\u5806\u5185\u5b58\u76f8\u5bf9\u66f4\u5927\uff0c\u53ef\u4ee5\u5728\u4ee3\u7801\u4e2d\u52a8\u6001\u5206\u914d\uff0c\u66f4\u5bb9\u6613\u788e\u7247\u5316\u3002\u56e0\u6b64\uff0c\u5806\u4e0a\u7684\u5206\u914d\u548c\u91ca\u653e\u64cd\u4f5c\u901a\u5e38\u6bd4\u6808\u4e0a\u7684\u6162\u3002
- \u5927\u5c0f\u9650\u5236\uff1a\u6808\u5185\u5b58\u76f8\u5bf9\u8f83\u5c0f\uff0c\u5806\u7684\u5927\u5c0f\u4e00\u822c\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\u3002\u56e0\u6b64\u5806\u66f4\u52a0\u9002\u5408\u5b58\u50a8\u5927\u578b\u6570\u7ec4\u3002
- \u7075\u6d3b\u6027\uff1a\u6808\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u9700\u8981\u5728\u7f16\u8bd1\u65f6\u786e\u5b9a\uff0c\u800c\u5806\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u53ef\u4ee5\u5728\u8fd0\u884c\u65f6\u52a8\u6001\u786e\u5b9a\u3002
\u4e3a\u4ec0\u4e48\u6570\u7ec4\u8981\u6c42\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u5374\u6ca1\u6709\u5f3a\u8c03\u540c\u7c7b\u578b\u5462\uff1f
\u94fe\u8868\u7531\u8282\u70b9\u7ec4\u6210\uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u5f15\u7528\uff08\u6307\u9488\uff09\u8fde\u63a5\uff0c\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5b58\u50a8\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f8b\u5982 int\u3001double\u3001string\u3001object \u7b49\u3002
\u76f8\u5bf9\u5730\uff0c\u6570\u7ec4\u5143\u7d20\u5219\u5fc5\u987b\u662f\u76f8\u540c\u7c7b\u578b\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u8ba1\u7b97\u504f\u79fb\u91cf\u6765\u83b7\u53d6\u5bf9\u5e94\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u540c\u65f6\u5305\u542b int \u548c long \u4e24\u79cd\u7c7b\u578b\uff0c\u5355\u4e2a\u5143\u7d20\u5206\u522b\u5360\u7528 4 bytes \u548c 8 bytes \uff0c\u6b64\u65f6\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5305\u542b\u4e86\u4e24\u79cd\u957f\u5ea6\u7684\u5143\u7d20\u3002
# \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\n\u5220\u9664\u8282\u70b9\u540e\uff0c\u662f\u5426\u9700\u8981\u628a
P.next\u8bbe\u4e3a \\(\\text{None}\\) \u5462\uff1f\u4e0d\u4fee\u6539
P.next\u4e5f\u53ef\u4ee5\u3002\u4ece\u8be5\u94fe\u8868\u7684\u89d2\u5ea6\u770b\uff0c\u4ece\u5934\u8282\u70b9\u904d\u5386\u5230\u5c3e\u8282\u70b9\u5df2\u7ecf\u4e0d\u4f1a\u9047\u5230P\u4e86\u3002\u8fd9\u610f\u5473\u7740\u8282\u70b9P\u5df2\u7ecf\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u4e86\uff0c\u6b64\u65f6\u8282\u70b9P\u6307\u5411\u54ea\u91cc\u90fd\u4e0d\u4f1a\u5bf9\u8be5\u94fe\u8868\u4ea7\u751f\u5f71\u54cd\u3002\u4ece\u5783\u573e\u56de\u6536\u7684\u89d2\u5ea6\u770b\uff0c\u5bf9\u4e8e Java\u3001Python\u3001Go \u7b49\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\u7684\u8bed\u8a00\u6765\u8bf4\uff0c\u8282\u70b9
P\u662f\u5426\u88ab\u56de\u6536\u53d6\u51b3\u4e8e\u662f\u5426\u4ecd\u5b58\u5728\u6307\u5411\u5b83\u7684\u5f15\u7528\uff0c\u800c\u4e0d\u662fP.next\u7684\u503c\u3002\u5728 C \u548c C++ \u7b49\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u624b\u52a8\u91ca\u653e\u8282\u70b9\u5185\u5b58\u3002\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u5220\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u67e5\u627e\u5143\u7d20\uff0c\u90a3\u4e3a\u4ec0\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f
\u5982\u679c\u662f\u5148\u67e5\u627e\u5143\u7d20\u3001\u518d\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u786e\u5b9e\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u94fe\u8868\u7684 \\(O(1)\\) \u589e\u5220\u7684\u4f18\u52bf\u53ef\u4ee5\u5728\u5176\u4ed6\u5e94\u7528\u4e0a\u5f97\u5230\u4f53\u73b0\u3002\u4f8b\u5982\uff0c\u53cc\u5411\u961f\u5217\u9002\u5408\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u7ef4\u62a4\u4e00\u4e2a\u6307\u9488\u53d8\u91cf\u59cb\u7ec8\u6307\u5411\u5934\u8282\u70b9\u3001\u5c3e\u8282\u70b9\uff0c\u6bcf\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002
\u56fe\u201c\u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f\u201d\u4e2d\uff0c\u6d45\u84dd\u8272\u7684\u5b58\u50a8\u8282\u70b9\u6307\u9488\u662f\u5360\u7528\u4e00\u5757\u5185\u5b58\u5730\u5740\u5417\uff1f\u8fd8\u662f\u548c\u8282\u70b9\u503c\u5404\u5360\u4e00\u534a\u5462\uff1f
\u8be5\u793a\u610f\u56fe\u53ea\u662f\u5b9a\u6027\u8868\u793a\uff0c\u5b9a\u91cf\u8868\u793a\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002
- \u4e0d\u540c\u7c7b\u578b\u7684\u8282\u70b9\u503c\u5360\u7528\u7684\u7a7a\u95f4\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 int\u3001long\u3001double \u548c\u5b9e\u4f8b\u5bf9\u8c61\u7b49\u3002
- \u6307\u9488\u53d8\u91cf\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u6839\u636e\u6240\u4f7f\u7528\u7684\u64cd\u4f5c\u7cfb\u7edf\u53ca\u7f16\u8bd1\u73af\u5883\u800c\u5b9a\uff0c\u5927\u591a\u4e3a 8 \u5b57\u8282\u6216 4 \u5b57\u8282\u3002
\u5728\u5217\u8868\u672b\u5c3e\u6dfb\u52a0\u5143\u7d20\u662f\u5426\u65f6\u65f6\u523b\u523b\u90fd\u4e3a \\(O(1)\\) \uff1f
\u5982\u679c\u6dfb\u52a0\u5143\u7d20\u65f6\u8d85\u51fa\u5217\u8868\u957f\u5ea6\uff0c\u5219\u9700\u8981\u5148\u6269\u5bb9\u5217\u8868\u518d\u6dfb\u52a0\u3002\u7cfb\u7edf\u4f1a\u7533\u8bf7\u4e00\u5757\u65b0\u7684\u5185\u5b58\uff0c\u5e76\u5c06\u539f\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u642c\u8fd0\u8fc7\u53bb\uff0c\u8fd9\u65f6\u5019\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4f1a\u662f \\(O(n)\\) \u3002
\u201c\u5217\u8868\u7684\u51fa\u73b0\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u201d\uff0c\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u662f\u6307\u989d\u5916\u589e\u52a0\u7684\u53d8\u91cf\u5982\u5bb9\u91cf\u3001\u957f\u5ea6\u3001\u6269\u5bb9\u500d\u6570\u6240\u5360\u7684\u5185\u5b58\u5417\uff1f
\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u4e3b\u8981\u6709\u4e24\u65b9\u9762\u542b\u4e49\uff1a\u4e00\u65b9\u9762\uff0c\u5217\u8868\u90fd\u4f1a\u8bbe\u5b9a\u4e00\u4e2a\u521d\u59cb\u957f\u5ea6\uff0c\u6211\u4eec\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u8fd9\u4e48\u591a\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u4e3a\u4e86\u9632\u6b62\u9891\u7e41\u6269\u5bb9\uff0c\u6269\u5bb9\u4e00\u822c\u4f1a\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u8fd9\u6837\u4e00\u6765\uff0c\u4e5f\u4f1a\u51fa\u73b0\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u4eec\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6ee1\u5b83\u4eec\u3002
\u5728 Python \u4e2d\u521d\u59cb\u5316
n = [1, 2, 3]\u540e\uff0c\u8fd9 3 \u4e2a\u5143\u7d20\u7684\u5730\u5740\u662f\u76f8\u8fde\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316m = [2, 1, 3]\u4f1a\u53d1\u73b0\u5b83\u4eec\u6bcf\u4e2a\u5143\u7d20\u7684 id \u5e76\u4e0d\u662f\u8fde\u7eed\u7684\uff0c\u800c\u662f\u5206\u522b\u8ddfn\u4e2d\u7684\u76f8\u540c\u3002\u8fd9\u4e9b\u5143\u7d20\u7684\u5730\u5740\u4e0d\u8fde\u7eed\uff0c\u90a3\u4e48m\u8fd8\u662f\u6570\u7ec4\u5417\uff1f\u5047\u5982\u628a\u5217\u8868\u5143\u7d20\u6362\u6210\u94fe\u8868\u8282\u70b9
n = [n1, n2, n3, n4, n5]\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u8fd9 5 \u4e2a\u8282\u70b9\u5bf9\u8c61\u4e5f\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u3002\u7136\u800c\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5217\u8868\u7d22\u5f15\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u8282\u70b9\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u8282\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5b58\u50a8\u7684\u662f\u8282\u70b9\u7684\u5f15\u7528\uff0c\u800c\u975e\u8282\u70b9\u672c\u8eab\u3002\u4e0e\u8bb8\u591a\u8bed\u8a00\u4e0d\u540c\uff0cPython \u4e2d\u7684\u6570\u5b57\u4e5f\u88ab\u5305\u88c5\u4e3a\u5bf9\u8c61\uff0c\u5217\u8868\u4e2d\u5b58\u50a8\u7684\u4e0d\u662f\u6570\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5bf9\u6570\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u4e24\u4e2a\u6570\u7ec4\u4e2d\u7684\u76f8\u540c\u6570\u5b57\u62e5\u6709\u540c\u4e00\u4e2a id \uff0c\u5e76\u4e14\u8fd9\u4e9b\u6570\u5b57\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002
C++ STL \u91cc\u9762\u7684
std::list\u5df2\u7ecf\u5b9e\u73b0\u4e86\u53cc\u5411\u94fe\u8868\uff0c\u4f46\u597d\u50cf\u4e00\u4e9b\u7b97\u6cd5\u4e66\u4e0a\u4e0d\u600e\u4e48\u76f4\u63a5\u4f7f\u7528\u5b83\uff0c\u662f\u4e0d\u662f\u56e0\u4e3a\u6709\u4ec0\u4e48\u5c40\u9650\u6027\u5462\uff1f\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u5f80\u5f80\u66f4\u9752\u7750\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u7b97\u6cd5\uff0c\u800c\u53ea\u5728\u5fc5\u8981\u65f6\u624d\u4f7f\u7528\u94fe\u8868\uff0c\u4e3b\u8981\u6709\u4e24\u4e2a\u539f\u56e0\u3002
- \u7a7a\u95f4\u5f00\u9500\uff1a\u7531\u4e8e\u6bcf\u4e2a\u5143\u7d20\u9700\u8981\u4e24\u4e2a\u989d\u5916\u7684\u6307\u9488\uff08\u4e00\u4e2a\u7528\u4e8e\u524d\u4e00\u4e2a\u5143\u7d20\uff0c\u4e00\u4e2a\u7528\u4e8e\u540e\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u6240\u4ee5
std::list\u901a\u5e38\u6bd4std::vector\u66f4\u5360\u7528\u7a7a\u95f4\u3002 - \u7f13\u5b58\u4e0d\u53cb\u597d\uff1a\u7531\u4e8e\u6570\u636e\u4e0d\u662f\u8fde\u7eed\u5b58\u653e\u7684\uff0c\u56e0\u6b64
std::list\u5bf9\u7f13\u5b58\u7684\u5229\u7528\u7387\u8f83\u4f4e\u3002\u4e00\u822c\u60c5\u51b5\u4e0b\uff0cstd::vector\u7684\u6027\u80fd\u4f1a\u66f4\u597d\u3002
\u53e6\u4e00\u65b9\u9762\uff0c\u5fc5\u8981\u4f7f\u7528\u94fe\u8868\u7684\u60c5\u51b5\u4e3b\u8981\u662f\u4e8c\u53c9\u6811\u548c\u56fe\u3002\u6808\u548c\u961f\u5217\u5f80\u5f80\u4f1a\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684
stack\u548cqueue\uff0c\u800c\u975e\u94fe\u8868\u3002\u521d\u59cb\u5316\u5217\u8868
res = [0] * self.size()\u64cd\u4f5c\uff0c\u4f1a\u5bfc\u81f4res\u7684\u6bcf\u4e2a\u5143\u7d20\u5f15\u7528\u76f8\u540c\u7684\u5730\u5740\u5417\uff1f\u4e0d\u4f1a\u3002\u4f46\u4e8c\u7ef4\u6570\u7ec4\u4f1a\u6709\u8fd9\u4e2a\u95ee\u9898\uff0c\u4f8b\u5982\u521d\u59cb\u5316\u4e8c\u7ef4\u5217\u8868
res = [[0] * self.size()]\uff0c\u5219\u591a\u6b21\u5f15\u7528\u4e86\u540c\u4e00\u4e2a\u5217\u8868[0]\u3002\u5728\u5220\u9664\u8282\u70b9\u4e2d\uff0c\u9700\u8981\u65ad\u5f00\u8be5\u8282\u70b9\u4e0e\u5176\u540e\u7ee7\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u6307\u5411\u5417\uff1f
\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\uff08\u505a\u9898\uff09\u7684\u89d2\u5ea6\u770b\uff0c\u4e0d\u65ad\u5f00\u6ca1\u6709\u5173\u7cfb\uff0c\u53ea\u8981\u4fdd\u8bc1\u7a0b\u5e8f\u7684\u903b\u8f91\u662f\u6b63\u786e\u7684\u5c31\u884c\u3002\u4ece\u6807\u51c6\u5e93\u7684\u89d2\u5ea6\u770b\uff0c\u65ad\u5f00\u66f4\u52a0\u5b89\u5168\u3001\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3002\u5982\u679c\u4e0d\u65ad\u5f00\uff0c\u5047\u8bbe\u88ab\u5220\u9664\u8282\u70b9\u672a\u88ab\u6b63\u5e38\u56de\u6536\uff0c\u90a3\u4e48\u5b83\u4f1a\u5f71\u54cd\u540e\u7ee7\u8282\u70b9\u7684\u5185\u5b58\u56de\u6536\u3002
"},{"location":"chapter_backtracking/","title":"\u7b2c 13 \u7ae0 \u00a0 \u56de\u6eaf","text":"Abstract
\u6211\u4eec\u5982\u540c\u8ff7\u5bab\u4e2d\u7684\u63a2\u7d22\u8005\uff0c\u5728\u524d\u8fdb\u7684\u9053\u8def\u4e0a\u53ef\u80fd\u4f1a\u9047\u5230\u56f0\u96be\u3002
\u56de\u6eaf\u7684\u529b\u91cf\u8ba9\u6211\u4eec\u80fd\u591f\u91cd\u65b0\u5f00\u59cb\uff0c\u4e0d\u65ad\u5c1d\u8bd5\uff0c\u6700\u7ec8\u627e\u5230\u901a\u5f80\u5149\u660e\u7684\u51fa\u53e3\u3002
"},{"location":"chapter_backtracking/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5
- 13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898
- 13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898
- 13.4 \u00a0 N \u7687\u540e\u95ee\u9898
- 13.5 \u00a0 \u5c0f\u7ed3
\u300c\u56de\u6eaf\u7b97\u6cd5 backtracking algorithm\u300d\u662f\u4e00\u79cd\u901a\u8fc7\u7a77\u4e3e\u6765\u89e3\u51b3\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4ece\u4e00\u4e2a\u521d\u59cb\u72b6\u6001\u51fa\u53d1\uff0c\u66b4\u529b\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5f53\u9047\u5230\u6b63\u786e\u7684\u89e3\u5219\u5c06\u5176\u8bb0\u5f55\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5c1d\u8bd5\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u90fd\u65e0\u6cd5\u627e\u5230\u89e3\u4e3a\u6b62\u3002
\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u6765\u904d\u5386\u89e3\u7a7a\u95f4\u3002\u5728\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5229\u7528\u524d\u5e8f\u904d\u5386\u6784\u9020\u4e00\u4e2a\u56de\u6eaf\u95ee\u9898\uff0c\u9010\u6b65\u4e86\u89e3\u56de\u6eaf\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002
\u4f8b\u9898\u4e00
\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\uff0c\u641c\u7d22\u5e76\u8bb0\u5f55\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u8282\u70b9\u5217\u8868\u3002
\u5bf9\u4e8e\u6b64\u9898\uff0c\u6211\u4eec\u524d\u5e8f\u904d\u5386\u8fd9\u68f5\u6811\uff0c\u5e76\u5224\u65ad\u5f53\u524d\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \uff0c\u82e5\u662f\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u7684\u503c\u52a0\u5165\u7ed3\u679c\u5217\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_i_compact.pyres\u4e4b\u4e2d\u3002\u76f8\u5173\u8fc7\u7a0b\u5b9e\u73b0\u5982\u56fe 13-1 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff1a
preorder_traversal_i_compact.cppdef pre_order(root: TreeNode):\n \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00\"\"\"\n if root is None:\n return\n if root.val == 7:\n # \u8bb0\u5f55\u89e3\n res.append(root)\n pre_order(root.left)\n pre_order(root.right)\n
preorder_traversal_i_compact.java/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n if (root == nullptr) {\n return;\n }\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n res.push_back(root);\n }\n preOrder(root->left);\n preOrder(root->right);\n}\n
preorder_traversal_i_compact.cs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\n if (root == null) {\n return;\n }\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(root);\n }\n preOrder(root.left);\n preOrder(root.right);\n}\n
preorder_traversal_i_compact.go/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid PreOrder(TreeNode? root) {\n if (root == null) {\n return;\n }\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.Add(root);\n }\n PreOrder(root.left);\n PreOrder(root.right);\n}\n
preorder_traversal_i_compact.swift/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n if root == nil {\n return\n }\n if (root.Val).(int) == 7 {\n // \u8bb0\u5f55\u89e3\n *res = append(*res, root)\n }\n preOrderI(root.Left, res)\n preOrderI(root.Right, res)\n}\n
preorder_traversal_i_compact.js/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n if root.val == 7 {\n // \u8bb0\u5f55\u89e3\n res.append(root)\n }\n preOrder(root: root.left)\n preOrder(root: root.right)\n}\n
preorder_traversal_i_compact.ts/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root, res) {\n if (root === null) {\n return;\n }\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push(root);\n }\n preOrder(root.left, res);\n preOrder(root.right, res);\n}\n
preorder_traversal_i_compact.dart/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n if (root === null) {\n return;\n }\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push(root);\n }\n preOrder(root.left, res);\n preOrder(root.right, res);\n}\n
preorder_traversal_i_compact.rs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode? root, List<TreeNode> res) {\n if (root == null) {\n return;\n }\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(root);\n }\n preOrder(root.left, res);\n preOrder(root.right, res);\n}\n
preorder_traversal_i_compact.c/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfn pre_order(res: &mut Vec<Rc<RefCell<TreeNode>>>, root: Option<Rc<RefCell<TreeNode>>>) {\n if root.is_none() {\n return;\n }\n if let Some(node) = root {\n if node.borrow().val == 7 {\n // \u8bb0\u5f55\u89e3\n res.push(node.clone());\n }\n pre_order(res, node.borrow().left.clone());\n pre_order(res, node.borrow().right.clone());\n }\n}\n
preorder_traversal_i_compact.zig/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n if (root == NULL) {\n return;\n }\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n res[resSize++] = root;\n }\n preOrder(root->left);\n preOrder(root->right);\n}\n[class]{}-[func]{preOrder}\n\u56fe 13-1 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u4e2d\u641c\u7d22\u8282\u70b9
"},{"location":"chapter_backtracking/backtracking_algorithm/#1311","title":"13.1.1 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000","text":"\u4e4b\u6240\u4ee5\u79f0\u4e4b\u4e3a\u56de\u6eaf\u7b97\u6cd5\uff0c\u662f\u56e0\u4e3a\u8be5\u7b97\u6cd5\u5728\u641c\u7d22\u89e3\u7a7a\u95f4\u65f6\u4f1a\u91c7\u7528\u201c\u5c1d\u8bd5\u201d\u4e0e\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u5f53\u7b97\u6cd5\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u9047\u5230\u67d0\u4e2a\u72b6\u6001\u65e0\u6cd5\u7ee7\u7eed\u524d\u8fdb\u6216\u65e0\u6cd5\u5f97\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u65f6\uff0c\u5b83\u4f1a\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u5c1d\u8bd5\u5176\u4ed6\u53ef\u80fd\u7684\u9009\u62e9\u3002
\u5bf9\u4e8e\u4f8b\u9898\u4e00\uff0c\u8bbf\u95ee\u6bcf\u4e2a\u8282\u70b9\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5c1d\u8bd5\u201d\uff0c\u800c\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u8fd4\u56de\u7236\u8282\u70b9\u7684
return\u5219\u8868\u793a\u201c\u56de\u9000\u201d\u3002\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u56de\u9000\u5e76\u4e0d\u4ec5\u4ec5\u5305\u62ec\u51fd\u6570\u8fd4\u56de\u3002\u4e3a\u89e3\u91ca\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u5bf9\u4f8b\u9898\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002
\u4f8b\u9898\u4e8c
\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\u3002
\u5728\u4f8b\u9898\u4e00\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u5217\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_ii_compact.pypath\u8bb0\u5f55\u8bbf\u95ee\u8fc7\u7684\u8282\u70b9\u8def\u5f84\u3002\u5f53\u8bbf\u95ee\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u590d\u5236path\u5e76\u6dfb\u52a0\u8fdb\u7ed3\u679c\u5217\u8868res\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0cres\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
preorder_traversal_ii_compact.cppdef pre_order(root: TreeNode):\n \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c\"\"\"\n if root is None:\n return\n # \u5c1d\u8bd5\n path.append(root)\n if root.val == 7:\n # \u8bb0\u5f55\u89e3\n res.append(list(path))\n pre_order(root.left)\n pre_order(root.right)\n # \u56de\u9000\n path.pop()\n
preorder_traversal_ii_compact.java/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n if (root == nullptr) {\n return;\n }\n // \u5c1d\u8bd5\n path.push_back(root);\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n res.push_back(path);\n }\n preOrder(root->left);\n preOrder(root->right);\n // \u56de\u9000\n path.pop_back();\n}\n
preorder_traversal_ii_compact.cs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\n if (root == null) {\n return;\n }\n // \u5c1d\u8bd5\n path.add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(new ArrayList<>(path));\n }\n preOrder(root.left);\n preOrder(root.right);\n // \u56de\u9000\n path.remove(path.size() - 1);\n}\n
preorder_traversal_ii_compact.go/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid PreOrder(TreeNode? root) {\n if (root == null) {\n return;\n }\n // \u5c1d\u8bd5\n path.Add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.Add(new List<TreeNode>(path));\n }\n PreOrder(root.left);\n PreOrder(root.right);\n // \u56de\u9000\n path.RemoveAt(path.Count - 1);\n}\n
preorder_traversal_ii_compact.swift/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n if root == nil {\n return\n }\n // \u5c1d\u8bd5\n *path = append(*path, root)\n if root.Val.(int) == 7 {\n // \u8bb0\u5f55\u89e3\n *res = append(*res, append([]*TreeNode{}, *path...))\n }\n preOrderII(root.Left, res, path)\n preOrderII(root.Right, res, path)\n // \u56de\u9000\n *path = (*path)[:len(*path)-1]\n}\n
preorder_traversal_ii_compact.js/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n // \u5c1d\u8bd5\n path.append(root)\n if root.val == 7 {\n // \u8bb0\u5f55\u89e3\n res.append(path)\n }\n preOrder(root: root.left)\n preOrder(root: root.right)\n // \u56de\u9000\n path.removeLast()\n}\n
preorder_traversal_ii_compact.ts/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(root, path, res) {\n if (root === null) {\n return;\n }\n // \u5c1d\u8bd5\n path.push(root);\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push([...path]);\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.pop();\n}\n
preorder_traversal_ii_compact.dart/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(\n root: TreeNode | null,\n path: TreeNode[],\n res: TreeNode[][]\n): void {\n if (root === null) {\n return;\n }\n // \u5c1d\u8bd5\n path.push(root);\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push([...path]);\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.pop();\n}\n
preorder_traversal_ii_compact.rs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(\n TreeNode? root,\n List<TreeNode> path,\n List<List<TreeNode>> res,\n) {\n if (root == null) {\n return;\n }\n\n // \u5c1d\u8bd5\n path.add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(List.from(path));\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.removeLast();\n}\n
preorder_traversal_ii_compact.c/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfn pre_order(res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>, path: &mut Vec<Rc<RefCell<TreeNode>>>, root: Option<Rc<RefCell<TreeNode>>>) {\n if root.is_none() {\n return;\n }\n if let Some(node) = root {\n // \u5c1d\u8bd5\n path.push(node.clone());\n if node.borrow().val == 7 {\n // \u8bb0\u5f55\u89e3\n res.push(path.clone());\n }\n pre_order(res, path, node.borrow().left.clone());\n pre_order(res, path, node.borrow().right.clone());\n // \u56de\u9000\n path.remove(path.len() - 1);\n }\n}\n
preorder_traversal_ii_compact.zig/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n if (root == NULL) {\n return;\n }\n // \u5c1d\u8bd5\n path[pathSize++] = root;\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n for (int i = 0; i < pathSize; ++i) {\n res[resSize][i] = path[i];\n }\n resSize++;\n }\n preOrder(root->left);\n preOrder(root->right);\n // \u56de\u9000\n pathSize--;\n}\n[class]{}-[func]{preOrder}\n\u5728\u6bcf\u6b21\u201c\u5c1d\u8bd5\u201d\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u5f53\u524d\u8282\u70b9\u6dfb\u52a0\u8fdb
path\u6765\u8bb0\u5f55\u8def\u5f84\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u4eec\u9700\u8981\u5c06\u8be5\u8282\u70b9\u4ecepath\u4e2d\u5f39\u51fa\uff0c\u4ee5\u6062\u590d\u672c\u6b21\u5c1d\u8bd5\u4e4b\u524d\u7684\u72b6\u6001\u3002\u89c2\u5bdf\u56fe 13-2 \u6240\u793a\u7684\u8fc7\u7a0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5c1d\u8bd5\u548c\u56de\u9000\u7406\u89e3\u4e3a\u201c\u524d\u8fdb\u201d\u4e0e\u201c\u64a4\u9500\u201d\uff0c\u4e24\u4e2a\u64cd\u4f5c\u4e92\u4e3a\u9006\u5411\u3002
<1><2><3><4><5><6><7><8><9><10><11>\u56fe 13-2 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000
"},{"location":"chapter_backtracking/backtracking_algorithm/#1312","title":"13.1.2 \u00a0 \u526a\u679d","text":"\u590d\u6742\u7684\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u4e00\u4e2a\u6216\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u7ea6\u675f\u6761\u4ef6\u901a\u5e38\u53ef\u7528\u4e8e\u201c\u526a\u679d\u201d\u3002
\u4f8b\u9898\u4e09
\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\uff0c\u5e76\u8981\u6c42\u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u3002
\u4e3a\u4e86\u6ee1\u8db3\u4ee5\u4e0a\u7ea6\u675f\u6761\u4ef6\uff0c\u6211\u4eec\u9700\u8981\u6dfb\u52a0\u526a\u679d\u64cd\u4f5c\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u82e5\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\uff0c\u5219\u63d0\u524d\u8fd4\u56de\uff0c\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_iii_compact.py
preorder_traversal_iii_compact.cppdef pre_order(root: TreeNode):\n \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09\"\"\"\n # \u526a\u679d\n if root is None or root.val == 3:\n return\n # \u5c1d\u8bd5\n path.append(root)\n if root.val == 7:\n # \u8bb0\u5f55\u89e3\n res.append(list(path))\n pre_order(root.left)\n pre_order(root.right)\n # \u56de\u9000\n path.pop()\n
preorder_traversal_iii_compact.java/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n // \u526a\u679d\n if (root == nullptr || root->val == 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.push_back(root);\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n res.push_back(path);\n }\n preOrder(root->left);\n preOrder(root->right);\n // \u56de\u9000\n path.pop_back();\n}\n
preorder_traversal_iii_compact.cs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n // \u526a\u679d\n if (root == null || root.val == 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(new ArrayList<>(path));\n }\n preOrder(root.left);\n preOrder(root.right);\n // \u56de\u9000\n path.remove(path.size() - 1);\n}\n
preorder_traversal_iii_compact.go/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid PreOrder(TreeNode? root) {\n // \u526a\u679d\n if (root == null || root.val == 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.Add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.Add(new List<TreeNode>(path));\n }\n PreOrder(root.left);\n PreOrder(root.right);\n // \u56de\u9000\n path.RemoveAt(path.Count - 1);\n}\n
preorder_traversal_iii_compact.swift/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n // \u526a\u679d\n if root == nil || root.Val == 3 {\n return\n }\n // \u5c1d\u8bd5\n *path = append(*path, root)\n if root.Val.(int) == 7 {\n // \u8bb0\u5f55\u89e3\n *res = append(*res, *path)\n }\n preOrderIII(root.Left, res, path)\n preOrderIII(root.Right, res, path)\n // \u56de\u9000\n *path = (*path)[:len(*path)-1]\n}\n
preorder_traversal_iii_compact.js/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrder(root: TreeNode?) {\n // \u526a\u679d\n guard let root = root, root.val != 3 else {\n return\n }\n // \u5c1d\u8bd5\n path.append(root)\n if root.val == 7 {\n // \u8bb0\u5f55\u89e3\n res.append(path)\n }\n preOrder(root: root.left)\n preOrder(root: root.right)\n // \u56de\u9000\n path.removeLast()\n}\n
preorder_traversal_iii_compact.ts/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(root, path, res) {\n // \u526a\u679d\n if (root === null || root.val === 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.push(root);\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push([...path]);\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.pop();\n}\n
preorder_traversal_iii_compact.dart/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(\n root: TreeNode | null,\n path: TreeNode[],\n res: TreeNode[][]\n): void {\n // \u526a\u679d\n if (root === null || root.val === 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.push(root);\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push([...path]);\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.pop();\n}\n
preorder_traversal_iii_compact.rs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(\n TreeNode? root,\n List<TreeNode> path,\n List<List<TreeNode>> res,\n) {\n if (root == null || root.val == 3) {\n return;\n }\n\n // \u5c1d\u8bd5\n path.add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(List.from(path));\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.removeLast();\n}\n
preorder_traversal_iii_compact.c/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfn pre_order(res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>, path: &mut Vec<Rc<RefCell<TreeNode>>>, root: Option<Rc<RefCell<TreeNode>>>) {\n // \u526a\u679d\n if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n return;\n }\n if let Some(node) = root {\n // \u5c1d\u8bd5\n path.push(node.clone());\n if node.borrow().val == 7 {\n // \u8bb0\u5f55\u89e3\n res.push(path.clone());\n }\n pre_order(res, path, node.borrow().left.clone());\n pre_order(res, path, node.borrow().right.clone());\n // \u56de\u9000\n path.remove(path.len() - 1);\n }\n}\n
preorder_traversal_iii_compact.zig/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n // \u526a\u679d\n if (root == NULL || root->val == 3) {\n return;\n }\n // \u5c1d\u8bd5\n path[pathSize++] = root;\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n for (int i = 0; i < pathSize; i++) {\n res[resSize][i] = path[i];\n }\n resSize++;\n }\n preOrder(root->left);\n preOrder(root->right);\n // \u56de\u9000\n pathSize--;\n}\n[class]{}-[func]{preOrder}\n\u201c\u526a\u679d\u201d\u662f\u4e00\u4e2a\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8bcd\u3002\u5982\u56fe 13-3 \u6240\u793a\uff0c\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u641c\u7d22\u5206\u652f\uff0c\u907f\u514d\u8bb8\u591a\u65e0\u610f\u4e49\u7684\u5c1d\u8bd5\uff0c\u4ece\u800c\u63d0\u9ad8\u4e86\u641c\u7d22\u6548\u7387\u3002
\u56fe 13-3 \u00a0 \u6839\u636e\u7ea6\u675f\u6761\u4ef6\u526a\u679d
"},{"location":"chapter_backtracking/backtracking_algorithm/#1313","title":"13.1.3 \u00a0 \u6846\u67b6\u4ee3\u7801","text":"\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c1d\u8bd5\u5c06\u56de\u6eaf\u7684\u201c\u5c1d\u8bd5\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u4f53\u6846\u67b6\u63d0\u70bc\u51fa\u6765\uff0c\u63d0\u5347\u4ee3\u7801\u7684\u901a\u7528\u6027\u3002
\u5728\u4ee5\u4e0b\u6846\u67b6\u4ee3\u7801\u4e2d\uff0c
PythonC++JavaC#GoSwiftJSTSDartRustCZigstate\u8868\u793a\u95ee\u9898\u7684\u5f53\u524d\u72b6\u6001\uff0cchoices\u8868\u793a\u5f53\u524d\u72b6\u6001\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9009\u62e9\uff1adef backtrack(state: State, choices: list[choice], res: list[state]):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\"\"\"\n # \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if is_solution(state):\n # \u8bb0\u5f55\u89e3\n record_solution(state, res)\n # \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices:\n # \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if is_valid(state, choice):\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n make_choice(state, choice)\n backtrack(state, choices, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undo_choice(state, choice)\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector<Choice *> &choices, vector<State *> &res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (Choice choice : choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice> choices, List<State> res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (Choice choice : choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid Backtrack(State state, List<Choice> choices, List<State> res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (IsSolution(state)) {\n // \u8bb0\u5f55\u89e3\n RecordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n foreach (Choice choice in choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (IsValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n MakeChoice(state, choice);\n Backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n UndoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if isSolution(state) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res)\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for _, choice := range choices {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if isValid(state, choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice)\n backtrack(state, choices, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice)\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if isSolution(state: state) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state: state, res: &res)\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if isValid(state: state, choice: choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state: &state, choice: choice)\n backtrack(state: &state, choices: choices, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state: &state, choice: choice)\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (let choice of choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (let choice of choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice>, List<State> res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (Choice choice in choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfn backtrack(state: &mut State, choices: &Vec<Choice>, res: &mut Vec<State>) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if is_solution(state) {\n // \u8bb0\u5f55\u89e3\n record_solution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if is_valid(state, choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n make_choice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undo_choice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res, numRes);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < numChoices; i++) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, &choices[i])) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, &choices[i]);\n backtrack(state, choices, numChoices, res, numRes);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, &choices[i]);\n }\n }\n}\n\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u57fa\u4e8e\u6846\u67b6\u4ee3\u7801\u6765\u89e3\u51b3\u4f8b\u9898\u4e09\u3002\u72b6\u6001
PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_iii_template.pystate\u4e3a\u8282\u70b9\u904d\u5386\u8def\u5f84\uff0c\u9009\u62e9choices\u4e3a\u5f53\u524d\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\uff0c\u7ed3\u679cres\u662f\u8def\u5f84\u5217\u8868\uff1a
preorder_traversal_iii_template.cppdef is_solution(state: list[TreeNode]) -> bool:\n \"\"\"\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3\"\"\"\n return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n \"\"\"\u8bb0\u5f55\u89e3\"\"\"\n res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -> bool:\n \"\"\"\u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5\"\"\"\n return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n \"\"\"\u66f4\u65b0\u72b6\u6001\"\"\"\n state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n \"\"\"\u6062\u590d\u72b6\u6001\"\"\"\n state.pop()\n\ndef backtrack(\n state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09\"\"\"\n # \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if is_solution(state):\n # \u8bb0\u5f55\u89e3\n record_solution(state, res)\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices:\n # \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if is_valid(state, choice):\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n make_choice(state, choice)\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, [choice.left, choice.right], res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undo_choice(state, choice)\n
preorder_traversal_iii_template.java/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(vector<TreeNode *> &state) {\n return !state.empty() && state.back()->val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(vector<TreeNode *> &state, vector<vector<TreeNode *>> &res) {\n res.push_back(state);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector<TreeNode *> &state, TreeNode *choice) {\n return choice != nullptr && choice->val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(vector<TreeNode *> &state, TreeNode *choice) {\n state.push_back(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(vector<TreeNode *> &state, TreeNode *choice) {\n state.pop_back();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(vector<TreeNode *> &state, vector<TreeNode *> &choices, vector<vector<TreeNode *>> &res) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (TreeNode *choice : choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n vector<TreeNode *> nextChoices{choice->left, choice->right};\n backtrack(state, nextChoices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n
preorder_traversal_iii_template.cs/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nboolean isSolution(List<TreeNode> state) {\n return !state.isEmpty() && state.get(state.size() - 1).val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\n res.add(new ArrayList<>(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List<TreeNode> state, TreeNode choice) {\n return choice != null && choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List<TreeNode> state, TreeNode choice) {\n state.add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List<TreeNode> state, TreeNode choice) {\n state.remove(state.size() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (TreeNode choice : choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, Arrays.asList(choice.left, choice.right), res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n
preorder_traversal_iii_template.go/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool IsSolution(List<TreeNode> state) {\n return state.Count != 0 && state[^1].val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid RecordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\n res.Add(new List<TreeNode>(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool IsValid(List<TreeNode> state, TreeNode choice) {\n return choice != null && choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid MakeChoice(List<TreeNode> state, TreeNode choice) {\n state.Add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid UndoChoice(List<TreeNode> state, TreeNode choice) {\n state.RemoveAt(state.Count - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid Backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (IsSolution(state)) {\n // \u8bb0\u5f55\u89e3\n RecordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n foreach (TreeNode choice in choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (IsValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n MakeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, [choice.left!, choice.right!], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n UndoChoice(state, choice);\n }\n }\n}\n
preorder_traversal_iii_template.swift/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\n return len(*state) != 0 && (*state)[len(*state)-1].Val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n *res = append(*res, *state)\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n return choice != nil && choice.Val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n *state = append(*state, choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n *state = (*state)[:len(*state)-1]\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if isSolution(state) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res)\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for _, choice := range *choices {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if isValid(state, choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n temp := make([]*TreeNode, 0)\n temp = append(temp, choice.Left, choice.Right)\n backtrackIII(state, &temp, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice)\n }\n }\n}\n
preorder_traversal_iii_template.js/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state: [TreeNode]) -> Bool {\n !state.isEmpty && state.last!.val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n res.append(state)\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -> Bool {\n choice != nil && choice!.val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n state.append(choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n state.removeLast()\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if isSolution(state: state) {\n recordSolution(state: state, res: &res)\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if isValid(state: state, choice: choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state: &state, choice: choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, choices: [choice.left, choice.right].compactMap { $0 }, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state: &state, choice: choice)\n }\n }\n}\n
preorder_traversal_iii_template.ts/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state) {\n return state && state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state, res) {\n res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\n return choice !== null && choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state, choice) {\n state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state) {\n state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(state, choices, res) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (const choice of choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, [choice.left, choice.right], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state);\n }\n }\n}\n
preorder_traversal_iii_template.dart/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\n return state && state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n return choice !== null && choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state: TreeNode[]): void {\n state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(\n state: TreeNode[],\n choices: TreeNode[],\n res: TreeNode[][]\n): void {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (const choice of choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, [choice.left, choice.right], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state);\n }\n }\n}\n
preorder_traversal_iii_template.rs/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(List<TreeNode> state) {\n return state.isNotEmpty && state.last.val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\n res.add(List.from(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(List<TreeNode> state, TreeNode? choice) {\n return choice != null && choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List<TreeNode> state, TreeNode? choice) {\n state.add(choice!);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List<TreeNode> state, TreeNode? choice) {\n state.removeLast();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(\n List<TreeNode> state,\n List<TreeNode?> choices,\n List<List<TreeNode>> res,\n) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (TreeNode? choice in choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, [choice!.left, choice.right], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n
preorder_traversal_iii_template.c/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfn is_solution(state: &mut Vec<Rc<RefCell<TreeNode>>>) -> bool {\n return !state.is_empty() && state.get(state.len() - 1).unwrap().borrow().val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfn record_solution(state: &mut Vec<Rc<RefCell<TreeNode>>>, res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>) {\n res.push(state.clone());\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfn is_valid(_: &mut Vec<Rc<RefCell<TreeNode>>>, choice: Rc<RefCell<TreeNode>>) -> bool {\n return choice.borrow().val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfn make_choice(state: &mut Vec<Rc<RefCell<TreeNode>>>, choice: Rc<RefCell<TreeNode>>) {\n state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfn undo_choice(state: &mut Vec<Rc<RefCell<TreeNode>>>, _: Rc<RefCell<TreeNode>>) {\n state.remove(state.len() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfn backtrack(state: &mut Vec<Rc<RefCell<TreeNode>>>, choices: &mut Vec<Rc<RefCell<TreeNode>>>, res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if is_solution(state) {\n // \u8bb0\u5f55\u89e3\n record_solution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if is_valid(state, choice.clone()) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n make_choice(state, choice.clone());\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, &mut vec![choice.borrow().left.clone().unwrap(), choice.borrow().right.clone().unwrap()], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undo_choice(state, choice.clone());\n }\n }\n}\n
preorder_traversal_iii_template.zig/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(void) {\n return pathSize > 0 && path[pathSize - 1]->val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(void) {\n for (int i = 0; i < pathSize; i++) {\n res[resSize][i] = path[i];\n }\n resSize++;\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(TreeNode *choice) {\n return choice != NULL && choice->val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(TreeNode *choice) {\n path[pathSize++] = choice;\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(void) {\n pathSize--;\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(TreeNode *choices[2]) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution()) {\n // \u8bb0\u5f55\u89e3\n recordSolution();\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < 2; i++) {\n TreeNode *choice = choices[i];\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n TreeNode *nextChoices[2] = {choice->left, choice->right};\n backtrack(nextChoices);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice();\n }\n }\n}\n[class]{}-[func]{isSolution}\n\n[class]{}-[func]{recordSolution}\n\n[class]{}-[func]{isValid}\n\n[class]{}-[func]{makeChoice}\n\n[class]{}-[func]{undoChoice}\n\n[class]{}-[func]{backtrack}\n\u6839\u636e\u9898\u610f\uff0c\u6211\u4eec\u5728\u627e\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u540e\u5e94\u8be5\u7ee7\u7eed\u641c\u7d22\uff0c\u56e0\u6b64\u9700\u8981\u5c06\u8bb0\u5f55\u89e3\u4e4b\u540e\u7684
return\u8bed\u53e5\u5220\u9664\u3002\u56fe 13-4 \u5bf9\u6bd4\u4e86\u4fdd\u7559\u6216\u5220\u9664return\u8bed\u53e5\u7684\u641c\u7d22\u8fc7\u7a0b\u3002\u56fe 13-4 \u00a0 \u4fdd\u7559\u4e0e\u5220\u9664 return \u7684\u641c\u7d22\u8fc7\u7a0b\u5bf9\u6bd4
\u76f8\u6bd4\u57fa\u4e8e\u524d\u5e8f\u904d\u5386\u7684\u4ee3\u7801\u5b9e\u73b0\uff0c\u57fa\u4e8e\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\u7684\u4ee3\u7801\u5b9e\u73b0\u867d\u7136\u663e\u5f97\u5570\u5506\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u56de\u6eaf\u95ee\u9898\u53ef\u4ee5\u5728\u8be5\u6846\u67b6\u4e0b\u89e3\u51b3\u3002\u6211\u4eec\u53ea\u9700\u6839\u636e\u5177\u4f53\u95ee\u9898\u6765\u5b9a\u4e49
"},{"location":"chapter_backtracking/backtracking_algorithm/#1314","title":"13.1.4 \u00a0 \u5e38\u7528\u672f\u8bed","text":"state\u548cchoices\uff0c\u5e76\u5b9e\u73b0\u6846\u67b6\u4e2d\u7684\u5404\u4e2a\u65b9\u6cd5\u5373\u53ef\u3002\u4e3a\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u7b97\u6cd5\u95ee\u9898\uff0c\u6211\u4eec\u603b\u7ed3\u4e00\u4e0b\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5e38\u7528\u672f\u8bed\u7684\u542b\u4e49\uff0c\u5e76\u5bf9\u7167\u4f8b\u9898\u4e09\u7ed9\u51fa\u5bf9\u5e94\u793a\u4f8b\uff0c\u5982\u8868 13-1 \u6240\u793a\u3002
\u8868 13-1 \u00a0 \u5e38\u89c1\u7684\u56de\u6eaf\u7b97\u6cd5\u672f\u8bed
\u540d\u8bcd \u5b9a\u4e49 \u4f8b\u9898\u4e09 \u89e3\uff08solution\uff09 \u89e3\u662f\u6ee1\u8db3\u95ee\u9898\u7279\u5b9a\u6761\u4ef6\u7684\u7b54\u6848\uff0c\u53ef\u80fd\u6709\u4e00\u4e2a\u6216\u591a\u4e2a \u6839\u8282\u70b9\u5230\u8282\u70b9 \\(7\\) \u7684\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u6240\u6709\u8def\u5f84 \u7ea6\u675f\u6761\u4ef6\uff08constraint\uff09 \u7ea6\u675f\u6761\u4ef6\u662f\u95ee\u9898\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u6761\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u526a\u679d \u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u8282\u70b9 \\(3\\) \u72b6\u6001\uff08state\uff09 \u72b6\u6001\u8868\u793a\u95ee\u9898\u5728\u67d0\u4e00\u65f6\u523b\u7684\u60c5\u51b5\uff0c\u5305\u62ec\u5df2\u7ecf\u505a\u51fa\u7684\u9009\u62e9 \u5f53\u524d\u5df2\u8bbf\u95ee\u7684\u8282\u70b9\u8def\u5f84\uff0c\u5373path\u8282\u70b9\u5217\u8868 \u5c1d\u8bd5\uff08attempt\uff09 \u5c1d\u8bd5\u662f\u6839\u636e\u53ef\u7528\u9009\u62e9\u6765\u63a2\u7d22\u89e3\u7a7a\u95f4\u7684\u8fc7\u7a0b\uff0c\u5305\u62ec\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\uff0c\u68c0\u67e5\u662f\u5426\u4e3a\u89e3 \u9012\u5f52\u8bbf\u95ee\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\uff0c\u5c06\u8282\u70b9\u6dfb\u52a0\u8fdbpath\uff0c\u5224\u65ad\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \u56de\u9000\uff08backtracking\uff09 \u56de\u9000\u6307\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u72b6\u6001\u65f6\uff0c\u64a4\u9500\u524d\u9762\u505a\u51fa\u7684\u9009\u62e9\uff0c\u56de\u5230\u4e0a\u4e00\u4e2a\u72b6\u6001 \u5f53\u8d8a\u8fc7\u53f6\u8282\u70b9\u3001\u7ed3\u675f\u8282\u70b9\u8bbf\u95ee\u3001\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\u7ec8\u6b62\u641c\u7d22\uff0c\u51fd\u6570\u8fd4\u56de \u526a\u679d\uff08pruning\uff09 \u526a\u679d\u662f\u6839\u636e\u95ee\u9898\u7279\u6027\u548c\u7ea6\u675f\u6761\u4ef6\u907f\u514d\u65e0\u610f\u4e49\u7684\u641c\u7d22\u8def\u5f84\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u7d22\u6548\u7387 \u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22Tip
\u95ee\u9898\u3001\u89e3\u3001\u72b6\u6001\u7b49\u6982\u5ff5\u662f\u901a\u7528\u7684\uff0c\u5728\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u4e2d\u90fd\u6709\u6d89\u53ca\u3002
"},{"location":"chapter_backtracking/backtracking_algorithm/#1315","title":"13.1.5 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u7b97\u6cd5\uff0c\u5b83\u5c1d\u8bd5\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\u76f4\u5230\u627e\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u4f18\u70b9\u5728\u4e8e\u80fd\u591f\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002
\u7136\u800c\uff0c\u5728\u5904\u7406\u5927\u89c4\u6a21\u6216\u8005\u590d\u6742\u95ee\u9898\u65f6\uff0c\u56de\u6eaf\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u53ef\u80fd\u96be\u4ee5\u63a5\u53d7\u3002
- \u65f6\u95f4\uff1a\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u9700\u8981\u904d\u5386\u72b6\u6001\u7a7a\u95f4\u7684\u6240\u6709\u53ef\u80fd\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u6307\u6570\u9636\u6216\u9636\u4e58\u9636\u3002
- \u7a7a\u95f4\uff1a\u5728\u9012\u5f52\u8c03\u7528\u4e2d\u9700\u8981\u4fdd\u5b58\u5f53\u524d\u7684\u72b6\u6001\uff08\u4f8b\u5982\u8def\u5f84\u3001\u7528\u4e8e\u526a\u679d\u7684\u8f85\u52a9\u53d8\u91cf\u7b49\uff09\uff0c\u5f53\u6df1\u5ea6\u5f88\u5927\u65f6\uff0c\u7a7a\u95f4\u9700\u6c42\u53ef\u80fd\u4f1a\u53d8\u5f97\u5f88\u5927\u3002
\u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u7684\u6700\u4f73\u89e3\u51b3\u65b9\u6848\u3002\u5bf9\u4e8e\u8fd9\u4e9b\u95ee\u9898\uff0c\u7531\u4e8e\u65e0\u6cd5\u9884\u6d4b\u54ea\u4e9b\u9009\u62e9\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u8fdb\u884c\u904d\u5386\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5173\u952e\u662f\u5982\u4f55\u4f18\u5316\u6548\u7387\uff0c\u5e38\u89c1\u7684\u6548\u7387\u4f18\u5316\u65b9\u6cd5\u6709\u4e24\u79cd\u3002
- \u526a\u679d\uff1a\u907f\u514d\u641c\u7d22\u90a3\u4e9b\u80af\u5b9a\u4e0d\u4f1a\u4ea7\u751f\u89e3\u7684\u8def\u5f84\uff0c\u4ece\u800c\u8282\u7701\u65f6\u95f4\u548c\u7a7a\u95f4\u3002
- \u542f\u53d1\u5f0f\u641c\u7d22\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8ba1\u503c\uff0c\u4ece\u800c\u4f18\u5148\u641c\u7d22\u6700\u6709\u53ef\u80fd\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\u3002
\u56de\u6eaf\u7b97\u6cd5\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u641c\u7d22\u95ee\u9898\u3001\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u548c\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u3002
\u641c\u7d22\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u89e3\u51b3\u65b9\u6848\u3002
- \u5168\u6392\u5217\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7ec4\u5408\u3002
- \u5b50\u96c6\u548c\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\u548c\u4e00\u4e2a\u76ee\u6807\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u4e3a\u76ee\u6807\u548c\u7684\u5b50\u96c6\u3002
- \u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5706\u76d8\uff0c\u8981\u6c42\u5c06\u6240\u6709\u5706\u76d8\u4ece\u4e00\u6839\u67f1\u5b50\u79fb\u52a8\u5230\u53e6\u4e00\u6839\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\uff0c\u4e14\u4e0d\u80fd\u5c06\u5927\u5706\u76d8\u653e\u5728\u5c0f\u5706\u76d8\u4e0a\u3002
\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u6240\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u89e3\u3002
- \\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e \\(n\\) \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u4e92\u4e0d\u653b\u51fb\u3002
- \u6570\u72ec\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7f51\u683c\u4e2d\u586b\u5165\u6570\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u4e2a \\(3 \\times 3\\) \u5b50\u7f51\u683c\u4e2d\u7684\u6570\u5b57\u4e0d\u91cd\u590d\u3002
- \u56fe\u7740\u8272\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u7528\u6700\u5c11\u7684\u989c\u8272\u7ed9\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u7740\u8272\uff0c\u4f7f\u5f97\u76f8\u90bb\u9876\u70b9\u989c\u8272\u4e0d\u540c\u3002
\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u4e00\u4e2a\u7ec4\u5408\u7a7a\u95f4\u4e2d\u627e\u5230\u6ee1\u8db3\u67d0\u4e9b\u6761\u4ef6\u7684\u6700\u4f18\u89e3\u3002
- 0-1 \u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u80cc\u5305\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u4ef7\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5185\uff0c\u9009\u62e9\u7269\u54c1\u4f7f\u5f97\u603b\u4ef7\u503c\u6700\u5927\u3002
- \u65c5\u884c\u5546\u95ee\u9898\uff1a\u5728\u4e00\u4e2a\u56fe\u4e2d\uff0c\u4ece\u4e00\u4e2a\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u6240\u6709\u5176\u4ed6\u70b9\u6070\u597d\u4e00\u6b21\u540e\u8fd4\u56de\u8d77\u70b9\uff0c\u6c42\u6700\u77ed\u8def\u5f84\u3002
- \u6700\u5927\u56e2\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u56fe\uff0c\u5373\u5b50\u56fe\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u90fd\u6709\u8fb9\u76f8\u8fde\u3002
\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff0c\u56de\u6eaf\u4e0d\u662f\u6700\u4f18\u89e3\u51b3\u65b9\u6848\u3002
- 0-1 \u80cc\u5305\u95ee\u9898\u901a\u5e38\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\uff0c\u4ee5\u8fbe\u5230\u66f4\u9ad8\u7684\u65f6\u95f4\u6548\u7387\u3002
- \u65c5\u884c\u5546\u662f\u4e00\u4e2a\u8457\u540d\u7684 NP-Hard \u95ee\u9898\uff0c\u5e38\u7528\u89e3\u6cd5\u6709\u9057\u4f20\u7b97\u6cd5\u548c\u8681\u7fa4\u7b97\u6cd5\u7b49\u3002
- \u6700\u5927\u56e2\u95ee\u9898\u662f\u56fe\u8bba\u4e2d\u7684\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\uff0c\u53ef\u7528\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u542f\u53d1\u5f0f\u7b97\u6cd5\u6765\u89e3\u51b3\u3002
Question
\u6839\u636e\u56fd\u9645\u8c61\u68cb\u7684\u89c4\u5219\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u51fb\u4e0e\u540c\u5904\u4e00\u884c\u3001\u4e00\u5217\u6216\u4e00\u6761\u659c\u7ebf\u4e0a\u7684\u68cb\u5b50\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u7687\u540e\u548c\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5bfb\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u95f4\u65e0\u6cd5\u76f8\u4e92\u653b\u51fb\u7684\u6446\u653e\u65b9\u6848\u3002
\u5982\u56fe 13-15 \u6240\u793a\uff0c\u5f53 \\(n = 4\\) \u65f6\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u4e24\u4e2a\u89e3\u3002\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\u5171\u6709 \\(n^2\\) \u4e2a\u683c\u5b50\uff0c\u7ed9\u51fa\u4e86\u6240\u6709\u7684\u9009\u62e9
choices\u3002\u5728\u9010\u4e2a\u653e\u7f6e\u7687\u540e\u7684\u8fc7\u7a0b\u4e2d\uff0c\u68cb\u76d8\u72b6\u6001\u5728\u4e0d\u65ad\u5730\u53d8\u5316\uff0c\u6bcf\u4e2a\u65f6\u523b\u7684\u68cb\u76d8\u5c31\u662f\u72b6\u6001state\u3002\u56fe 13-15 \u00a0 4 \u7687\u540e\u95ee\u9898\u7684\u89e3
\u56fe 13-16 \u5c55\u793a\u4e86\u672c\u9898\u7684\u4e09\u4e2a\u7ea6\u675f\u6761\u4ef6\uff1a\u591a\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u3001\u540c\u4e00\u5bf9\u89d2\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u89d2\u7ebf\u5206\u4e3a\u4e3b\u5bf9\u89d2\u7ebf
\\\u548c\u6b21\u5bf9\u89d2\u7ebf/\u4e24\u79cd\u3002\u56fe 13-16 \u00a0 n \u7687\u540e\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6
"},{"location":"chapter_backtracking/n_queens_problem/#1","title":"1. \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565","text":"\u7687\u540e\u7684\u6570\u91cf\u548c\u68cb\u76d8\u7684\u884c\u6570\u90fd\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u6211\u4eec\u5bb9\u6613\u5f97\u5230\u4e00\u4e2a\u63a8\u8bba\uff1a\u68cb\u76d8\u6bcf\u884c\u90fd\u5141\u8bb8\u4e14\u53ea\u5141\u8bb8\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002
\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\uff0c\u76f4\u81f3\u6700\u540e\u4e00\u884c\u7ed3\u675f\u3002
\u56fe 13-17 \u6240\u793a\u4e3a \\(4\\) \u7687\u540e\u95ee\u9898\u7684\u9010\u884c\u653e\u7f6e\u8fc7\u7a0b\u3002\u53d7\u753b\u5e45\u9650\u5236\uff0c\u56fe 13-17 \u4ec5\u5c55\u5f00\u4e86\u7b2c\u4e00\u884c\u7684\u5176\u4e2d\u4e00\u4e2a\u641c\u7d22\u5206\u652f\uff0c\u5e76\u4e14\u5c06\u4e0d\u6ee1\u8db3\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u65b9\u6848\u90fd\u8fdb\u884c\u4e86\u526a\u679d\u3002
\u56fe 13-17 \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565
\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9010\u884c\u653e\u7f6e\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73b0\u591a\u4e2a\u7687\u540e\u7684\u6240\u6709\u641c\u7d22\u5206\u652f\u3002
"},{"location":"chapter_backtracking/n_queens_problem/#2","title":"2. \u00a0 \u5217\u4e0e\u5bf9\u89d2\u7ebf\u526a\u679d","text":"\u4e3a\u4e86\u6ee1\u8db3\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5e03\u5c14\u578b\u6570\u7ec4
cols\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u51b3\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u4eec\u901a\u8fc7cols\u5c06\u5df2\u6709\u7687\u540e\u7684\u5217\u8fdb\u884c\u526a\u679d\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u52a8\u6001\u66f4\u65b0cols\u7684\u72b6\u6001\u3002\u90a3\u4e48\uff0c\u5982\u4f55\u5904\u7406\u5bf9\u89d2\u7ebf\u7ea6\u675f\u5462\uff1f\u8bbe\u68cb\u76d8\u4e2d\u67d0\u4e2a\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\((row, col)\\) \uff0c\u9009\u5b9a\u77e9\u9635\u4e2d\u7684\u67d0\u6761\u4e3b\u5bf9\u89d2\u7ebf\uff0c\u6211\u4eec\u53d1\u73b0\u8be5\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u51cf\u5217\u7d22\u5f15\u90fd\u76f8\u7b49\uff0c\u5373\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684 \\(row - col\\) \u4e3a\u6052\u5b9a\u503c\u3002
\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u4e24\u4e2a\u683c\u5b50\u6ee1\u8db3 \\(row_1 - col_1 = row_2 - col_2\\) \uff0c\u5219\u5b83\u4eec\u4e00\u5b9a\u5904\u5728\u540c\u4e00\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u3002\u5229\u7528\u8be5\u89c4\u5f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u56fe 13-18 \u6240\u793a\u7684\u6570\u7ec4
diags1\u8bb0\u5f55\u6bcf\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002\u540c\u7406\uff0c\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 \\(row + col\\) \u662f\u6052\u5b9a\u503c\u3002\u6211\u4eec\u540c\u6837\u4e5f\u53ef\u4ee5\u501f\u52a9\u6570\u7ec4
diags2\u6765\u5904\u7406\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u56fe 13-18 \u00a0 \u5904\u7406\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f
"},{"location":"chapter_backtracking/n_queens_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u8bf7\u6ce8\u610f\uff0c\\(n\\) \u7ef4\u65b9\u9635\u4e2d \\(row - col\\) \u7684\u8303\u56f4\u662f \\([-n + 1, n - 1]\\) \uff0c\\(row + col\\) \u7684\u8303\u56f4\u662f \\([0, 2n - 2]\\) \uff0c\u6240\u4ee5\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\u7684\u6570\u91cf\u90fd\u4e3a \\(2n - 1\\) \uff0c\u5373\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig n_queens.pydiags1\u548cdiags2\u7684\u957f\u5ea6\u90fd\u4e3a \\(2n - 1\\) \u3002
n_queens.cppdef backtrack(\n row: int,\n n: int,\n state: list[list[str]],\n res: list[list[list[str]]],\n cols: list[bool],\n diags1: list[bool],\n diags2: list[bool],\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e\"\"\"\n # \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n:\n res.append([list(row) for row in state])\n return\n # \u904d\u5386\u6240\u6709\u5217\n for col in range(n):\n # \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n diag1 = row - col + n - 1\n diag2 = row + col\n # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n # \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\"\n cols[col] = diags1[diag1] = diags2[diag2] = True\n # \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2)\n # \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\"\n cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -> list[list[list[str]]]:\n \"\"\"\u6c42\u89e3 N \u7687\u540e\"\"\"\n # \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n state = [[\"#\" for _ in range(n)] for _ in range(n)]\n cols = [False] * n # \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n diags1 = [False] * (2 * n - 1) # \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n diags2 = [False] * (2 * n - 1) # \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n res = []\n backtrack(0, n, state, res, cols, diags1, diags2)\n\n return res\n
n_queens.java/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, vector<vector<string>> &state, vector<vector<vector<string>>> &res, vector<bool> &cols,\n vector<bool> &diags1, vector<bool> &diags2) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\";\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\";\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nvector<vector<vector<string>>> nQueens(int n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n vector<vector<string>> state(n, vector<string>(n, \"#\"));\n vector<bool> cols(n, false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n vector<bool> diags1(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n vector<bool> diags2(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n vector<vector<vector<string>>> res;\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n\n return res;\n}\n
n_queens.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, List<List<String>> state, List<List<List<String>>> res,\n boolean[] cols, boolean[] diags1, boolean[] diags2) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n List<List<String>> copyState = new ArrayList<>();\n for (List<String> sRow : state) {\n copyState.add(new ArrayList<>(sRow));\n }\n res.add(copyState);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state.get(row).set(col, \"Q\");\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state.get(row).set(col, \"#\");\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<String>>> nQueens(int n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n List<List<String>> state = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n List<String> row = new ArrayList<>();\n for (int j = 0; j < n; j++) {\n row.add(\"#\");\n }\n state.add(row);\n }\n boolean[] cols = new boolean[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n boolean[] diags1 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n boolean[] diags2 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n List<List<List<String>>> res = new ArrayList<>();\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n\n return res;\n}\n
n_queens.go/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid Backtrack(int row, int n, List<List<string>> state, List<List<List<string>>> res,\n bool[] cols, bool[] diags1, bool[] diags2) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n List<List<string>> copyState = [];\n foreach (List<string> sRow in state) {\n copyState.Add(new List<string>(sRow));\n }\n res.Add(copyState);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\";\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\";\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<string>>> NQueens(int n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n List<List<string>> state = [];\n for (int i = 0; i < n; i++) {\n List<string> row = [];\n for (int j = 0; j < n; j++) {\n row.Add(\"#\");\n }\n state.Add(row);\n }\n bool[] cols = new bool[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n bool[] diags1 = new bool[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n bool[] diags2 = new bool[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n List<List<List<string>>> res = [];\n\n Backtrack(0, n, state, res, cols, diags1, diags2);\n\n return res;\n}\n
n_queens.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n {\n newState := make([][]string, len(*state))\n for i, _ := range newState {\n newState[i] = make([]string, len((*state)[0]))\n copy(newState[i], (*state)[i])\n\n }\n *res = append(*res, newState)\n }\n // \u904d\u5386\u6240\u6709\u5217\n for col := 0; col < n; col++ {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n diag1 := row - col + n - 1\n diag2 := row + col\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if !(*cols)[col] && !(*diags1)[diag1] && !(*diags2)[diag2] {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n (*state)[row][col] = \"Q\"\n (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row+1, n, state, res, cols, diags1, diags2)\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n (*state)[row][col] = \"#\"\n (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n }\n }\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n {\n newState := make([][]string, len(*state))\n for i, _ := range newState {\n newState[i] = make([]string, len((*state)[0]))\n copy(newState[i], (*state)[i])\n\n }\n *res = append(*res, newState)\n }\n // \u904d\u5386\u6240\u6709\u5217\n for col := 0; col < n; col++ {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n diag1 := row - col + n - 1\n diag2 := row + col\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if !(*cols)[col] && !(*diags1)[diag1] && !(*diags2)[diag2] {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n (*state)[row][col] = \"Q\"\n (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row+1, n, state, res, cols, diags1, diags2)\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n (*state)[row][col] = \"#\"\n (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n }\n }\n}\n\nfunc nQueens(n int) [][][]string {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n state := make([][]string, n)\n for i := 0; i < n; i++ {\n row := make([]string, n)\n for i := 0; i < n; i++ {\n row[i] = \"#\"\n }\n state[i] = row\n }\n // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n cols := make([]bool, n)\n diags1 := make([]bool, 2*n-1)\n diags2 := make([]bool, 2*n-1)\n res := make([][][]string, 0)\n backtrack(0, n, &state, &res, &cols, &diags1, &diags2)\n return res\n}\n
n_queens.js/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u5217\n for col in 0 ..< n {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n let diag1 = row - col + n - 1\n let diag2 = row + col\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if !cols[col] && !diags1[diag1] && !diags2[diag2] {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\"\n cols[col] = true\n diags1[diag1] = true\n diags2[diag2] = true\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row: row + 1, n: n, state: &state, res: &res, cols: &cols, diags1: &diags1, diags2: &diags2)\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\"\n cols[col] = false\n diags1[diag1] = false\n diags2[diag2] = false\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfunc nQueens(n: Int) -> [[[String]]] {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n var cols = Array(repeating: false, count: n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n var diags1 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n var diags2 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n var res: [[[String]]] = []\n\n backtrack(row: 0, n: n, state: &state, res: &res, cols: &cols, diags1: &diags1, diags2: &diags2)\n\n return res\n}\n
n_queens.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row === n) {\n res.push(state.map((row) => row.slice()));\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (let col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n const diag1 = row - col + n - 1;\n const diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = 'Q';\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = '#';\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n const state = Array.from({ length: n }, () => Array(n).fill('#'));\n const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n const res = [];\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n return res;\n}\n
n_queens.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(\n row: number,\n n: number,\n state: string[][],\n res: string[][][],\n cols: boolean[],\n diags1: boolean[],\n diags2: boolean[]\n): void {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row === n) {\n res.push(state.map((row) => row.slice()));\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (let col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n const diag1 = row - col + n - 1;\n const diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = 'Q';\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = '#';\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n const state = Array.from({ length: n }, () => Array(n).fill('#'));\n const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n const res: string[][][] = [];\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n return res;\n}\n
n_queens.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(\n int row,\n int n,\n List<List<String>> state,\n List<List<List<String>>> res,\n List<bool> cols,\n List<bool> diags1,\n List<bool> diags2,\n) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n List<List<String>> copyState = [];\n for (List<String> sRow in state) {\n copyState.add(List.from(sRow));\n }\n res.add(copyState);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\";\n cols[col] = true;\n diags1[diag1] = true;\n diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\";\n cols[col] = false;\n diags1[diag1] = false;\n diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<String>>> nQueens(int n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n List<List<String>> state = List.generate(n, (index) => List.filled(n, \"#\"));\n List<bool> cols = List.filled(n, false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n List<bool> diags1 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n List<bool> diags2 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n List<List<List<String>>> res = [];\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n\n return res;\n}\n
n_queens.c/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfn backtrack(row: usize, n: usize, state: &mut Vec<Vec<String>>, res: &mut Vec<Vec<Vec<String>>>,\n cols: &mut [bool], diags1: &mut [bool], diags2: &mut [bool]) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n {\n let mut copy_state: Vec<Vec<String>> = Vec::new();\n for s_row in state.clone() {\n copy_state.push(s_row);\n }\n res.push(copy_state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for col in 0..n {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n let diag1 = row + n - 1 - col;\n let diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if !cols[col] && !diags1[diag1] && !diags2[diag2] {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state.get_mut(row).unwrap()[col] = \"Q\".into();\n (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state.get_mut(row).unwrap()[col] = \"#\".into();\n (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfn n_queens(n: usize) -> Vec<Vec<Vec<String>>> {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n let mut state: Vec<Vec<String>> = Vec::new();\n for _ in 0..n {\n let mut row: Vec<String> = Vec::new();\n for _ in 0..n {\n row.push(\"#\".into());\n }\n state.push(row);\n }\n let mut cols = vec![false; n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n let mut diags1 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n let mut diags2 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n let mut res: Vec<Vec<Vec<String>>> = Vec::new();\n\n backtrack(0, n, &mut state, &mut res, &mut cols, &mut diags1, &mut diags2);\n\n res\n}\n
n_queens.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n res[*resSize] = (char **)malloc(sizeof(char *) * n);\n for (int i = 0; i < n; ++i) {\n res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n strcpy(res[*resSize][i], state[i]);\n }\n (*resSize)++;\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = 'Q';\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = '#';\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nchar ***nQueens(int n, int *returnSize) {\n char state[MAX_SIZE][MAX_SIZE];\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n for (int i = 0; i < n; ++i) {\n for (int j = 0; j < n; ++j) {\n state[i][j] = '#';\n }\n state[i][n] = '\\0';\n }\n bool cols[MAX_SIZE] = {false}; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n bool diags1[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n bool diags2[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n\n char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n *returnSize = 0;\n backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n return res;\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{nQueens}\n\u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u8651\u5217\u7ea6\u675f\uff0c\u5219\u4ece\u7b2c\u4e00\u884c\u5230\u6700\u540e\u4e00\u884c\u5206\u522b\u6709 \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u4e2a\u9009\u62e9\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!)\\) \u3002\u5b9e\u9645\u4e0a\uff0c\u6839\u636e\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u526a\u679d\u4e5f\u80fd\u591f\u5927\u5e45\u7f29\u5c0f\u641c\u7d22\u7a7a\u95f4\uff0c\u56e0\u800c\u641c\u7d22\u6548\u7387\u5f80\u5f80\u4f18\u4e8e\u4ee5\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u3002
\u6570\u7ec4
"},{"location":"chapter_backtracking/permutations_problem/","title":"13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898","text":"state\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff0c\u6570\u7ec4cols\u3001diags1\u548cdiags2\u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5168\u6392\u5217\u95ee\u9898\u662f\u56de\u6eaf\u7b97\u6cd5\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u7684\u5b9a\u4e49\u662f\u5728\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff08\u5982\u4e00\u4e2a\u6570\u7ec4\u6216\u5b57\u7b26\u4e32\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u627e\u51fa\u5176\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002
\u8868 13-2 \u5217\u4e3e\u4e86\u51e0\u4e2a\u793a\u4f8b\u6570\u636e\uff0c\u5305\u62ec\u8f93\u5165\u6570\u7ec4\u548c\u5bf9\u5e94\u7684\u6240\u6709\u6392\u5217\u3002
\u8868 13-2 \u00a0 \u5168\u6392\u5217\u793a\u4f8b
\u8f93\u5165\u6570\u7ec4 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1321","title":"13.2.1 \u00a0 \u65e0\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"Question
\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u5176\u4e2d\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002
\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\u3002\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u4eec\u5148\u9009\u62e9 \\(1\\) \uff0c\u518d\u9009\u62e9 \\(3\\) \uff0c\u6700\u540e\u9009\u62e9 \\(2\\) \uff0c\u5219\u83b7\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u9500\u4e00\u4e2a\u9009\u62e9\uff0c\u4e4b\u540e\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002
\u4ece\u56de\u6eaf\u4ee3\u7801\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9009\u96c6\u5408
choices\u662f\u8f93\u5165\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72b6\u6001state\u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9009\u62e9\u7684\u5143\u7d20\u3002\u8bf7\u6ce8\u610f\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u5141\u8bb8\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u56e0\u6b64state\u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u5e94\u8be5\u662f\u552f\u4e00\u7684\u3002\u5982\u56fe 13-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u641c\u7d22\u8fc7\u7a0b\u5c55\u5f00\u6210\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u5f53\u524d\u72b6\u6001
state\u3002\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u7ecf\u8fc7\u4e09\u8f6e\u9009\u62e9\u540e\u5230\u8fbe\u53f6\u8282\u70b9\uff0c\u6bcf\u4e2a\u53f6\u8282\u70b9\u90fd\u5bf9\u5e94\u4e00\u4e2a\u6392\u5217\u3002\u56fe 13-5 \u00a0 \u5168\u6392\u5217\u7684\u9012\u5f52\u6811
"},{"location":"chapter_backtracking/permutations_problem/#1","title":"1. \u00a0 \u91cd\u590d\u9009\u62e9\u526a\u679d","text":"\u4e3a\u4e86\u5b9e\u73b0\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u6211\u4eec\u8003\u8651\u5f15\u5165\u4e00\u4e2a\u5e03\u5c14\u578b\u6570\u7ec4
selected\uff0c\u5176\u4e2dselected[i]\u8868\u793achoices[i]\u662f\u5426\u5df2\u88ab\u9009\u62e9\uff0c\u5e76\u57fa\u4e8e\u5b83\u5b9e\u73b0\u4ee5\u4e0b\u526a\u679d\u64cd\u4f5c\u3002- \u5728\u505a\u51fa\u9009\u62e9
choice[i]\u540e\uff0c\u6211\u4eec\u5c31\u5c06selected[i]\u8d4b\u503c\u4e3a \\(\\text{True}\\) \uff0c\u4ee3\u8868\u5b83\u5df2\u88ab\u9009\u62e9\u3002 - \u904d\u5386\u9009\u62e9\u5217\u8868
choices\u65f6\uff0c\u8df3\u8fc7\u6240\u6709\u5df2\u88ab\u9009\u62e9\u7684\u8282\u70b9\uff0c\u5373\u526a\u679d\u3002
\u5982\u56fe 13-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u7b2c\u4e00\u8f6e\u9009\u62e9 1 \uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9 3 \uff0c\u7b2c\u4e09\u8f6e\u9009\u62e9 2 \uff0c\u5219\u9700\u8981\u5728\u7b2c\u4e8c\u8f6e\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f6e\u526a\u6389\u5143\u7d20 1 \u548c\u5143\u7d20 3 \u7684\u5206\u652f\u3002
\u56fe 13-6 \u00a0 \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b
\u89c2\u5bdf\u56fe 13-6 \u53d1\u73b0\uff0c\u8be5\u526a\u679d\u64cd\u4f5c\u5c06\u641c\u7d22\u7a7a\u95f4\u5927\u5c0f\u4ece \\(O(n^n)\\) \u51cf\u5c0f\u81f3 \\(O(n!)\\) \u3002
"},{"location":"chapter_backtracking/permutations_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u60f3\u6e05\u695a\u4ee5\u4e0a\u4fe1\u606f\u4e4b\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u4ee3\u7801\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u4e3a\u4e86\u7f29\u77ed\u6574\u4f53\u4ee3\u7801\uff0c\u6211\u4eec\u4e0d\u5355\u72ec\u5b9e\u73b0\u6846\u67b6\u4ee3\u7801\u4e2d\u7684\u5404\u4e2a\u51fd\u6570\uff0c\u800c\u662f\u5c06\u5b83\u4eec\u5c55\u5f00\u5728
PythonC++JavaC#GoSwiftJSTSDartRustCZig permutations_i.pybacktrack()\u51fd\u6570\u4e2d\uff1a
permutations_i.cppdef backtrack(\n state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if len(state) == len(choices):\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for i, choice in enumerate(choices):\n # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if not selected[i]:\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = True\n state.append(choice)\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = False\n state.pop()\n\ndef permutations_i(nums: list[int]) -> list[list[int]]:\n \"\"\"\u5168\u6392\u5217 I\"\"\"\n res = []\n backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n return res\n
permutations_i.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.size() == choices.size()) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.size(); i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.push_back(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop_back();\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nvector<vector<int>> permutationsI(vector<int> nums) {\n vector<int> state;\n vector<bool> selected(nums.size(), false);\n vector<vector<int>> res;\n backtrack(state, nums, selected, res);\n return res;\n}\n
permutations_i.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List<Integer> state, int[] choices, boolean[] selected, List<List<Integer>> res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.size() == choices.length) {\n res.add(new ArrayList<Integer>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.remove(state.size() - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nList<List<Integer>> permutationsI(int[] nums) {\n List<List<Integer>> res = new ArrayList<List<Integer>>();\n backtrack(new ArrayList<Integer>(), nums, new boolean[nums.length], res);\n return res;\n}\n
permutations_i.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid Backtrack(List<int> state, int[] choices, bool[] selected, List<List<int>> res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.Count == choices.Length) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.Length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.Add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.RemoveAt(state.Count - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nList<List<int>> PermutationsI(int[] nums) {\n List<List<int>> res = [];\n Backtrack([], nums, new bool[nums.Length], res);\n return res;\n}\n
permutations_i.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if len(*state) == len(*choices) {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i := 0; i < len(*choices); i++ {\n choice := (*choices)[i]\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if !(*selected)[i] {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n (*selected)[i] = true\n *state = append(*state, choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackI(state, choices, selected, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n (*selected)[i] = false\n *state = (*state)[:len(*state)-1]\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\n res := make([][]int, 0)\n state := make([]int, 0)\n selected := make([]bool, len(nums))\n backtrackI(&state, &nums, &selected, &res)\n return res\n}\n
permutations_i.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if state.count == choices.count {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (i, choice) in choices.enumerated() {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if !selected[i] {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true\n state.append(choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, choices: choices, selected: &selected, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false\n state.removeLast()\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -> [[Int]] {\n var state: [Int] = []\n var selected = Array(repeating: false, count: nums.count)\n var res: [[Int]] = []\n backtrack(state: &state, choices: nums, selected: &selected, res: &res)\n return res\n}\n
permutations_i.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length === choices.length) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n choices.forEach((choice, i) => {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop();\n }\n });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\n const res = [];\n backtrack([], nums, Array(nums.length).fill(false), res);\n return res;\n}\n
permutations_i.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\n state: number[],\n choices: number[],\n selected: boolean[],\n res: number[][]\n): void {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length === choices.length) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n choices.forEach((choice, i) => {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop();\n }\n });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\n const res: number[][] = [];\n backtrack([], nums, Array(nums.length).fill(false), res);\n return res;\n}\n
permutations_i.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(\n List<int> state,\n List<int> choices,\n List<bool> selected,\n List<List<int>> res,\n) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length == choices.length) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.removeLast();\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nList<List<int>> permutationsI(List<int> nums) {\n List<List<int>> res = [];\n backtrack([], nums, List.filled(nums.length, false), res);\n return res;\n}\n
permutations_i.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfn backtrack(mut state: Vec<i32>, choices: &[i32], selected: &mut [bool], res: &mut Vec<Vec<i32>>) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if state.len() == choices.len() {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i in 0..choices.len() {\n let choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if !selected[i] {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.remove(state.len() - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nfn permutations_i(nums: &mut [i32]) -> Vec<Vec<i32>> {\n let mut res = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n backtrack(Vec::new(), nums, &mut vec![false; nums.len()], &mut res);\n res\n}\n
permutations_i.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (stateSize == choicesSize) {\n res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n for (int i = 0; i < choicesSize; i++) {\n res[*resSize][i] = state[i];\n }\n (*resSize)++;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choicesSize; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state[stateSize] = choice;\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n int *state = (int *)malloc(numsSize * sizeof(int));\n bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n for (int i = 0; i < numsSize; i++) {\n selected[i] = false;\n }\n int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n *returnSize = 0;\n\n backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n free(state);\n free(selected);\n\n return res;\n}\n
"},{"location":"chapter_backtracking/permutations_problem/#1322","title":"13.2.2 \u00a0 \u8003\u8651\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsI}\nQuestion
\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u590d\u7684\u6392\u5217\u3002
\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 1, 2]\\) \u3002\u4e3a\u4e86\u65b9\u4fbf\u533a\u5206\u4e24\u4e2a\u91cd\u590d\u5143\u7d20 \\(1\\) \uff0c\u6211\u4eec\u5c06\u7b2c\u4e8c\u4e2a \\(1\\) \u8bb0\u4e3a \\(\\hat{1}\\) \u3002
\u5982\u56fe 13-7 \u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u662f\u91cd\u590d\u7684\u3002
\u56fe 13-7 \u00a0 \u91cd\u590d\u6392\u5217
\u90a3\u4e48\u5982\u4f55\u53bb\u9664\u91cd\u590d\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u76f4\u63a5\u5bf9\u6392\u5217\u7ed3\u679c\u8fdb\u884c\u53bb\u91cd\u3002\u7136\u800c\u8fd9\u6837\u505a\u4e0d\u591f\u4f18\u96c5\uff0c\u56e0\u4e3a\u751f\u6210\u91cd\u590d\u6392\u5217\u7684\u641c\u7d22\u5206\u652f\u6ca1\u6709\u5fc5\u8981\uff0c\u5e94\u5f53\u63d0\u524d\u8bc6\u522b\u5e76\u526a\u679d\uff0c\u8fd9\u6837\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002
"},{"location":"chapter_backtracking/permutations_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"\u89c2\u5bdf\u56fe 13-8 \uff0c\u5728\u7b2c\u4e00\u8f6e\u4e2d\uff0c\u9009\u62e9 \\(1\\) \u6216\u9009\u62e9 \\(\\hat{1}\\) \u662f\u7b49\u4ef7\u7684\uff0c\u5728\u8fd9\u4e24\u4e2a\u9009\u62e9\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u590d\u7684\u3002\u56e0\u6b64\u5e94\u8be5\u628a \\(\\hat{1}\\) \u526a\u679d\u3002
\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(2\\) \u4e4b\u540e\uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u5e94\u5c06\u7b2c\u4e8c\u8f6e\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002
\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u5728\u67d0\u4e00\u8f6e\u9009\u62e9\u4e2d\uff0c\u4fdd\u8bc1\u591a\u4e2a\u76f8\u7b49\u7684\u5143\u7d20\u4ec5\u88ab\u9009\u62e9\u4e00\u6b21\u3002
\u56fe 13-8 \u00a0 \u91cd\u590d\u6392\u5217\u526a\u679d
"},{"location":"chapter_backtracking/permutations_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5728\u4e0a\u4e00\u9898\u7684\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u8003\u8651\u5728\u6bcf\u4e00\u8f6e\u9009\u62e9\u4e2d\u5f00\u542f\u4e00\u4e2a\u54c8\u5e0c\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig permutations_ii.pyduplicated\uff0c\u7528\u4e8e\u8bb0\u5f55\u8be5\u8f6e\u4e2d\u5df2\u7ecf\u5c1d\u8bd5\u8fc7\u7684\u5143\u7d20\uff0c\u5e76\u5c06\u91cd\u590d\u5143\u7d20\u526a\u679d\uff1a
permutations_ii.cppdef backtrack(\n state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if len(state) == len(choices):\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n duplicated = set[int]()\n for i, choice in enumerate(choices):\n # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if not selected[i] and choice not in duplicated:\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice) # \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = True\n state.append(choice)\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = False\n state.pop()\n\ndef permutations_ii(nums: list[int]) -> list[list[int]]:\n \"\"\"\u5168\u6392\u5217 II\"\"\"\n res = []\n backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n return res\n
permutations_ii.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.size() == choices.size()) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n unordered_set<int> duplicated;\n for (int i = 0; i < choices.size(); i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && duplicated.find(choice) == duplicated.end()) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.emplace(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.push_back(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop_back();\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nvector<vector<int>> permutationsII(vector<int> nums) {\n vector<int> state;\n vector<bool> selected(nums.size(), false);\n vector<vector<int>> res;\n backtrack(state, nums, selected, res);\n return res;\n}\n
permutations_ii.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List<Integer> state, int[] choices, boolean[] selected, List<List<Integer>> res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.size() == choices.length) {\n res.add(new ArrayList<Integer>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n Set<Integer> duplicated = new HashSet<Integer>();\n for (int i = 0; i < choices.length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.contains(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.remove(state.size() - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nList<List<Integer>> permutationsII(int[] nums) {\n List<List<Integer>> res = new ArrayList<List<Integer>>();\n backtrack(new ArrayList<Integer>(), nums, new boolean[nums.length], res);\n return res;\n}\n
permutations_ii.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid Backtrack(List<int> state, int[] choices, bool[] selected, List<List<int>> res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.Count == choices.Length) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n HashSet<int> duplicated = [];\n for (int i = 0; i < choices.Length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.Contains(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.Add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.Add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.RemoveAt(state.Count - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nList<List<int>> PermutationsII(int[] nums) {\n List<List<int>> res = [];\n Backtrack([], nums, new bool[nums.Length], res);\n return res;\n}\n
permutations_ii.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if len(*state) == len(*choices) {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n duplicated := make(map[int]struct{}, 0)\n for i := 0; i < len(*choices); i++ {\n choice := (*choices)[i]\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if _, ok := duplicated[choice]; !ok && !(*selected)[i] {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n duplicated[choice] = struct{}{}\n (*selected)[i] = true\n *state = append(*state, choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackI(state, choices, selected, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n (*selected)[i] = false\n *state = (*state)[:len(*state)-1]\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\n res := make([][]int, 0)\n state := make([]int, 0)\n selected := make([]bool, len(nums))\n backtrackII(&state, &nums, &selected, &res)\n return res\n}\n
permutations_ii.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if state.count == choices.count {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n var duplicated: Set<Int> = []\n for (i, choice) in choices.enumerated() {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if !selected[i], !duplicated.contains(choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.insert(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true\n state.append(choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, choices: choices, selected: &selected, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false\n state.removeLast()\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -> [[Int]] {\n var state: [Int] = []\n var selected = Array(repeating: false, count: nums.count)\n var res: [[Int]] = []\n backtrack(state: &state, choices: nums, selected: &selected, res: &res)\n return res\n}\n
permutations_ii.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length === choices.length) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n const duplicated = new Set();\n choices.forEach((choice, i) => {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.has(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop();\n }\n });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\n const res = [];\n backtrack([], nums, Array(nums.length).fill(false), res);\n return res;\n}\n
permutations_ii.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\n state: number[],\n choices: number[],\n selected: boolean[],\n res: number[][]\n): void {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length === choices.length) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n const duplicated = new Set();\n choices.forEach((choice, i) => {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.has(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop();\n }\n });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\n const res: number[][] = [];\n backtrack([], nums, Array(nums.length).fill(false), res);\n return res;\n}\n
permutations_ii.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(\n List<int> state,\n List<int> choices,\n List<bool> selected,\n List<List<int>> res,\n) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length == choices.length) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n Set<int> duplicated = {};\n for (int i = 0; i < choices.length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.contains(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.removeLast();\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nList<List<int>> permutationsII(List<int> nums) {\n List<List<int>> res = [];\n backtrack([], nums, List.filled(nums.length, false), res);\n return res;\n}\n
permutations_ii.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfn backtrack(mut state: Vec<i32>, choices: &[i32], selected: &mut [bool], res: &mut Vec<Vec<i32>>) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if state.len() == choices.len() {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n let mut duplicated = HashSet::<i32>::new();\n for i in 0..choices.len() {\n let choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if !selected[i] && !duplicated.contains(&choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.insert(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.remove(state.len() - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nfn permutations_ii(nums: &mut [i32]) -> Vec<Vec<i32>> {\n let mut res = Vec::new();\n backtrack(Vec::new(), nums, &mut vec![false; nums.len()], &mut res);\n res\n}\n
permutations_ii.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (stateSize == choicesSize) {\n res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n for (int i = 0; i < choicesSize; i++) {\n res[*resSize][i] = state[i];\n }\n (*resSize)++;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n bool duplicated[MAX_SIZE] = {false};\n for (int i = 0; i < choicesSize; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated[choice]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated[choice] = true; // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state[stateSize] = choice;\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n int *state = (int *)malloc(numsSize * sizeof(int));\n bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n for (int i = 0; i < numsSize; i++) {\n selected[i] = false;\n }\n int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n *returnSize = 0;\n\n backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n free(state);\n free(selected);\n\n return res;\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsII}\n\u5047\u8bbe\u5143\u7d20\u4e24\u4e24\u4e4b\u95f4\u4e92\u4e0d\u76f8\u540c\uff0c\u5219 \\(n\\) \u4e2a\u5143\u7d20\u5171\u6709 \\(n!\\) \u79cd\u6392\u5217\uff08\u9636\u4e58\uff09\uff1b\u5728\u8bb0\u5f55\u7ed3\u679c\u65f6\uff0c\u9700\u8981\u590d\u5236\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5217\u8868\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!n)\\) \u3002
\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002
"},{"location":"chapter_backtracking/permutations_problem/#3","title":"3. \u00a0 \u4e24\u79cd\u526a\u679d\u5bf9\u6bd4","text":"selected\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u540c\u4e00\u65f6\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u4e2aduplicated\uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u867d\u7136
selected\u548cduplicated\u90fd\u7528\u4e8e\u526a\u679d\uff0c\u4f46\u4e24\u8005\u7684\u76ee\u6807\u4e0d\u540c\u3002- \u91cd\u590d\u9009\u62e9\u526a\u679d\uff1a\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\u4e2d\u53ea\u6709\u4e00\u4e2a
selected\u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5f53\u524d\u72b6\u6001\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u5176\u4f5c\u7528\u662f\u9632\u6b62choices\u4e2d\u7684\u4efb\u4e00\u5143\u7d20\u5728state\u4e2d\u91cd\u590d\u51fa\u73b0\u3002 - \u76f8\u7b49\u5143\u7d20\u526a\u679d\uff1a\u6bcf\u8f6e\u9009\u62e9\uff08\u6bcf\u4e2a\u8c03\u7528\u7684
backtrack\u51fd\u6570\uff09\u90fd\u5305\u542b\u4e00\u4e2aduplicated\u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5728\u672c\u8f6e\u904d\u5386\uff08for\u5faa\u73af\uff09\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9009\u62e9\u8fc7\uff0c\u5176\u4f5c\u7528\u662f\u4fdd\u8bc1\u76f8\u7b49\u7684\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002
\u56fe 13-9 \u5c55\u793a\u4e86\u4e24\u4e2a\u526a\u679d\u6761\u4ef6\u7684\u751f\u6548\u8303\u56f4\u3002\u6ce8\u610f\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u9009\u62e9\uff0c\u4ece\u6839\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\u6784\u6210\u4e00\u4e2a\u6392\u5217\u3002
\u56fe 13-9 \u00a0 \u4e24\u79cd\u526a\u679d\u6761\u4ef6\u7684\u4f5c\u7528\u8303\u56f4
"},{"location":"chapter_backtracking/subset_sum_problem/","title":"13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898","text":""},{"location":"chapter_backtracking/subset_sum_problem/#1331","title":"13.3.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4
nums\u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570target\uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8etarget\u3002\u7ed9\u5b9a\u6570\u7ec4\u65e0\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ef\u4ee5\u88ab\u9009\u53d6\u591a\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6807\u6574\u6570 \\(9\\) \uff0c\u89e3\u4e3a \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002
- \u8f93\u5165\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u91cd\u590d\u9009\u53d6\u3002
- \u5b50\u96c6\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u6bd4\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u4e2a\u5b50\u96c6\u3002
\u7c7b\u4f3c\u4e8e\u5168\u6392\u5217\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\uff0c\u5e76\u5728\u9009\u62e9\u8fc7\u7a0b\u4e2d\u5b9e\u65f6\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u5f53\u5143\u7d20\u548c\u7b49\u4e8e
target\u65f6\uff0c\u5c31\u5c06\u5b50\u96c6\u8bb0\u5f55\u81f3\u7ed3\u679c\u5217\u8868\u3002\u800c\u4e0e\u5168\u6392\u5217\u95ee\u9898\u4e0d\u540c\u7684\u662f\uff0c\u672c\u9898\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u9009\u53d6\uff0c\u56e0\u6b64\u65e0\u987b\u501f\u52a9
PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_i_naive.pyselected\u5e03\u5c14\u5217\u8868\u6765\u8bb0\u5f55\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u53ef\u4ee5\u5bf9\u5168\u6392\u5217\u4ee3\u7801\u8fdb\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u9898\u4ee3\u7801\uff1a
subset_sum_i_naive.cppdef backtrack(\n state: list[int],\n target: int,\n total: int,\n choices: list[int],\n res: list[list[int]],\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if total == target:\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for i in range(len(choices)):\n # \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if total + choices[i] > target:\n continue\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.append(choices[i])\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -> list[list[int]]:\n \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09\"\"\"\n state = [] # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n total = 0 # \u5b50\u96c6\u548c\n res = [] # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res)\n return res\n
subset_sum_i_naive.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector<int> &state, int target, int total, vector<int> &choices, vector<vector<int>> &res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (size_t i = 0; i < choices.size(); i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.push_back(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop_back();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvector<vector<int>> subsetSumINaive(vector<int> &nums, int target) {\n vector<int> state; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n int total = 0; // \u5b50\u96c6\u548c\n vector<vector<int>> res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<Integer> state, int target, int total, int[] choices, List<List<Integer>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n res.add(new ArrayList<>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.remove(state.size() - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<Integer>> subsetSumINaive(int[] nums, int target) {\n List<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n int total = 0; // \u5b50\u96c6\u548c\n List<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List<int> state, int target, int total, int[] choices, List<List<int>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.Length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.Add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.RemoveAt(state.Count - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<int>> SubsetSumINaive(int[] nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n int total = 0; // \u5b50\u96c6\u548c\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n Backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == total {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i := 0; i < len(*choices); i++ {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if total+(*choices)[i] > target {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n *state = append(*state, (*choices)[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n *state = (*state)[:len(*state)-1]\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n total := 0 // \u5b50\u96c6\u548c\n res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrackSubsetSumINaive(total, target, &state, &nums, &res)\n return res\n}\n
subset_sum_i_naive.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if total == target {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i in stride(from: 0, to: choices.count, by: 1) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if total + choices[i] > target {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.append(choices[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, target: target, total: total + choices[i], choices: choices, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast()\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums: [Int], target: Int) -> [[Int]] {\n var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n let total = 0 // \u5b50\u96c6\u548c\n var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state: &state, target: target, total: total, choices: nums, res: &res)\n return res\n}\n
subset_sum_i_naive.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, total, choices, res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total === target) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (let i = 0; i < choices.length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums, target) {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n const total = 0; // \u5b50\u96c6\u548c\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n state: number[],\n target: number,\n total: number,\n choices: number[],\n res: number[][]\n): void {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total === target) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (let i = 0; i < choices.length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n const total = 0; // \u5b50\u96c6\u548c\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n List<int> state,\n int target,\n int total,\n List<int> choices,\n List<List<int>> res,\n) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<int>> subsetSumINaive(List<int> nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n int total = 0; // \u5143\u7d20\u548c\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(mut state: Vec<i32>, target: i32, total: i32, choices: &[i32], res: &mut Vec<Vec<i32>>) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if total == target {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i in 0..choices.len() {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if total + choices[i] > target {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfn subset_sum_i_naive(nums: &[i32], target: i32) -> Vec<Vec<i32>> {\n let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n let total = 0; // \u5b50\u96c6\u548c\n let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, &mut res);\n res\n}\n
subset_sum_i_naive.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n for (int i = 0; i < stateSize; i++) {\n res[resSize][i] = state[i];\n }\n resColSizes[resSize++] = stateSize;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choicesSize; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state[stateSize++] = choices[i];\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(target, total + choices[i], choices, choicesSize);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n stateSize--;\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n resSize = 0; // \u521d\u59cb\u5316\u89e3\u7684\u6570\u91cf\u4e3a0\n backtrack(target, 0, nums, numsSize);\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumINaive}\n\u5411\u4ee5\u4e0a\u4ee3\u7801\u8f93\u5165\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u8f93\u51fa\u7ed3\u679c\u4e3a \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u867d\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u4e3a \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u590d\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002
\u8fd9\u662f\u56e0\u4e3a\u641c\u7d22\u8fc7\u7a0b\u662f\u533a\u5206\u9009\u62e9\u987a\u5e8f\u7684\uff0c\u7136\u800c\u5b50\u96c6\u4e0d\u533a\u5206\u9009\u62e9\u987a\u5e8f\u3002\u5982\u56fe 13-10 \u6240\u793a\uff0c\u5148\u9009 \\(4\\) \u540e\u9009 \\(5\\) \u4e0e\u5148\u9009 \\(5\\) \u540e\u9009 \\(4\\) \u662f\u4e0d\u540c\u7684\u5206\u652f\uff0c\u4f46\u5bf9\u5e94\u540c\u4e00\u4e2a\u5b50\u96c6\u3002
\u56fe 13-10 \u00a0 \u5b50\u96c6\u641c\u7d22\u4e0e\u8d8a\u754c\u526a\u679d
\u4e3a\u4e86\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u601d\u8def\u662f\u5bf9\u7ed3\u679c\u5217\u8868\u8fdb\u884c\u53bb\u91cd\u3002\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u6709\u4e24\u65b9\u9762\u539f\u56e0\u3002
- \u5f53\u6570\u7ec4\u5143\u7d20\u8f83\u591a\uff0c\u5c24\u5176\u662f\u5f53
target\u8f83\u5927\u65f6\uff0c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u4ea7\u751f\u5927\u91cf\u7684\u91cd\u590d\u5b50\u96c6\u3002 - \u6bd4\u8f83\u5b50\u96c6\uff08\u6570\u7ec4\uff09\u7684\u5f02\u540c\u975e\u5e38\u8017\u65f6\uff0c\u9700\u8981\u5148\u6392\u5e8f\u6570\u7ec4\uff0c\u518d\u6bd4\u8f83\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684\u5f02\u540c\u3002
\u6211\u4eec\u8003\u8651\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u901a\u8fc7\u526a\u679d\u8fdb\u884c\u53bb\u91cd\u3002\u89c2\u5bdf\u56fe 13-11 \uff0c\u91cd\u590d\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u987a\u5e8f\u9009\u62e9\u6570\u7ec4\u5143\u7d20\u65f6\u4ea7\u751f\u7684\uff0c\u4f8b\u5982\u4ee5\u4e0b\u60c5\u51b5\u3002
- \u5f53\u7b2c\u4e00\u8f6e\u548c\u7b2c\u4e8c\u8f6e\u5206\u522b\u9009\u62e9 \\(3\\) \u548c \\(4\\) \u65f6\uff0c\u4f1a\u751f\u6210\u5305\u542b\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8bb0\u4e3a \\([3, 4, \\dots]\\) \u3002
- \u4e4b\u540e\uff0c\u5f53\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(4\\) \u65f6\uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \uff0c\u56e0\u4e3a\u8be5\u9009\u62e9\u4ea7\u751f\u7684\u5b50\u96c6 \\([4, 3, \\dots]\\) \u548c\u7b2c
1.\u6b65\u4e2d\u751f\u6210\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002
\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e00\u5c42\u7684\u9009\u62e9\u90fd\u662f\u4ece\u5de6\u5230\u53f3\u88ab\u9010\u4e2a\u5c1d\u8bd5\u7684\uff0c\u56e0\u6b64\u8d8a\u9760\u53f3\u7684\u5206\u652f\u88ab\u526a\u6389\u7684\u8d8a\u591a\u3002
- \u524d\u4e24\u8f6e\u9009\u62e9 \\(3\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([3, 5, \\dots]\\) \u3002
- \u524d\u4e24\u8f6e\u9009\u62e9 \\(4\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([4, 5, \\dots]\\) \u3002
- \u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(5\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u4e3a\u5b50\u96c6 \\([5, 3, \\dots]\\) \u548c \\([5, 4, \\dots]\\) \u4e0e\u7b2c
1.\u6b65\u548c\u7b2c2.\u6b65\u4e2d\u63cf\u8ff0\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002
\u56fe 13-11 \u00a0 \u4e0d\u540c\u9009\u62e9\u987a\u5e8f\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6
\u603b\u7ed3\u6765\u770b\uff0c\u7ed9\u5b9a\u8f93\u5165\u6570\u7ec4 \\([x_1, x_2, \\dots, x_n]\\) \uff0c\u8bbe\u641c\u7d22\u8fc7\u7a0b\u4e2d\u7684\u9009\u62e9\u5e8f\u5217\u4e3a \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\) \uff0c\u5219\u8be5\u9009\u62e9\u5e8f\u5217\u9700\u8981\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4e0d\u6ee1\u8db3\u8be5\u6761\u4ef6\u7684\u9009\u62e9\u5e8f\u5217\u90fd\u4f1a\u9020\u6210\u91cd\u590d\uff0c\u5e94\u5f53\u526a\u679d\u3002
"},{"location":"chapter_backtracking/subset_sum_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u4e3a\u5b9e\u73b0\u8be5\u526a\u679d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u53d8\u91cf
start\uff0c\u7528\u4e8e\u6307\u793a\u904d\u5386\u8d77\u59cb\u70b9\u3002\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i\\) \u5f00\u59cb\u904d\u5386\u3002\u8fd9\u6837\u505a\u5c31\u53ef\u4ee5\u8ba9\u9009\u62e9\u5e8f\u5217\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4ece\u800c\u4fdd\u8bc1\u5b50\u96c6\u552f\u4e00\u3002\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u8fd8\u5bf9\u4ee3\u7801\u8fdb\u884c\u4e86\u4ee5\u4e0b\u4e24\u9879\u4f18\u5316\u3002
- \u5728\u5f00\u542f\u641c\u7d22\u524d\uff0c\u5148\u5c06\u6570\u7ec4
nums\u6392\u5e8f\u3002\u5728\u904d\u5386\u6240\u6709\u9009\u62e9\u65f6\uff0c\u5f53\u5b50\u96c6\u548c\u8d85\u8fc7target\u65f6\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\uff0c\u56e0\u4e3a\u540e\u8fb9\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7target\u3002 - \u7701\u53bb\u5143\u7d20\u548c\u53d8\u91cf
total\uff0c\u901a\u8fc7\u5728target\u4e0a\u6267\u884c\u51cf\u6cd5\u6765\u7edf\u8ba1\u5143\u7d20\u548c\uff0c\u5f53target\u7b49\u4e8e \\(0\\) \u65f6\u8bb0\u5f55\u89e3\u3002
subset_sum_i.cppdef backtrack(\n state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0:\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for i in range(start, len(choices)):\n # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0:\n break\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.append(choices[i])\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -> list[list[int]]:\n \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\n state = [] # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort() # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n start = 0 # \u904d\u5386\u8d77\u59cb\u70b9\n res = [] # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res)\n return res\n
subset_sum_i.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choices.size(); i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push_back(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop_back();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector<vector<int>> subsetSumI(vector<int> &nums, int target) {\n vector<int> state; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n vector<vector<int>> res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<Integer> state, int target, int[] choices, int start, List<List<Integer>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.add(new ArrayList<>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.remove(state.size() - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<Integer>> subsetSumI(int[] nums, int target) {\n List<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List<int> state, int target, int[] choices, int start, List<List<int>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choices.Length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.Add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.RemoveAt(state.Count - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<int>> SubsetSumI(int[] nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n Backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for i := start; i < len(*choices); i++ {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target-(*choices)[i] < 0 {\n break\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n *state = append(*state, (*choices)[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n *state = (*state)[:len(*state)-1]\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\n state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n sort.Ints(nums) // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n start := 0 // \u904d\u5386\u8d77\u59cb\u70b9\n res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrackSubsetSumI(start, target, &state, &nums, &res)\n return res\n}\n
subset_sum_i.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for i in stride(from: start, to: choices.count, by: 1) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0 {\n break\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.append(choices[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, target: target - choices[i], choices: choices, start: i, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast()\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums: [Int], target: Int) -> [[Int]] {\n var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state: &state, target: target, choices: nums, start: start, res: &res)\n return res\n}\n
subset_sum_i.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, choices, start, res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target === 0) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (let i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums, target) {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort((a, b) => a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n state: number[],\n target: number,\n choices: number[],\n start: number,\n res: number[][]\n): void {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target === 0) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (let i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort((a, b) => a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n List<int> state,\n int target,\n List<int> choices,\n int start,\n List<List<int>> res,\n) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<int>> subsetSumI(List<int> nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(mut state: Vec<i32>, target: i32, choices: &[i32], start: usize, res: &mut Vec<Vec<i32>>) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for i in start..choices.len() {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0 {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfn subset_sum_i(nums: &mut [i32], target: i32) -> Vec<Vec<i32>> {\n let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, &mut res);\n res\n}\n
subset_sum_i.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n for (int i = 0; i < stateSize; ++i) {\n res[resSize][i] = state[i];\n }\n resColSizes[resSize++] = stateSize;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choicesSize; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state[stateSize] = choices[i];\n stateSize++;\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(target - choices[i], choices, choicesSize, i);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n stateSize--;\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n qsort(nums, numsSize, sizeof(int), cmp); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n backtrack(target, nums, numsSize, start);\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumI}\n\u56fe 13-12 \u6240\u793a\u4e3a\u5c06\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u8f93\u5165\u4ee5\u4e0a\u4ee3\u7801\u540e\u7684\u6574\u4f53\u56de\u6eaf\u8fc7\u7a0b\u3002
\u56fe 13-12 \u00a0 \u5b50\u96c6\u548c I \u56de\u6eaf\u8fc7\u7a0b
"},{"location":"chapter_backtracking/subset_sum_problem/#1332","title":"13.3.2 \u00a0 \u8003\u8651\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4
nums\u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570target\uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8etarget\u3002\u7ed9\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u53ef\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002\u76f8\u6bd4\u4e8e\u4e0a\u9898\uff0c\u672c\u9898\u7684\u8f93\u5165\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd9\u5f15\u5165\u4e86\u65b0\u7684\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u7ed9\u5b9a\u6570\u7ec4 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u5219\u73b0\u6709\u4ee3\u7801\u7684\u8f93\u51fa\u7ed3\u679c\u4e3a \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u51fa\u73b0\u4e86\u91cd\u590d\u5b50\u96c6\u3002
\u9020\u6210\u8fd9\u79cd\u91cd\u590d\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f6e\u4e2d\u88ab\u591a\u6b21\u9009\u62e9\u3002\u5728\u56fe 13-13 \u4e2d\uff0c\u7b2c\u4e00\u8f6e\u5171\u6709\u4e09\u4e2a\u9009\u62e9\uff0c\u5176\u4e2d\u4e24\u4e2a\u90fd\u4e3a \\(4\\) \uff0c\u4f1a\u4ea7\u751f\u4e24\u4e2a\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\uff0c\u4ece\u800c\u8f93\u51fa\u91cd\u590d\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f6e\u7684\u4e24\u4e2a \\(4\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002
\u56fe 13-13 \u00a0 \u76f8\u7b49\u5143\u7d20\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6
"},{"location":"chapter_backtracking/subset_sum_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5b9e\u73b0\u65b9\u5f0f\u6bd4\u8f83\u5de7\u5999\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u90bb\u7684\u3002\u8fd9\u610f\u5473\u7740\u5728\u67d0\u8f6e\u9009\u62e9\u4e2d\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u4e0e\u5176\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u5219\u8bf4\u660e\u5b83\u5df2\u7ecf\u88ab\u9009\u62e9\u8fc7\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u8fc7\u5f53\u524d\u5143\u7d20\u3002
\u4e0e\u6b64\u540c\u65f6\uff0c\u672c\u9898\u89c4\u5b9a\u6bcf\u4e2a\u6570\u7ec4\u5143\u7d20\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5e78\u8fd0\u7684\u662f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5229\u7528\u53d8\u91cf
"},{"location":"chapter_backtracking/subset_sum_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_ii.pystart\u6765\u6ee1\u8db3\u8be5\u7ea6\u675f\uff1a\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i + 1\\) \u5f00\u59cb\u5411\u540e\u904d\u5386\u3002\u8fd9\u6837\u65e2\u80fd\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u590d\u9009\u62e9\u5143\u7d20\u3002
subset_sum_ii.cppdef backtrack(\n state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0:\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n # \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for i in range(start, len(choices)):\n # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0:\n break\n # \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if i > start and choices[i] == choices[i - 1]:\n continue\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.append(choices[i])\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -> list[list[int]]:\n \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\n state = [] # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort() # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n start = 0 # \u904d\u5386\u8d77\u59cb\u70b9\n res = [] # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res)\n return res\n
subset_sum_ii.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choices.size(); i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push_back(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop_back();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector<vector<int>> subsetSumII(vector<int> &nums, int target) {\n vector<int> state; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n vector<vector<int>> res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List<Integer> state, int target, int[] choices, int start, List<List<Integer>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.add(new ArrayList<>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.remove(state.size() - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<Integer>> subsetSumII(int[] nums, int target) {\n List<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid Backtrack(List<int> state, int target, int[] choices, int start, List<List<int>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choices.Length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.Add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.RemoveAt(state.Count - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<int>> SubsetSumII(int[] nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n Backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for i := start; i < len(*choices); i++ {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target-(*choices)[i] < 0 {\n break\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if i > start && (*choices)[i] == (*choices)[i-1] {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n *state = append(*state, (*choices)[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n *state = (*state)[:len(*state)-1]\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\n state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n sort.Ints(nums) // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n start := 0 // \u904d\u5386\u8d77\u59cb\u70b9\n res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrackSubsetSumII(start, target, &state, &nums, &res)\n return res\n}\n
subset_sum_ii.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for i in stride(from: start, to: choices.count, by: 1) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0 {\n break\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if i > start, choices[i] == choices[i - 1] {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.append(choices[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, target: target - choices[i], choices: choices, start: i + 1, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast()\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums: [Int], target: Int) -> [[Int]] {\n var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state: &state, target: target, choices: nums, start: start, res: &res)\n return res\n}\n
subset_sum_ii.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(state, target, choices, start, res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target === 0) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (let i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] === choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums, target) {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort((a, b) => a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(\n state: number[],\n target: number,\n choices: number[],\n start: number,\n res: number[][]\n): void {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target === 0) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (let i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] === choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort((a, b) => a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(\n List<int> state,\n int target,\n List<int> choices,\n int start,\n List<List<int>> res,\n) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<int>> subsetSumII(List<int> nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfn backtrack(mut state: Vec<i32>, target: i32, choices: &[i32], start: usize, res: &mut Vec<Vec<i32>>) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for i in start..choices.len() {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0 {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if i > start && choices[i] == choices[i - 1] {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfn subset_sum_ii(nums: &mut [i32], target: i32) -> Vec<Vec<i32>> {\n let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, &mut res);\n res\n}\n
subset_sum_ii.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n for (int i = 0; i < stateSize; i++) {\n res[resSize][i] = state[i];\n }\n resColSizes[resSize++] = stateSize;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choicesSize; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\n if (target - choices[i] < 0) {\n continue;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state[stateSize] = choices[i];\n stateSize++;\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(target - choices[i], choices, choicesSize, i + 1);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n stateSize--;\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n qsort(nums, numsSize, sizeof(int), cmp);\n // \u5f00\u59cb\u56de\u6eaf\n backtrack(target, nums, numsSize, 0);\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumII}\n\u56fe 13-14 \u5c55\u793a\u4e86\u6570\u7ec4 \\([4, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u8fc7\u7a0b\uff0c\u5171\u5305\u542b\u56db\u79cd\u526a\u679d\u64cd\u4f5c\u3002\u8bf7\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u6ce8\u91ca\u76f8\u7ed3\u5408\uff0c\u7406\u89e3\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u79cd\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002
\u56fe 13-14 \u00a0 \u5b50\u96c6\u548c II \u56de\u6eaf\u8fc7\u7a0b
"},{"location":"chapter_backtracking/summary/","title":"13.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_backtracking/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"- \u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u662f\u7a77\u4e3e\u6cd5\uff0c\u901a\u8fc7\u5bf9\u89e3\u7a7a\u95f4\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6765\u5bfb\u627e\u7b26\u5408\u6761\u4ef6\u7684\u89e3\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u9047\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u5219\u8bb0\u5f55\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u904d\u5386\u5b8c\u6210\u540e\u7ed3\u675f\u3002
- \u56de\u6eaf\u7b97\u6cd5\u7684\u641c\u7d22\u8fc7\u7a0b\u5305\u62ec\u5c1d\u8bd5\u4e0e\u56de\u9000\u4e24\u4e2a\u90e8\u5206\u3002\u5b83\u901a\u8fc7\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u6765\u5c1d\u8bd5\u5404\u79cd\u9009\u62e9\uff0c\u5f53\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u60c5\u51b5\u65f6\uff0c\u5219\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002\u5c1d\u8bd5\u4e0e\u56de\u9000\u662f\u4e24\u4e2a\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002
- \u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u5b83\u4eec\u53ef\u7528\u4e8e\u5b9e\u73b0\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7ed3\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002
- \u56de\u6eaf\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u4e8e\u89e3\u51b3\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u3002\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u867d\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u7b97\u6cd5\u89e3\u51b3\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u6548\u7387\u66f4\u9ad8\u6216\u6548\u679c\u66f4\u597d\u7684\u89e3\u6cd5\u3002
- \u5168\u6392\u5217\u95ee\u9898\u65e8\u5728\u641c\u7d22\u7ed9\u5b9a\u96c6\u5408\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5143\u7d20\u662f\u5426\u88ab\u9009\u62e9\uff0c\u526a\u6389\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\u7684\u641c\u7d22\u5206\u652f\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002
- \u5728\u5168\u6392\u5217\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff0c\u5219\u6700\u7ec8\u7ed3\u679c\u4f1a\u51fa\u73b0\u91cd\u590d\u6392\u5217\u3002\u6211\u4eec\u9700\u8981\u7ea6\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u8fd9\u901a\u5e38\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002
- \u5b50\u96c6\u548c\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u7ed9\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u4e3a\u76ee\u6807\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u800c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u8f93\u51fa\u6240\u6709\u987a\u5e8f\u7684\u7ed3\u679c\uff0c\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002\u6211\u4eec\u5728\u56de\u6eaf\u524d\u5c06\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8bbe\u7f6e\u4e00\u4e2a\u53d8\u91cf\u6765\u6307\u793a\u6bcf\u4e00\u8f6e\u7684\u904d\u5386\u8d77\u59cb\u70b9\uff0c\u4ece\u800c\u5c06\u751f\u6210\u91cd\u590d\u5b50\u96c6\u7684\u641c\u7d22\u5206\u652f\u8fdb\u884c\u526a\u679d\u3002
- \u5bf9\u4e8e\u5b50\u96c6\u548c\u95ee\u9898\uff0c\u6570\u7ec4\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u4f1a\u4ea7\u751f\u91cd\u590d\u96c6\u5408\u3002\u6211\u4eec\u5229\u7528\u6570\u7ec4\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\uff0c\u901a\u8fc7\u5224\u65ad\u76f8\u90bb\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5b9e\u73b0\u526a\u679d\uff0c\u4ece\u800c\u786e\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u4e2d\u4e00\u6b21\u3002
- \\(n\\) \u7687\u540e\u95ee\u9898\u65e8\u5728\u5bfb\u627e\u5c06 \\(n\\) \u4e2a\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76d8\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u4e24\u4e24\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u5bf9\u65b9\u3002\u8be5\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\u6709\u884c\u7ea6\u675f\u3001\u5217\u7ea6\u675f\u3001\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u4e3a\u6ee1\u8db3\u884c\u7ea6\u675f\uff0c\u6211\u4eec\u91c7\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8bc1\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002
- \u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u5904\u7406\u65b9\u5f0f\u7c7b\u4f3c\u3002\u5bf9\u4e8e\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u4ece\u800c\u6307\u793a\u9009\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5bf9\u4e8e\u5bf9\u89d2\u7ebf\u7ea6\u675f\uff0c\u6211\u4eec\u501f\u52a9\u4e24\u4e2a\u6570\u7ec4\u6765\u5206\u522b\u8bb0\u5f55\u8be5\u4e3b\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96be\u70b9\u5728\u4e8e\u627e\u5904\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5bf9\u89d2\u7ebf\u4e0a\u683c\u5b50\u6ee1\u8db3\u7684\u884c\u5217\u7d22\u5f15\u89c4\u5f8b\u3002
\u600e\u4e48\u7406\u89e3\u56de\u6eaf\u548c\u9012\u5f52\u7684\u5173\u7cfb\uff1f
\u603b\u7684\u6765\u770b\uff0c\u56de\u6eaf\u662f\u4e00\u79cd\u201c\u7b97\u6cd5\u7b56\u7565\u201d\uff0c\u800c\u9012\u5f52\u66f4\u50cf\u662f\u4e00\u4e2a\u201c\u5de5\u5177\u201d\u3002
- \u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002\u7136\u800c\uff0c\u56de\u6eaf\u662f\u9012\u5f52\u7684\u5e94\u7528\u573a\u666f\u4e4b\u4e00\uff0c\u662f\u9012\u5f52\u5728\u641c\u7d22\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u3002
- \u9012\u5f52\u7684\u7ed3\u6784\u4f53\u73b0\u4e86\u201c\u5b50\u95ee\u9898\u5206\u89e3\u201d\u7684\u89e3\u9898\u8303\u5f0f\uff0c\u5e38\u7528\u4e8e\u89e3\u51b3\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\uff08\u8bb0\u5fc6\u5316\u9012\u5f52\uff09\u7b49\u95ee\u9898\u3002
Abstract
\u590d\u6742\u5ea6\u5206\u6790\u72b9\u5982\u6d69\u701a\u7684\u7b97\u6cd5\u5b87\u5b99\u4e2d\u7684\u65f6\u7a7a\u5411\u5bfc\u3002
\u5b83\u5e26\u9886\u6211\u4eec\u5728\u65f6\u95f4\u4e0e\u7a7a\u95f4\u8fd9\u4e24\u4e2a\u7ef4\u5ea6\u4e0a\u6df1\u5165\u63a2\u7d22\uff0c\u5bfb\u627e\u66f4\u4f18\u96c5\u7684\u89e3\u51b3\u65b9\u6848\u3002
"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30
- 2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52
- 2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6
- 2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6
- 2.5 \u00a0 \u5c0f\u7ed3
\u5728\u7b97\u6cd5\u4e2d\uff0c\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u662f\u5f88\u5e38\u89c1\u7684\uff0c\u5b83\u4e0e\u590d\u6742\u5ea6\u5206\u6790\u606f\u606f\u76f8\u5173\u3002\u56e0\u6b64\uff0c\u5728\u4ecb\u7ecd\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e4b\u524d\uff0c\u6211\u4eec\u5148\u6765\u4e86\u89e3\u5982\u4f55\u5728\u7a0b\u5e8f\u4e2d\u5b9e\u73b0\u91cd\u590d\u6267\u884c\u4efb\u52a1\uff0c\u5373\u4e24\u79cd\u57fa\u672c\u7684\u7a0b\u5e8f\u63a7\u5236\u7ed3\u6784\uff1a\u8fed\u4ee3\u3001\u9012\u5f52\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","title":"2.2.1 \u00a0 \u8fed\u4ee3","text":"\u300c\u8fed\u4ee3 iteration\u300d\u662f\u4e00\u79cd\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u7684\u63a7\u5236\u7ed3\u6784\u3002\u5728\u8fed\u4ee3\u4e2d\uff0c\u7a0b\u5e8f\u4f1a\u5728\u6ee1\u8db3\u4e00\u5b9a\u7684\u6761\u4ef6\u4e0b\u91cd\u590d\u6267\u884c\u67d0\u6bb5\u4ee3\u7801\uff0c\u76f4\u5230\u8fd9\u4e2a\u6761\u4ef6\u4e0d\u518d\u6ee1\u8db3\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","title":"1. \u00a0 for \u5faa\u73af","text":"for\u5faa\u73af\u662f\u6700\u5e38\u89c1\u7684\u8fed\u4ee3\u5f62\u5f0f\u4e4b\u4e00\uff0c\u9002\u5408\u5728\u9884\u5148\u77e5\u9053\u8fed\u4ee3\u6b21\u6570\u65f6\u4f7f\u7528\u3002\u4ee5\u4e0b\u51fd\u6570\u57fa\u4e8e
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyfor\u5faa\u73af\u5b9e\u73b0\u4e86\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff0c\u6c42\u548c\u7ed3\u679c\u4f7f\u7528\u53d8\u91cfres\u8bb0\u5f55\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0cPython \u4e2drange(a, b)\u5bf9\u5e94\u7684\u533a\u95f4\u662f\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u7684\uff0c\u5bf9\u5e94\u7684\u904d\u5386\u8303\u56f4\u4e3a \\(a, a + 1, \\dots, b-1\\) \uff1a
iteration.cppdef for_loop(n: int) -> int:\n \"\"\"for \u5faa\u73af\"\"\"\n res = 0\n # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in range(1, n + 1):\n res += i\n return res\n
iteration.java/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; ++i) {\n res += i;\n }\n return res;\n}\n
iteration.cs/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.go/* for \u5faa\u73af */\nint ForLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.swift/* for \u5faa\u73af */\nfunc forLoop(n int) int {\n res := 0\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i := 1; i <= n; i++ {\n res += i\n }\n return res\n}\n
iteration.js/* for \u5faa\u73af */\nfunc forLoop(n: Int) -> Int {\n var res = 0\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in 1 ... n {\n res += i\n }\n return res\n}\n
iteration.ts/* for \u5faa\u73af */\nfunction forLoop(n) {\n let res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.dart/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\n let res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.rs/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.c/* for \u5faa\u73af */\nfn for_loop(n: i32) -> i32 {\n let mut res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in 1..=n {\n res += i;\n }\n res\n} \n
iteration.zig/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\n var res: i32 = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (1..n+1) |i| {\n res = res + @as(i32, @intCast(i));\n }\n return res;\n} \n\u56fe 2-1 \u662f\u8be5\u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe\u3002
\u56fe 2-1 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe
\u6b64\u6c42\u548c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u6210\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u63cf\u8ff0\u7684\u5c31\u662f\u8fd9\u4e2a\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u76f8\u5173\u5185\u5bb9\u5c06\u4f1a\u5728\u4e0b\u4e00\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","title":"2. \u00a0 while \u5faa\u73af","text":"\u4e0e
for\u5faa\u73af\u7c7b\u4f3c\uff0cwhile\u5faa\u73af\u4e5f\u662f\u4e00\u79cd\u5b9e\u73b0\u8fed\u4ee3\u7684\u65b9\u6cd5\u3002\u5728while\u5faa\u73af\u4e2d\uff0c\u7a0b\u5e8f\u6bcf\u8f6e\u90fd\u4f1a\u5148\u68c0\u67e5\u6761\u4ef6\uff0c\u5982\u679c\u6761\u4ef6\u4e3a\u771f\uff0c\u5219\u7ee7\u7eed\u6267\u884c\uff0c\u5426\u5219\u5c31\u7ed3\u675f\u5faa\u73af\u3002\u4e0b\u9762\u6211\u4eec\u7528
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pywhile\u5faa\u73af\u6765\u5b9e\u73b0\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff1a
iteration.cppdef while_loop(n: int) -> int:\n \"\"\"while \u5faa\u73af\"\"\"\n res = 0\n i = 1 # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n:\n res += i\n i += 1 # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n return res\n
iteration.java/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.cs/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.go/* while \u5faa\u73af */\nint WhileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.swift/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\n res := 0\n // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n i := 1\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++\n }\n return res\n}\n
iteration.js/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -> Int {\n var res = 0\n var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n {\n res += i\n i += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res\n}\n
iteration.ts/* while \u5faa\u73af */\nfunction whileLoop(n) {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.dart/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.rs/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.c/* while \u5faa\u73af */\nfn while_loop(n: i32) -> i32 {\n let mut res = 0;\n let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n {\n res += i;\n i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n res\n}\n
iteration.zig/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\n var res: i32 = 0;\n var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += @intCast(i);\n i += 1;\n }\n return res;\n}\nwhile\u5faa\u73af\u6bd4for\u5faa\u73af\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\u3002\u5728while\u5faa\u73af\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u5730\u8bbe\u8ba1\u6761\u4ef6\u53d8\u91cf\u7684\u521d\u59cb\u5316\u548c\u66f4\u65b0\u6b65\u9aa4\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6761\u4ef6\u53d8\u91cf \\(i\\) \u6bcf\u8f6e\u8fdb\u884c\u4e24\u6b21\u66f4\u65b0\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u4e0d\u592a\u65b9\u4fbf\u7528
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyfor\u5faa\u73af\u5b9e\u73b0\uff1a
iteration.cppdef while_loop_ii(n: int) -> int:\n \"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\n res = 0\n i = 1 # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n # \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n:\n res += i\n # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1\n i *= 2\n return res\n
iteration.java/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.cs/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.go/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, 4, 5...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1; \n i *= 2;\n }\n return res;\n}\n
iteration.swift/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n res := 0\n // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n i := 1\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n for i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++\n i *= 2\n }\n return res\n}\n
iteration.js/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -> Int {\n var res = 0\n var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1\n i *= 2\n }\n return res\n}\n
iteration.ts/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.dart/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.rs/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.c/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -> i32 {\n let mut res = 0;\n let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1;\n i *= 2;\n }\n res\n}\n
iteration.zig/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n// while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n var res: i32 = 0;\n var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += @intCast(i);\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1;\n i *= 2;\n }\n return res;\n}\n\u603b\u7684\u6765\u8bf4\uff0c
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","title":"3. \u00a0 \u5d4c\u5957\u5faa\u73af","text":"for\u5faa\u73af\u7684\u4ee3\u7801\u66f4\u52a0\u7d27\u51d1\uff0cwhile\u5faa\u73af\u66f4\u52a0\u7075\u6d3b\uff0c\u4e24\u8005\u90fd\u53ef\u4ee5\u5b9e\u73b0\u8fed\u4ee3\u7ed3\u6784\u3002\u9009\u62e9\u4f7f\u7528\u54ea\u4e00\u4e2a\u5e94\u8be5\u6839\u636e\u7279\u5b9a\u95ee\u9898\u7684\u9700\u6c42\u6765\u51b3\u5b9a\u3002\u6211\u4eec\u53ef\u4ee5\u5728\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\u5185\u5d4c\u5957\u53e6\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\uff0c\u4e0b\u9762\u4ee5
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyfor\u5faa\u73af\u4e3a\u4f8b\uff1a
iteration.cppdef nested_for_loop(n: int) -> str:\n \"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\n res = \"\"\n # \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in range(1, n + 1):\n # \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in range(1, n + 1):\n res += f\"({i}, {j}), \"\n return res\n
iteration.java/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\n ostringstream res;\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; ++i) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; ++j) {\n res << \"(\" << i << \", \" << j << \"), \";\n }\n }\n return res.str();\n}\n
iteration.cs/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n StringBuilder res = new StringBuilder();\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res.append(\"(\" + i + \", \" + j + \"), \");\n }\n }\n return res.toString();\n}\n
iteration.go/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\n StringBuilder res = new();\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res.Append($\"({i}, {j}), \");\n }\n }\n return res.ToString();\n}\n
iteration.swift/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\n res := \"\"\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i := 1; i <= n; i++ {\n for j := 1; j <= n; j++ {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n res += fmt.Sprintf(\"(%d, %d), \", i, j)\n }\n }\n return res\n}\n
iteration.js/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -> String {\n var res = \"\"\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in 1 ... n {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in 1 ... n {\n res.append(\"(\\(i), \\(j)), \")\n }\n }\n return res\n}\n
iteration.ts/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\n let res = '';\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (let j = 1; j <= n; j++) {\n res += `(${i}, ${j}), `;\n }\n }\n return res;\n}\n
iteration.dart/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\n let res = '';\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (let j = 1; j <= n; j++) {\n res += `(${i}, ${j}), `;\n }\n }\n return res;\n}\n
iteration.rs/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n String res = \"\";\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res += \"($i, $j), \";\n }\n }\n return res;\n}\n
iteration.c/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -> String {\n let mut res = vec![];\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in 1..=n {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in 1..=n {\n res.push(format!(\"({}, {}), \", i, j));\n }\n }\n res.join(\"\")\n}\n
iteration.zig/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n // n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\n int size = n * n * 26 + 1;\n char *res = malloc(size * sizeof(char));\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n char tmp[26];\n snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n strncat(res, tmp, size - strlen(res) - 1);\n }\n }\n return res;\n}\n// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n var res = std.ArrayList(u8).init(allocator);\n defer res.deinit();\n var buffer: [20]u8 = undefined;\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (1..n+1) |i| {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (1..n+1) |j| {\n var _str = try std.fmt.bufPrint(&buffer, \"({d}, {d}), \", .{i, j});\n try res.appendSlice(_str);\n }\n }\n return res.toOwnedSlice();\n}\n\u56fe 2-2 \u662f\u8be5\u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe\u3002
\u56fe 2-2 \u00a0 \u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe
\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e \\(n^2\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u548c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u201c\u5e73\u65b9\u5173\u7cfb\u201d\u3002
\u6211\u4eec\u53ef\u4ee5\u7ee7\u7eed\u6dfb\u52a0\u5d4c\u5957\u5faa\u73af\uff0c\u6bcf\u4e00\u6b21\u5d4c\u5957\u90fd\u662f\u4e00\u6b21\u201c\u5347\u7ef4\u201d\uff0c\u5c06\u4f1a\u4f7f\u65f6\u95f4\u590d\u6742\u5ea6\u63d0\u9ad8\u81f3\u201c\u7acb\u65b9\u5173\u7cfb\u201d\u201c\u56db\u6b21\u65b9\u5173\u7cfb\u201d\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","title":"2.2.2 \u00a0 \u9012\u5f52","text":"\u300c\u9012\u5f52 recursion\u300d\u662f\u4e00\u79cd\u7b97\u6cd5\u7b56\u7565\uff0c\u901a\u8fc7\u51fd\u6570\u8c03\u7528\u81ea\u8eab\u6765\u89e3\u51b3\u95ee\u9898\u3002\u5b83\u4e3b\u8981\u5305\u542b\u4e24\u4e2a\u9636\u6bb5\u3002
- \u9012\uff1a\u7a0b\u5e8f\u4e0d\u65ad\u6df1\u5165\u5730\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u4f20\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\uff0c\u76f4\u5230\u8fbe\u5230\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u3002
- \u5f52\uff1a\u89e6\u53d1\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u540e\uff0c\u7a0b\u5e8f\u4ece\u6700\u6df1\u5c42\u7684\u9012\u5f52\u51fd\u6570\u5f00\u59cb\u9010\u5c42\u8fd4\u56de\uff0c\u6c47\u805a\u6bcf\u4e00\u5c42\u7684\u7ed3\u679c\u3002
\u800c\u4ece\u5b9e\u73b0\u7684\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u4ee3\u7801\u4e3b\u8981\u5305\u542b\u4e09\u4e2a\u8981\u7d20\u3002
- \u7ec8\u6b62\u6761\u4ef6\uff1a\u7528\u4e8e\u51b3\u5b9a\u4ec0\u4e48\u65f6\u5019\u7531\u201c\u9012\u201d\u8f6c\u201c\u5f52\u201d\u3002
- \u9012\u5f52\u8c03\u7528\uff1a\u5bf9\u5e94\u201c\u9012\u201d\uff0c\u51fd\u6570\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u8f93\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\u3002
- \u8fd4\u56de\u7ed3\u679c\uff1a\u5bf9\u5e94\u201c\u5f52\u201d\uff0c\u5c06\u5f53\u524d\u9012\u5f52\u5c42\u7ea7\u7684\u7ed3\u679c\u8fd4\u56de\u81f3\u4e0a\u4e00\u5c42\u3002
\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u53ea\u9700\u8c03\u7528\u51fd\u6570
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyrecur(n)\uff0c\u5c31\u53ef\u4ee5\u5b8c\u6210 \\(1 + 2 + \\dots + n\\) \u7684\u8ba1\u7b97\uff1a
recursion.cppdef recur(n: int) -> int:\n \"\"\"\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if n == 1:\n return 1\n # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n res = recur(n - 1)\n # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n
recursion.java/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.cs/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.go/* \u9012\u5f52 */\nint Recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = Recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.swift/* \u9012\u5f52 */\nfunc recur(n int) int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n res := recur(n - 1)\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n}\n
recursion.js/* \u9012\u5f52 */\nfunc recur(n: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n let res = recur(n: n - 1)\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n}\n
recursion.ts/* \u9012\u5f52 */\nfunction recur(n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n const res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.dart/* \u9012\u5f52 */\nfunction recur(n: number): number {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n const res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.rs/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.c/* \u9012\u5f52 */\nfn recur(n: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1;\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n let res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n n + res\n}\n
recursion.zig/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1) {\n return 1;\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n var res: i32 = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n\u56fe 2-3 \u5c55\u793a\u4e86\u8be5\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b\u3002
\u56fe 2-3 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b
\u867d\u7136\u4ece\u8ba1\u7b97\u89d2\u5ea6\u770b\uff0c\u8fed\u4ee3\u4e0e\u9012\u5f52\u53ef\u4ee5\u5f97\u5230\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u4f46\u5b83\u4eec\u4ee3\u8868\u4e86\u4e24\u79cd\u5b8c\u5168\u4e0d\u540c\u7684\u601d\u8003\u548c\u89e3\u51b3\u95ee\u9898\u7684\u8303\u5f0f\u3002
- \u8fed\u4ee3\uff1a\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u4ece\u6700\u57fa\u7840\u7684\u6b65\u9aa4\u5f00\u59cb\uff0c\u7136\u540e\u4e0d\u65ad\u91cd\u590d\u6216\u7d2f\u52a0\u8fd9\u4e9b\u6b65\u9aa4\uff0c\u76f4\u5230\u4efb\u52a1\u5b8c\u6210\u3002
- \u9012\u5f52\uff1a\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u8fd9\u4e9b\u5b50\u95ee\u9898\u548c\u539f\u95ee\u9898\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f\u3002\u63a5\u4e0b\u6765\u5c06\u5b50\u95ee\u9898\u7ee7\u7eed\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u57fa\u672c\u60c5\u51b5\u65f6\u505c\u6b62\uff08\u57fa\u672c\u60c5\u51b5\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff09\u3002
\u4ee5\u4e0a\u8ff0\u6c42\u548c\u51fd\u6570\u4e3a\u4f8b\uff0c\u8bbe\u95ee\u9898 \\(f(n) = 1 + 2 + \\dots + n\\) \u3002
- \u8fed\u4ee3\uff1a\u5728\u5faa\u73af\u4e2d\u6a21\u62df\u6c42\u548c\u8fc7\u7a0b\uff0c\u4ece \\(1\\) \u904d\u5386\u5230 \\(n\\) \uff0c\u6bcf\u8f6e\u6267\u884c\u6c42\u548c\u64cd\u4f5c\uff0c\u5373\u53ef\u6c42\u5f97 \\(f(n)\\) \u3002
- \u9012\u5f52\uff1a\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(f(n) = n + f(n-1)\\) \uff0c\u4e0d\u65ad\uff08\u9012\u5f52\u5730\uff09\u5206\u89e3\u4e0b\u53bb\uff0c\u76f4\u81f3\u57fa\u672c\u60c5\u51b5 \\(f(1) = 1\\) \u65f6\u7ec8\u6b62\u3002
\u9012\u5f52\u51fd\u6570\u6bcf\u6b21\u8c03\u7528\u81ea\u8eab\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u4e3a\u65b0\u5f00\u542f\u7684\u51fd\u6570\u5206\u914d\u5185\u5b58\uff0c\u4ee5\u5b58\u50a8\u5c40\u90e8\u53d8\u91cf\u3001\u8c03\u7528\u5730\u5740\u548c\u5176\u4ed6\u4fe1\u606f\u7b49\u3002\u8fd9\u5c06\u5bfc\u81f4\u4e24\u65b9\u9762\u7684\u7ed3\u679c\u3002
- \u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u90fd\u5b58\u50a8\u5728\u79f0\u4e3a\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u7684\u5185\u5b58\u533a\u57df\u4e2d\uff0c\u76f4\u81f3\u51fd\u6570\u8fd4\u56de\u540e\u624d\u4f1a\u88ab\u91ca\u653e\u3002\u56e0\u6b64\uff0c\u9012\u5f52\u901a\u5e38\u6bd4\u8fed\u4ee3\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002
- \u9012\u5f52\u8c03\u7528\u51fd\u6570\u4f1a\u4ea7\u751f\u989d\u5916\u7684\u5f00\u9500\u3002\u56e0\u6b64\u9012\u5f52\u901a\u5e38\u6bd4\u5faa\u73af\u7684\u65f6\u95f4\u6548\u7387\u66f4\u4f4e\u3002
\u5982\u56fe 2-4 \u6240\u793a\uff0c\u5728\u89e6\u53d1\u7ec8\u6b62\u6761\u4ef6\u524d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684\u9012\u5f52\u51fd\u6570\uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002
\u56fe 2-4 \u00a0 \u9012\u5f52\u8c03\u7528\u6df1\u5ea6
\u5728\u5b9e\u9645\u4e2d\uff0c\u7f16\u7a0b\u8bed\u8a00\u5141\u8bb8\u7684\u9012\u5f52\u6df1\u5ea6\u901a\u5e38\u662f\u6709\u9650\u7684\uff0c\u8fc7\u6df1\u7684\u9012\u5f52\u53ef\u80fd\u5bfc\u81f4\u6808\u6ea2\u51fa\u9519\u8bef\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","title":"2. \u00a0 \u5c3e\u9012\u5f52","text":"\u6709\u8da3\u7684\u662f\uff0c\u5982\u679c\u51fd\u6570\u5728\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u6b65\u624d\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5219\u8be5\u51fd\u6570\u53ef\u4ee5\u88ab\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u4f18\u5316\uff0c\u4f7f\u5176\u5728\u7a7a\u95f4\u6548\u7387\u4e0a\u4e0e\u8fed\u4ee3\u76f8\u5f53\u3002\u8fd9\u79cd\u60c5\u51b5\u88ab\u79f0\u4e3a\u300c\u5c3e\u9012\u5f52 tail recursion\u300d\u3002
- \u666e\u901a\u9012\u5f52\uff1a\u5f53\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u7684\u51fd\u6570\u540e\uff0c\u9700\u8981\u7ee7\u7eed\u6267\u884c\u4ee3\u7801\uff0c\u56e0\u6b64\u7cfb\u7edf\u9700\u8981\u4fdd\u5b58\u4e0a\u4e00\u5c42\u8c03\u7528\u7684\u4e0a\u4e0b\u6587\u3002
- \u5c3e\u9012\u5f52\uff1a\u9012\u5f52\u8c03\u7528\u662f\u51fd\u6570\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u4e2a\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u540e\uff0c\u65e0\u987b\u7ee7\u7eed\u6267\u884c\u5176\u4ed6\u64cd\u4f5c\uff0c\u56e0\u6b64\u7cfb\u7edf\u65e0\u987b\u4fdd\u5b58\u4e0a\u4e00\u5c42\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u3002
\u4ee5\u8ba1\u7b97 \\(1 + 2 + \\dots + n\\) \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u7ed3\u679c\u53d8\u91cf
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyres\u8bbe\u4e3a\u51fd\u6570\u53c2\u6570\uff0c\u4ece\u800c\u5b9e\u73b0\u5c3e\u9012\u5f52\uff1a
recursion.cppdef tail_recur(n, res):\n \"\"\"\u5c3e\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if n == 0:\n return res\n # \u5c3e\u9012\u5f52\u8c03\u7528\n return tail_recur(n - 1, res + n)\n
recursion.java/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.cs/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.go/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return TailRecur(n - 1, res + n);\n}\n
recursion.swift/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n-1, res+n)\n}\n
recursion.js/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n: n - 1, res: res + n)\n}\n
recursion.ts/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.dart/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.rs/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.c/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res;\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n tail_recur(n - 1, res + n)\n}\n
recursion.zig/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0) {\n return res;\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n\u5c3e\u9012\u5f52\u7684\u6267\u884c\u8fc7\u7a0b\u5982\u56fe 2-5 \u6240\u793a\u3002\u5bf9\u6bd4\u666e\u901a\u9012\u5f52\u548c\u5c3e\u9012\u5f52\uff0c\u4e24\u8005\u7684\u6c42\u548c\u64cd\u4f5c\u7684\u6267\u884c\u70b9\u662f\u4e0d\u540c\u7684\u3002
- \u666e\u901a\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u6bcf\u5c42\u8fd4\u56de\u540e\u90fd\u8981\u518d\u6267\u884c\u4e00\u6b21\u6c42\u548c\u64cd\u4f5c\u3002
- \u5c3e\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u53ea\u9700\u5c42\u5c42\u8fd4\u56de\u3002
\u56fe 2-5 \u00a0 \u5c3e\u9012\u5f52\u8fc7\u7a0b
Tip
\u8bf7\u6ce8\u610f\uff0c\u8bb8\u591a\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u5e76\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\u3002\u4f8b\u5982\uff0cPython \u9ed8\u8ba4\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u56e0\u6b64\u5373\u4f7f\u51fd\u6570\u662f\u5c3e\u9012\u5f52\u5f62\u5f0f\uff0c\u4ecd\u7136\u53ef\u80fd\u4f1a\u9047\u5230\u6808\u6ea2\u51fa\u95ee\u9898\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","title":"3. \u00a0 \u9012\u5f52\u6811","text":"\u5f53\u5904\u7406\u4e0e\u201c\u5206\u6cbb\u201d\u76f8\u5173\u7684\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u9012\u5f52\u5f80\u5f80\u6bd4\u8fed\u4ee3\u7684\u601d\u8def\u66f4\u52a0\u76f4\u89c2\u3001\u4ee3\u7801\u66f4\u52a0\u6613\u8bfb\u3002\u4ee5\u201c\u6590\u6ce2\u90a3\u5951\u6570\u5217\u201d\u4e3a\u4f8b\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u5217 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\) \uff0c\u6c42\u8be5\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u3002
\u8bbe\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u4e3a \\(f(n)\\) \uff0c\u6613\u5f97\u4e24\u4e2a\u7ed3\u8bba\u3002
- \u6570\u5217\u7684\u524d\u4e24\u4e2a\u6570\u5b57\u4e3a \\(f(1) = 0\\) \u548c \\(f(2) = 1\\) \u3002
- \u6570\u5217\u4e2d\u7684\u6bcf\u4e2a\u6570\u5b57\u662f\u524d\u4e24\u4e2a\u6570\u5b57\u7684\u548c\uff0c\u5373 \\(f(n) = f(n - 1) + f(n - 2)\\) \u3002
\u6309\u7167\u9012\u63a8\u5173\u7cfb\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5c06\u524d\u4e24\u4e2a\u6570\u5b57\u4f5c\u4e3a\u7ec8\u6b62\u6761\u4ef6\uff0c\u4fbf\u53ef\u5199\u51fa\u9012\u5f52\u4ee3\u7801\u3002\u8c03\u7528
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyfib(n)\u5373\u53ef\u5f97\u5230\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\uff1a
recursion.cppdef fib(n: int) -> int:\n \"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 or n == 2:\n return n - 1\n # \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n res = fib(n - 1) + fib(n - 2)\n # \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n
recursion.java/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.cs/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.go/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = Fib(n - 1) + Fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.swift/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n res := fib(n-1) + fib(n-2)\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n}\n
recursion.js/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n let res = fib(n: n - 1) + fib(n: n - 2)\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n}\n
recursion.ts/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n === 1 || n === 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n const res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.dart/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n === 1 || n === 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n const res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.rs/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.c/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1;\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n let res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c\n res\n}\n
recursion.zig/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 or n == 2) {\n return n - 1;\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n var res: i32 = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u6211\u4eec\u5728\u51fd\u6570\u5185\u9012\u5f52\u8c03\u7528\u4e86\u4e24\u4e2a\u51fd\u6570\uff0c\u8fd9\u610f\u5473\u7740\u4ece\u4e00\u4e2a\u8c03\u7528\u4ea7\u751f\u4e86\u4e24\u4e2a\u8c03\u7528\u5206\u652f\u3002\u5982\u56fe 2-6 \u6240\u793a\uff0c\u8fd9\u6837\u4e0d\u65ad\u9012\u5f52\u8c03\u7528\u4e0b\u53bb\uff0c\u6700\u7ec8\u5c06\u4ea7\u751f\u4e00\u68f5\u5c42\u6570\u4e3a \\(n\\) \u7684\u300c\u9012\u5f52\u6811 recursion tree\u300d\u3002
\u56fe 2-6 \u00a0 \u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u9012\u5f52\u6811
\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9012\u5f52\u4f53\u73b0\u4e86\u201c\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u5b50\u95ee\u9898\u201d\u7684\u601d\u7ef4\u8303\u5f0f\uff0c\u8fd9\u79cd\u5206\u6cbb\u7b56\u7565\u81f3\u5173\u91cd\u8981\u3002
- \u4ece\u7b97\u6cd5\u89d2\u5ea6\u770b\uff0c\u641c\u7d22\u3001\u6392\u5e8f\u3001\u56de\u6eaf\u3001\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u7b49\u8bb8\u591a\u91cd\u8981\u7b97\u6cd5\u7b56\u7565\u76f4\u63a5\u6216\u95f4\u63a5\u5730\u5e94\u7528\u4e86\u8fd9\u79cd\u601d\u7ef4\u65b9\u5f0f\u3002
- \u4ece\u6570\u636e\u7ed3\u6784\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u5929\u7136\u9002\u5408\u5904\u7406\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u76f8\u5173\u95ee\u9898\uff0c\u56e0\u4e3a\u5b83\u4eec\u975e\u5e38\u9002\u5408\u7528\u5206\u6cbb\u601d\u60f3\u8fdb\u884c\u5206\u6790\u3002
\u603b\u7ed3\u4ee5\u4e0a\u5185\u5bb9\uff0c\u5982\u8868 2-1 \u6240\u793a\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5b9e\u73b0\u3001\u6027\u80fd\u548c\u9002\u7528\u6027\u4e0a\u6709\u6240\u4e0d\u540c\u3002
\u8868 2-1 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52\u7279\u70b9\u5bf9\u6bd4
\u8fed\u4ee3 \u9012\u5f52 \u5b9e\u73b0\u65b9\u5f0f \u5faa\u73af\u7ed3\u6784 \u51fd\u6570\u8c03\u7528\u81ea\u8eab \u65f6\u95f4\u6548\u7387 \u6548\u7387\u901a\u5e38\u8f83\u9ad8\uff0c\u65e0\u51fd\u6570\u8c03\u7528\u5f00\u9500 \u6bcf\u6b21\u51fd\u6570\u8c03\u7528\u90fd\u4f1a\u4ea7\u751f\u5f00\u9500 \u5185\u5b58\u4f7f\u7528 \u901a\u5e38\u4f7f\u7528\u56fa\u5b9a\u5927\u5c0f\u7684\u5185\u5b58\u7a7a\u95f4 \u7d2f\u79ef\u51fd\u6570\u8c03\u7528\u53ef\u80fd\u4f7f\u7528\u5927\u91cf\u7684\u6808\u5e27\u7a7a\u95f4 \u9002\u7528\u95ee\u9898 \u9002\u7528\u4e8e\u7b80\u5355\u5faa\u73af\u4efb\u52a1\uff0c\u4ee3\u7801\u76f4\u89c2\u3001\u53ef\u8bfb\u6027\u597d \u9002\u7528\u4e8e\u5b50\u95ee\u9898\u5206\u89e3\uff0c\u5982\u6811\u3001\u56fe\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u7b49\uff0c\u4ee3\u7801\u7ed3\u6784\u7b80\u6d01\u3001\u6e05\u6670Tip
\u5982\u679c\u611f\u89c9\u4ee5\u4e0b\u5185\u5bb9\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5728\u8bfb\u5b8c\u201c\u6808\u201d\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\u3002
\u90a3\u4e48\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5177\u6709\u4ec0\u4e48\u5185\u5728\u8054\u7cfb\u5462\uff1f\u4ee5\u4e0a\u8ff0\u9012\u5f52\u51fd\u6570\u4e3a\u4f8b\uff0c\u6c42\u548c\u64cd\u4f5c\u5728\u9012\u5f52\u7684\u201c\u5f52\u201d\u9636\u6bb5\u8fdb\u884c\u3002\u8fd9\u610f\u5473\u7740\u6700\u521d\u88ab\u8c03\u7528\u7684\u51fd\u6570\u5b9e\u9645\u4e0a\u662f\u6700\u540e\u5b8c\u6210\u5176\u6c42\u548c\u64cd\u4f5c\u7684\uff0c\u8fd9\u79cd\u5de5\u4f5c\u673a\u5236\u4e0e\u6808\u7684\u201c\u5148\u5165\u540e\u51fa\u201d\u539f\u5219\u5f02\u66f2\u540c\u5de5\u3002
\u4e8b\u5b9e\u4e0a\uff0c\u201c\u8c03\u7528\u6808\u201d\u548c\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u8fd9\u7c7b\u9012\u5f52\u672f\u8bed\u5df2\u7ecf\u6697\u793a\u4e86\u9012\u5f52\u4e0e\u6808\u4e4b\u95f4\u7684\u5bc6\u5207\u5173\u7cfb\u3002
- \u9012\uff1a\u5f53\u51fd\u6570\u88ab\u8c03\u7528\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5728\u201c\u8c03\u7528\u6808\u201d\u4e0a\u4e3a\u8be5\u51fd\u6570\u5206\u914d\u65b0\u7684\u6808\u5e27\uff0c\u7528\u4e8e\u5b58\u50a8\u51fd\u6570\u7684\u5c40\u90e8\u53d8\u91cf\u3001\u53c2\u6570\u3001\u8fd4\u56de\u5730\u5740\u7b49\u6570\u636e\u3002
- \u5f52\uff1a\u5f53\u51fd\u6570\u5b8c\u6210\u6267\u884c\u5e76\u8fd4\u56de\u65f6\uff0c\u5bf9\u5e94\u7684\u6808\u5e27\u4f1a\u88ab\u4ece\u201c\u8c03\u7528\u6808\u201d\u4e0a\u79fb\u9664\uff0c\u6062\u590d\u4e4b\u524d\u51fd\u6570\u7684\u6267\u884c\u73af\u5883\u3002
\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u8c03\u7528\u6808\u7684\u884c\u4e3a\uff0c\u4ece\u800c\u5c06\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u5f62\u5f0f\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py
recursion.cppdef for_loop_recur(n: int) -> int:\n \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n # \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack = []\n res = 0\n # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in range(n, 0, -1):\n # \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.append(i)\n # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while stack:\n # \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop()\n # res = 1+2+3+...+n\n return res\n
recursion.java/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack<int> stack;\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.empty()) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.top();\n stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.cs/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n Stack<Integer> stack = new Stack<>();\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.isEmpty()) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.go/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n Stack<int> stack = new();\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.Push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.Count > 0) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.Pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.swift/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack := list.New()\n res := 0\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i := n; i > 0; i-- {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.PushBack(i)\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n for stack.Len() != 0 {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.Back().Value.(int)\n stack.Remove(stack.Back())\n }\n // res = 1+2+3+...+n\n return res\n}\n
recursion.js/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -> Int {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n var stack: [Int] = []\n var res = 0\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in stride(from: n, to: 0, by: -1) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.append(i)\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while !stack.isEmpty {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.removeLast()\n }\n // res = 1+2+3+...+n\n return res\n}\n
recursion.ts/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n const stack = [];\n let res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (let i = 1; i <= n; i++) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.length) { \n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.dart/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \n const stack: number[] = [];\n let res: number = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (let i = 1; i <= n; i++) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.length) { \n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.rs/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n List<int> stack = [];\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.add(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.isEmpty) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.removeLast();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.c/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -> i32 {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n let mut stack = Vec::new();\n let mut res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in (1..=n).rev() {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while !stack.is_empty() {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop().unwrap();\n }\n // res = 1+2+3+...+n\n res\n}\n
recursion.zig/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n int stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\n int top = -1; // \u6808\u9876\u7d22\u5f15\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack[1 + top++] = i;\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (top >= 0) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack[top--];\n }\n // res = 1+2+3+...+n\n return res;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n var stack: [n]i32 = undefined;\n var res: i32 = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n var i: usize = n;\n while (i > 0) {\n stack[i - 1] = @intCast(i);\n i -= 1;\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n var index: usize = n;\n while (index > 0) {\n index -= 1;\n res += stack[index];\n }\n // res = 1+2+3+...+n\n return res;\n}\n\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u5f53\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u540e\uff0c\u4ee3\u7801\u53d8\u5f97\u66f4\u52a0\u590d\u6742\u4e86\u3002\u5c3d\u7ba1\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4e92\u76f8\u8f6c\u5316\uff0c\u4f46\u4e0d\u4e00\u5b9a\u503c\u5f97\u8fd9\u6837\u505a\uff0c\u6709\u4ee5\u4e0b\u4e24\u70b9\u539f\u56e0\u3002
- \u8f6c\u5316\u540e\u7684\u4ee3\u7801\u53ef\u80fd\u66f4\u52a0\u96be\u4ee5\u7406\u89e3\uff0c\u53ef\u8bfb\u6027\u66f4\u5dee\u3002
- \u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\u7684\u884c\u4e3a\u53ef\u80fd\u975e\u5e38\u56f0\u96be\u3002
\u603b\u4e4b\uff0c\u9009\u62e9\u8fed\u4ee3\u8fd8\u662f\u9012\u5f52\u53d6\u51b3\u4e8e\u7279\u5b9a\u95ee\u9898\u7684\u6027\u8d28\u3002\u5728\u7f16\u7a0b\u5b9e\u8df5\u4e2d\uff0c\u6743\u8861\u4e24\u8005\u7684\u4f18\u52a3\u5e76\u6839\u636e\u60c5\u5883\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u81f3\u5173\u91cd\u8981\u3002
"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":"\u5728\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u8ffd\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\u3002
- \u627e\u5230\u95ee\u9898\u89e3\u6cd5\uff1a\u7b97\u6cd5\u9700\u8981\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u5185\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002
- \u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\uff1a\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u5c3d\u53ef\u80fd\u9ad8\u6548\u7684\u7b97\u6cd5\u3002
\u4e5f\u5c31\u662f\u8bf4\uff0c\u5728\u80fd\u591f\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u5df2\u6210\u4e3a\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\uff0c\u5b83\u5305\u62ec\u4ee5\u4e0b\u4e24\u4e2a\u7ef4\u5ea6\u3002
- \u65f6\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002
- \u7a7a\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u7684\u5927\u5c0f\u3002
\u7b80\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bbe\u8ba1\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u3002\u800c\u6709\u6548\u5730\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u53ea\u6709\u8fd9\u6837\u6211\u4eec\u624d\u80fd\u5c06\u5404\u79cd\u7b97\u6cd5\u8fdb\u884c\u5bf9\u6bd4\uff0c\u8fdb\u800c\u6307\u5bfc\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u4f18\u5316\u8fc7\u7a0b\u3002
\u6548\u7387\u8bc4\u4f30\u65b9\u6cd5\u4e3b\u8981\u5206\u4e3a\u4e24\u79cd\uff1a\u5b9e\u9645\u6d4b\u8bd5\u3001\u7406\u8bba\u4f30\u7b97\u3002
"},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1 \u00a0 \u5b9e\u9645\u6d4b\u8bd5","text":"\u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5
A\u548c\u7b97\u6cd5B\uff0c\u5b83\u4eec\u90fd\u80fd\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u8fd9\u4e24\u4e2a\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u8fd0\u884c\u8fd9\u4e24\u4e2a\u7b97\u6cd5\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u5b83\u4eec\u7684\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u4e5f\u5b58\u5728\u8f83\u5927\u7684\u5c40\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u7b97\u6cd5\u7684\u6027\u80fd\u3002\u6bd4\u5982\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5
A\u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5B\u77ed\uff1b\u4f46\u5728\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u53ef\u80fd\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u8fdb\u884c\u6d4b\u8bd5\uff0c\u7edf\u8ba1\u5e73\u5747\u6548\u7387\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u8868\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5
"},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2 \u00a0 \u7406\u8bba\u4f30\u7b97","text":"A\u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5B\u77ed\uff1b\u800c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u53ef\u80fd\u6070\u6070\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u5f97\u5230\u6709\u8bf4\u670d\u529b\u7684\u7ed3\u8bba\uff0c\u6211\u4eec\u9700\u8981\u6d4b\u8bd5\u5404\u79cd\u89c4\u6a21\u7684\u8f93\u5165\u6570\u636e\uff0c\u800c\u8fd9\u9700\u8981\u8017\u8d39\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u3002\u7531\u4e8e\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u8f83\u5927\u7684\u5c40\u9650\u6027\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\u6765\u8bc4\u4f30\u7b97\u6cd5\u7684\u6548\u7387\u3002\u8fd9\u79cd\u4f30\u7b97\u65b9\u6cd5\u88ab\u79f0\u4e3a\u300c\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 asymptotic complexity analysis\u300d\uff0c\u7b80\u79f0\u300c\u590d\u6742\u5ea6\u5206\u6790\u300d\u3002
\u590d\u6742\u5ea6\u5206\u6790\u80fd\u591f\u4f53\u73b0\u7b97\u6cd5\u8fd0\u884c\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u5b83\u63cf\u8ff0\u4e86\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\uff0c\u7b97\u6cd5\u6267\u884c\u6240\u9700\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u5b9a\u4e49\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\u3002
- \u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u201d\u5206\u522b\u5bf9\u5e94\u300c\u65f6\u95f4\u590d\u6742\u5ea6 time complexity\u300d\u548c\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u3002
- \u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\u201d\u610f\u5473\u7740\u590d\u6742\u5ea6\u53cd\u6620\u4e86\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u4f53\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002
- \u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u5173\u6ce8\u7684\u4e0d\u662f\u8fd0\u884c\u65f6\u95f4\u6216\u5360\u7528\u7a7a\u95f4\u7684\u5177\u4f53\u503c\uff0c\u800c\u662f\u65f6\u95f4\u6216\u7a7a\u95f4\u589e\u957f\u7684\u201c\u5feb\u6162\u201d\u3002
\u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\uff0c\u4f53\u73b0\u5728\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u9762\u3002
- \u5b83\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002
- \u5b83\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002
Tip
\u5982\u679c\u4f60\u4ecd\u5bf9\u590d\u6742\u5ea6\u7684\u6982\u5ff5\u611f\u5230\u56f0\u60d1\uff0c\u65e0\u987b\u62c5\u5fc3\uff0c\u6211\u4eec\u4f1a\u5728\u540e\u7eed\u7ae0\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002
\u590d\u6742\u5ea6\u5206\u6790\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u628a\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u4f7f\u6211\u4eec\u53ef\u4ee5\u8861\u91cf\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u5bf9\u6bd4\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002
\u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u770b\uff0c\u590d\u6742\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9002\u5408\u4f5c\u4e3a\u6700\u5148\u4ecb\u7ecd\u7684\u5185\u5bb9\u3002\u7136\u800c\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u96be\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002
\u7efc\u4e0a\u6240\u8ff0\uff0c\u5efa\u8bae\u4f60\u5728\u6df1\u5165\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5148\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u5efa\u7acb\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u4ee5\u4fbf\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002
"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u5e38\u7c7b\u4f3c\uff0c\u53ea\u9700\u5c06\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u66ff\u6362\u4e3a\u201c\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u201d\u3002
"},{"location":"chapter_computational_complexity/space_complexity/#241","title":"2.4.1 \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002
- \u8f93\u5165\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\u3002
- \u6682\u5b58\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\u3002
- \u8f93\u51fa\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\u3002
\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u7edf\u8ba1\u8303\u56f4\u662f\u201c\u6682\u5b58\u7a7a\u95f4\u201d\u52a0\u4e0a\u201c\u8f93\u51fa\u7a7a\u95f4\u201d\u3002
\u6682\u5b58\u7a7a\u95f4\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002
- \u6682\u5b58\u6570\u636e\uff1a\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5404\u79cd\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u7b49\u3002
- \u6808\u5e27\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u5728\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\u90fd\u4f1a\u5728\u6808\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u540e\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002
- \u6307\u4ee4\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8ba1\u3002
\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5e8f\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u7edf\u8ba1\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u8f93\u51fa\u6570\u636e\u4e09\u90e8\u5206\uff0c\u5982\u56fe 2-15 \u6240\u793a\u3002
\u56fe 2-15 \u00a0 \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4
\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZigclass Node:\n \"\"\"\u7c7b\"\"\"\n def __init__(self, x: int):\n self.val: int = x # \u8282\u70b9\u503c\n self.next: Node | None = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\ndef function() -> int:\n \"\"\"\u51fd\u6570\"\"\"\n # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n\ndef algorithm(n) -> int: # \u8f93\u5165\u6570\u636e\n A = 0 # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\n b = 0 # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n node = Node(0) # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n c = function() # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return A + b + c # \u8f93\u51fa\u6570\u636e\n/* \u7ed3\u6784\u4f53 */\nstruct Node {\n int val;\n Node *next;\n Node(int x) : val(x), next(nullptr) {}\n};\n\n/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n const int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n Node* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n int c = func(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n int val;\n Node next;\n Node(int x) { val = x; }\n}\n\n/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n final int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n Node node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n int c = function(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node(int x) {\n int val = x;\n Node next;\n}\n\n/* \u51fd\u6570 */\nint Function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint Algorithm(int n) { // \u8f93\u5165\u6570\u636e\n const int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n Node node = new(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n int c = Function(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7ed3\u6784\u4f53 */\ntype node struct {\n val int\n next *node\n}\n\n/* \u521b\u5efa node \u7ed3\u6784\u4f53 */\nfunc newNode(val int) *node {\n return &node{val: val}\n}\n\n/* \u51fd\u6570 */\nfunc function() int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n}\n\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\n const a = 0 // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n b := 0 // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n newNode(0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n c := function() // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n var val: Int\n var next: Node?\n\n init(x: Int) {\n val = x\n }\n}\n\n/* \u51fd\u6570 */\nfunc function() -> Int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n}\n\nfunc algorithm(n: Int) -> Int { // \u8f93\u5165\u6570\u636e\n let a = 0 // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n var b = 0 // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n let node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n let c = function() // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n val;\n next;\n constructor(val) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = null; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\nfunction algorithm(n) { // \u8f93\u5165\u6570\u636e\n const a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n let b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n const node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n const c = constFunc(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n val: number;\n next: Node | null;\n constructor(val?: number) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = null; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\n const a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n let b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n const node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n const c = constFunc(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n int val;\n Node next;\n Node(this.val, [this.next]);\n}\n\n/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n const int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n Node node = Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n int c = function(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u7ed3\u6784\u4f53 */\nstruct Node {\n val: i32,\n next: Option<Rc<RefCell<Node>>>,\n}\n\n/* \u521b\u5efa Node \u7ed3\u6784\u4f53 */\nimpl Node {\n fn new(val: i32) -> Self {\n Self { val: val, next: None }\n }\n}\n\n/* \u51fd\u6570 */\nfn function() -> i32 { \n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nfn algorithm(n: i32) -> i32 { // \u8f93\u5165\u6570\u636e\n const a: i32 = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n let mut b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n let node = Node::new(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n let c = function(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n const int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n int c = func(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n
"},{"location":"chapter_computational_complexity/space_complexity/#242","title":"2.4.2 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"\n\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u9700\u5c06\u7edf\u8ba1\u5bf9\u8c61\u4ece\u201c\u64cd\u4f5c\u6570\u91cf\u201d\u8f6c\u4e3a\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002
\u800c\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u3002\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u9879\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u786e\u4fdd\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002
\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\u3002
- \u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\uff1a\u5f53 \\(n < 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u5f53 \\(n > 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4
nums\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002 - \u4ee5\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\uff1a\u4f8b\u5982\uff0c\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u5360\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4
nums\u65f6\uff0c\u7a0b\u5e8f\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
def algorithm(n: int):\n a = 0 # O(1)\n b = [0] * 10000 # O(1)\n if n > 10:\n nums = [0] * n # O(n)\nvoid algorithm(int n) {\n int a = 0; // O(1)\n vector<int> b(10000); // O(1)\n if (n > 10)\n vector<int> nums(n); // O(n)\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n int[] b = new int[10000]; // O(1)\n if (n > 10)\n int[] nums = new int[n]; // O(n)\n}\nvoid Algorithm(int n) {\n int a = 0; // O(1)\n int[] b = new int[10000]; // O(1)\n if (n > 10) {\n int[] nums = new int[n]; // O(n)\n }\n}\nfunc algorithm(n int) {\n a := 0 // O(1)\n b := make([]int, 10000) // O(1)\n var nums []int\n if n > 10 {\n nums := make([]int, n) // O(n)\n }\n fmt.Println(a, b, nums)\n}\nfunc algorithm(n: Int) {\n let a = 0 // O(1)\n let b = Array(repeating: 0, count: 10000) // O(1)\n if n > 10 {\n let nums = Array(repeating: 0, count: n) // O(n)\n }\n}\nfunction algorithm(n) {\n const a = 0; // O(1)\n const b = new Array(10000); // O(1)\n if (n > 10) {\n const nums = new Array(n); // O(n)\n }\n}\nfunction algorithm(n: number): void {\n const a = 0; // O(1)\n const b = new Array(10000); // O(1)\n if (n > 10) {\n const nums = new Array(n); // O(n)\n }\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n List<int> b = List.filled(10000, 0); // O(1)\n if (n > 10) {\n List<int> nums = List.filled(n, 0); // O(n)\n }\n}\nfn algorithm(n: i32) {\n let a = 0; // O(1)\n let b = [0; 10000]; // O(1)\n if n > 10 {\n let nums = vec![0; n as usize]; // O(n)\n }\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n int b[10000]; // O(1)\n if (n > 10)\n int nums[n] = {0}; // O(n)\n}\n\n\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef function() -> int:\n # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n\ndef loop(n: int):\n \"\"\"\u5faa\u73af O(1)\"\"\"\n for _ in range(n):\n function()\n\ndef recur(n: int) -> int:\n \"\"\"\u9012\u5f52 O(n)\"\"\"\n if n == 1: return\n return recur(n - 1)\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nint Function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid Loop(int n) {\n for (int i = 0; i < n; i++) {\n Function();\n }\n}\n/* \u9012\u5f52 O(n) */\nint Recur(int n) {\n if (n == 1) return 1;\n return Recur(n - 1);\n}\nfunc function() int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\n for i := 0; i < n; i++ {\n function()\n }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\n if n == 1 {\n return\n }\n recur(n - 1)\n}\n@discardableResult\nfunc function() -> Int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\n for _ in 0 ..< n {\n function()\n }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\n if n == 1 {\n return\n }\n recur(n: n - 1)\n}\nfunction constFunc() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\n if (n === 1) return;\n return recur(n - 1);\n}\nfunction constFunc(): number {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\n if (n === 1) return;\n return recur(n - 1);\n}\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nfn function() -> i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nfn loop(n: i32) {\n for i in 0..n {\n function();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(n: i32) {\n if n == 1 {\n return;\n }\n recur(n - 1);\n}\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\n\n\u51fd\u6570
loop()\u548crecur()\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u4f46\u7a7a\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u3002- \u51fd\u6570
loop()\u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21function()\uff0c\u6bcf\u8f6e\u4e2d\u7684function()\u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002 - \u9012\u5f52\u51fd\u6570
recur()\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684recur()\uff0c\u4ece\u800c\u5360\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002
\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u56fe 2-16 \u5c55\u793a\u4e86\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09\u3002
\\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n^2) < O(2^n) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} \\end{aligned} \\]\u56fe 2-16 \u00a0 \u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b
"},{"location":"chapter_computational_complexity/space_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"\u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002
\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u56e0\u6b64\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef function() -> int:\n \"\"\"\u51fd\u6570\"\"\"\n # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n\ndef constant(n: int):\n \"\"\"\u5e38\u6570\u9636\"\"\"\n # \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n a = 0\n nums = [0] * 10000\n node = ListNode(0)\n # \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in range(n):\n c = 0\n # \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in range(n):\n function()\n
space_complexity.java/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const int a = 0;\n int b = 0;\n vector<int> nums(10000);\n ListNode node(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n
space_complexity.cs/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n final int a = 0;\n int b = 0;\n int[] nums = new int[10000];\n ListNode node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n
space_complexity.go/* \u51fd\u6570 */\nint Function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n int a = 0;\n int b = 0;\n int[] nums = new int[10000];\n ListNode node = new(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n Function();\n }\n}\n
space_complexity.swift/* \u51fd\u6570 */\nfunc function() int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0\n b := 0\n nums := make([]int, 10000)\n ListNode := newNode(0)\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n var c int\n for i := 0; i < n; i++ {\n c = 0\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for i := 0; i < n; i++ {\n function()\n }\n fmt.Println(a, b, nums, c, ListNode)\n}\n
space_complexity.js/* \u51fd\u6570 */\n@discardableResult\nfunc function() -> Int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n let a = 0\n var b = 0\n let nums = Array(repeating: 0, count: 10000)\n let node = ListNode(x: 0)\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in 0 ..< n {\n let c = 0\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in 0 ..< n {\n function()\n }\n}\n
space_complexity.ts/* \u51fd\u6570 */\nfunction constFunc() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0;\n const b = 0;\n const nums = new Array(10000);\n const node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n const c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n
space_complexity.dart/* \u51fd\u6570 */\nfunction constFunc(): number {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0;\n const b = 0;\n const nums = new Array(10000);\n const node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n const c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n
space_complexity.rs/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n final int a = 0;\n int b = 0;\n List<int> nums = List.filled(10000, 0);\n ListNode node = ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (var i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (var i = 0; i < n; i++) {\n function();\n }\n}\n
space_complexity.c/* \u51fd\u6570 */\nfn function() ->i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const A: i32 = 0;\n let b = 0;\n let nums = vec![0; 10000];\n let node = ListNode::new(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for i in 0..n {\n let c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for i in 0..n {\n function();\n }\n}\n
space_complexity.zig/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const int a = 0;\n int b = 0;\n int nums[1000];\n ListNode *node = newListNode(0);\n free(node);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n
"},{"location":"chapter_computational_complexity/space_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"// \u51fd\u6570\nfn function() i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a: i32 = 0;\n var b: i32 = 0;\n var nums = [_]i32{0}**10000;\n var node = inc.ListNode(i32){.val = 0};\n var i: i32 = 0;\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n while (i < n) : (i += 1) {\n var c: i32 = 0;\n _ = c;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n i = 0;\n while (i < n) : (i += 1) {\n _ = function();\n }\n _ = a;\n _ = b;\n _ = nums;\n _ = node;\n}\n\u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef linear(n: int):\n \"\"\"\u7ebf\u6027\u9636\"\"\"\n # \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n nums = [0] * n\n # \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n hmap = dict[int, str]()\n for i in range(n):\n hmap[i] = str(i)\n
space_complexity.java/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n vector<int> nums(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n vector<ListNode> nodes;\n for (int i = 0; i < n; i++) {\n nodes.push_back(ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n unordered_map<int, string> map;\n for (int i = 0; i < n; i++) {\n map[i] = to_string(i);\n }\n}\n
space_complexity.cs/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int[] nums = new int[n];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n nodes.add(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Map<Integer, String> map = new HashMap<>();\n for (int i = 0; i < n; i++) {\n map.put(i, String.valueOf(i));\n }\n}\n
space_complexity.go/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int[] nums = new int[n];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = [];\n for (int i = 0; i < n; i++) {\n nodes.Add(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Dictionary<int, string> map = [];\n for (int i = 0; i < n; i++) {\n map.Add(i, i.ToString());\n }\n}\n
space_complexity.swift/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n _ = make([]int, n)\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var nodes []*node\n for i := 0; i < n; i++ {\n nodes = append(nodes, newNode(i))\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n m := make(map[int]string, n)\n for i := 0; i < n; i++ {\n m[i] = strconv.Itoa(i)\n }\n}\n
space_complexity.js/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n let nums = Array(repeating: 0, count: n)\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let nodes = (0 ..< n).map { ListNode(x: $0) }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, \"\\($0)\") })\n}\n
space_complexity.ts/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n const nums = new Array(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const nodes = [];\n for (let i = 0; i < n; i++) {\n nodes.push(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const map = new Map();\n for (let i = 0; i < n; i++) {\n map.set(i, i.toString());\n }\n}\n
space_complexity.dart/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n const nums = new Array(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const nodes: ListNode[] = [];\n for (let i = 0; i < n; i++) {\n nodes.push(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const map = new Map();\n for (let i = 0; i < n; i++) {\n map.set(i, i.toString());\n }\n}\n
space_complexity.rs/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n List<int> nums = List.filled(n, 0);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = [];\n for (var i = 0; i < n; i++) {\n nodes.add(ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Map<int, String> map = HashMap();\n for (var i = 0; i < n; i++) {\n map.putIfAbsent(i, () => i.toString());\n }\n}\n
space_complexity.c/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n let mut nums = vec![0; n as usize];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let mut nodes = Vec::new();\n for i in 0..n {\n nodes.push(ListNode::new(i))\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let mut map = HashMap::new();\n for i in 0..n {\n map.insert(i, i.to_string());\n }\n}\n
space_complexity.zig/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n int key;\n int val;\n UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int *nums = malloc(sizeof(int) * n);\n free(nums);\n\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n ListNode **nodes = malloc(sizeof(ListNode *) * n);\n for (int i = 0; i < n; i++) {\n nodes[i] = newListNode(i);\n }\n // \u5185\u5b58\u91ca\u653e\n for (int i = 0; i < n; i++) {\n free(nodes[i]);\n }\n free(nodes);\n\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n HashTable *h = NULL;\n for (int i = 0; i < n; i++) {\n HashTable *tmp = malloc(sizeof(HashTable));\n tmp->key = i;\n tmp->val = i;\n HASH_ADD_INT(h, key, tmp);\n }\n\n // \u5185\u5b58\u91ca\u653e\n HashTable *curr, *tmp;\n HASH_ITER(hh, h, curr, tmp) {\n HASH_DEL(h, curr);\n free(curr);\n }\n}\n// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n var nums = [_]i32{0}**n;\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n defer nodes.deinit();\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n try nodes.append(i);\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n defer map.deinit();\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n defer std.heap.page_allocator.free(string);\n try map.put(i, string);\n }\n _ = nums;\n}\n\u5982\u56fe 2-17 \u6240\u793a\uff0c\u6b64\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5373\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.pylinear_recur()\u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\uff1a
space_complexity.cppdef linear_recur(n: int):\n \"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n print(\"\u9012\u5f52 n =\", n)\n if n == 1:\n return\n linear_recur(n - 1)\n
space_complexity.java/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n cout << \"\u9012\u5f52 n = \" << n << endl;\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n
space_complexity.cs/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n System.out.println(\"\u9012\u5f52 n = \" + n);\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n
space_complexity.go/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\n Console.WriteLine(\"\u9012\u5f52 n = \" + n);\n if (n == 1) return;\n LinearRecur(n - 1);\n}\n
space_complexity.swift/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\n fmt.Println(\"\u9012\u5f52 n =\", n)\n if n == 1 {\n return\n }\n spaceLinearRecur(n - 1)\n}\n
space_complexity.js/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\n print(\"\u9012\u5f52 n = \\(n)\")\n if n == 1 {\n return\n }\n linearRecur(n: n - 1)\n}\n
space_complexity.ts/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\n console.log(`\u9012\u5f52 n = ${n}`);\n if (n === 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.dart/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\n console.log(`\u9012\u5f52 n = ${n}`);\n if (n === 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.rs/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n print('\u9012\u5f52 n = $n');\n if (n == 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.c/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\n println!(\"\u9012\u5f52 n = {}\", n);\n if n == 1 {return};\n linear_recur(n - 1);\n}\n
space_complexity.zig/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n printf(\"\u9012\u5f52 n = %d\\r\\n\", n);\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\n std.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\n if (n == 1) return;\n linearRecur(n - 1);\n}\n\u56fe 2-17 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6
"},{"location":"chapter_computational_complexity/space_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"\u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u77e9\u9635\u548c\u56fe\uff0c\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef quadratic(n: int):\n \"\"\"\u5e73\u65b9\u9636\"\"\"\n # \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n num_matrix = [[0] * n for _ in range(n)]\n
space_complexity.java/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n vector<vector<int>> numMatrix;\n for (int i = 0; i < n; i++) {\n vector<int> tmp;\n for (int j = 0; j < n; j++) {\n tmp.push_back(0);\n }\n numMatrix.push_back(tmp);\n }\n}\n
space_complexity.cs/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n int[][] numMatrix = new int[n][n];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<Integer>> numList = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n List<Integer> tmp = new ArrayList<>();\n for (int j = 0; j < n; j++) {\n tmp.add(0);\n }\n numList.add(tmp);\n }\n}\n
space_complexity.go/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n int[,] numMatrix = new int[n, n];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numList = [];\n for (int i = 0; i < n; i++) {\n List<int> tmp = [];\n for (int j = 0; j < n; j++) {\n tmp.Add(0);\n }\n numList.Add(tmp);\n }\n}\n
space_complexity.swift/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n numMatrix := make([][]int, n)\n for i := 0; i < n; i++ {\n numMatrix[i] = make([]int, n)\n }\n}\n
space_complexity.js/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n
space_complexity.ts/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numMatrix = Array(n)\n .fill(null)\n .map(() => Array(n).fill(null));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numList = [];\n for (let i = 0; i < n; i++) {\n const tmp = [];\n for (let j = 0; j < n; j++) {\n tmp.push(0);\n }\n numList.push(tmp);\n }\n}\n
space_complexity.dart/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numMatrix = Array(n)\n .fill(null)\n .map(() => Array(n).fill(null));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numList = [];\n for (let i = 0; i < n; i++) {\n const tmp = [];\n for (let j = 0; j < n; j++) {\n tmp.push(0);\n }\n numList.push(tmp);\n }\n}\n
space_complexity.rs/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numMatrix = List.generate(n, (_) => List.filled(n, 0));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numList = [];\n for (var i = 0; i < n; i++) {\n List<int> tmp = [];\n for (int j = 0; j < n; j++) {\n tmp.add(0);\n }\n numList.add(tmp);\n }\n}\n
space_complexity.c/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n let num_matrix = vec![vec![0; n as usize]; n as usize];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n let mut num_list = Vec::new();\n for i in 0..n {\n let mut tmp = Vec::new();\n for j in 0..n {\n tmp.push(0);\n }\n num_list.push(tmp);\n }\n}\n
space_complexity.zig/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n int **numMatrix = malloc(sizeof(int *) * n);\n for (int i = 0; i < n; i++) {\n int *tmp = malloc(sizeof(int) * n);\n for (int j = 0; j < n; j++) {\n tmp[j] = 0;\n }\n numMatrix[i] = tmp;\n }\n\n // \u5185\u5b58\u91ca\u653e\n for (int i = 0; i < n; i++) {\n free(numMatrix[i]);\n }\n free(numMatrix);\n}\n// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n defer nodes.deinit();\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n defer tmp.deinit();\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n try tmp.append(0);\n }\n try nodes.append(tmp);\n }\n}\n\u5982\u56fe 2-18 \u6240\u793a\uff0c\u8be5\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5728\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(n / 2\\) \uff0c\u56e0\u6b64\u603b\u4f53\u5360\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef quadratic_recur(n: int) -> int:\n \"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n <= 0:\n return 0\n # \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n nums = [0] * n\n return quadratic_recur(n - 1)\n
space_complexity.java/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n vector<int> nums(n);\n cout << \"\u9012\u5f52 n = \" << n << \" \u4e2d\u7684 nums \u957f\u5ea6 = \" << nums.size() << endl;\n return quadraticRecur(n - 1);\n}\n
space_complexity.cs/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n int[] nums = new int[n];\n System.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\n return quadraticRecur(n - 1);\n}\n
space_complexity.go/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\n if (n <= 0) return 0;\n int[] nums = new int[n];\n Console.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\n return QuadraticRecur(n - 1);\n}\n
space_complexity.swift/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n if n <= 0 {\n return 0\n }\n nums := make([]int, n)\n fmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\n return spaceQuadraticRecur(n - 1)\n}\n
space_complexity.js/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -> Int {\n if n <= 0 {\n return 0\n }\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n let nums = Array(repeating: 0, count: n)\n print(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\n return quadraticRecur(n: n - 1)\n}\n
space_complexity.ts/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\n if (n <= 0) return 0;\n const nums = new Array(n);\n console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n return quadraticRecur(n - 1);\n}\n
space_complexity.dart/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\n if (n <= 0) return 0;\n const nums = new Array(n);\n console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n return quadraticRecur(n - 1);\n}\n
space_complexity.rs/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0) return 0;\n List<int> nums = List.filled(n, 0);\n print('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\n return quadraticRecur(n - 1);\n}\n
space_complexity.c/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -> i32 {\n if n <= 0 {return 0};\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n let nums = vec![0; n as usize];\n println!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\n return quadratic_recur(n - 1);\n}\n
space_complexity.zig/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n int *nums = malloc(sizeof(int) * n);\n printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\n int res = quadraticRecur(n - 1);\n free(nums);\n return res;\n}\n// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n if (n <= 0) return 0;\n var nums = [_]i32{0}**n;\n std.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\n return quadraticRecur(n - 1);\n}\n\u56fe 2-18 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6
"},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"\u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u89c2\u5bdf\u56fe 2-19 \uff0c\u5c42\u6570\u4e3a \\(n\\) \u7684\u201c\u6ee1\u4e8c\u53c9\u6811\u201d\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u5360\u7528 \\(O(2^n)\\) \u7a7a\u95f4\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef build_tree(n: int) -> TreeNode | None:\n \"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\n if n == 0:\n return None\n root = TreeNode(0)\n root.left = build_tree(n - 1)\n root.right = build_tree(n - 1)\n return root\n
space_complexity.java/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n if (n == 0)\n return nullptr;\n TreeNode *root = new TreeNode(0);\n root->left = buildTree(n - 1);\n root->right = buildTree(n - 1);\n return root;\n}\n
space_complexity.cs/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\n if (n == 0)\n return null;\n TreeNode root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.go/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\n if (n == 0) return null;\n TreeNode root = new(0) {\n left = BuildTree(n - 1),\n right = BuildTree(n - 1)\n };\n return root;\n}\n
space_complexity.swift/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\n if n == 0 {\n return nil\n }\n root := newTreeNode(0)\n root.left = buildTree(n - 1)\n root.right = buildTree(n - 1)\n return root\n}\n
space_complexity.js/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -> TreeNode? {\n if n == 0 {\n return nil\n }\n let root = TreeNode(x: 0)\n root.left = buildTree(n: n - 1)\n root.right = buildTree(n: n - 1)\n return root\n}\n
space_complexity.ts/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\n if (n === 0) return null;\n const root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.dart/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n if (n === 0) return null;\n const root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.rs/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\n if (n == 0) return null;\n TreeNode root = TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.c/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -> Option<Rc<RefCell<TreeNode>>> {\n if n == 0 {return None};\n let root = TreeNode::new(0);\n root.borrow_mut().left = build_tree(n - 1);\n root.borrow_mut().right = build_tree(n - 1);\n return Some(root);\n}\n
space_complexity.zig/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n if (n == 0)\n return NULL;\n TreeNode *root = newTreeNode(0);\n root->left = buildTree(n - 1);\n root->right = buildTree(n - 1);\n return root;\n}\n// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n if (n == 0) return null;\n const root = try mem_allocator.create(inc.TreeNode(i32));\n root.init(0);\n root.left = try buildTree(mem_allocator, n - 1);\n root.right = try buildTree(mem_allocator, n - 1);\n return root;\n}\n\u56fe 2-19 \u00a0 \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6
"},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"\u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u3002\u4f8b\u5982\u5f52\u5e76\u6392\u5e8f\uff0c\u8f93\u5165\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u6bcf\u8f6e\u9012\u5f52\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6808\u5e27\u7a7a\u95f4\u3002
\u518d\u4f8b\u5982\u5c06\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\uff0c\u8f93\u5165\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\log_{10} n + 1\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\log_{10} n + 1\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n + 1) = O(\\log n)\\) \u3002
"},{"location":"chapter_computational_complexity/space_complexity/#244","title":"2.4.4 \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u8fbe\u5230\u6700\u4f18\u3002\u7136\u800c\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u975e\u5e38\u56f0\u96be\u3002
\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u5c06\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff1b\u53cd\u4e4b\uff0c\u5219\u79f0\u4e3a\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002
\u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\uff0c\u56e0\u6b64\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u5f53\u7136\uff0c\u5728\u6570\u636e\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u63a7\u5236\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u975e\u5e38\u91cd\u8981\u3002
"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_computational_complexity/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"\u7b97\u6cd5\u6548\u7387\u8bc4\u4f30
- \u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e24\u4e2a\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\u3002
- \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6d88\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5f71\u54cd\uff0c\u4e14\u4f1a\u8017\u8d39\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002
- \u590d\u6742\u5ea6\u5206\u6790\u53ef\u4ee5\u6d88\u9664\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u80fd\u591f\u63ed\u793a\u7b97\u6cd5\u5728\u4e0d\u540c\u6570\u636e\u89c4\u6a21\u4e0b\u7684\u6548\u7387\u3002
\u65f6\u95f4\u590d\u6742\u5ea6
- \u65f6\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f93\u5165\u7684\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u3002
- \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u5411\u6b63\u65e0\u7a77\u65f6\uff0c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u7684\u589e\u957f\u7ea7\u522b\u3002
- \u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002
- \u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n \\log n)\\)\u3001\\(O(n^2)\\)\u3001\\(O(2^n)\\) \u548c \\(O(n!)\\) \u7b49\u3002
- \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u4e00\u822c\u9700\u8981\u6ee1\u8db3\u4e25\u683c\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002
- \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53cd\u6620\u7b97\u6cd5\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u5206\u5e03\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002
\u7a7a\u95f4\u590d\u6742\u5ea6
- \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u4f5c\u7528\u7c7b\u4f3c\u4e8e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\u3002
- \u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u7eb3\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u6307\u4ee4\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u901a\u5e38\u4ec5\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u5f71\u54cd\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
- \u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u523b\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
- \u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n^2)\\) \u548c \\(O(2^n)\\) \u7b49\u3002
\u5c3e\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u5417\uff1f
\u7406\u8bba\u4e0a\uff0c\u5c3e\u9012\u5f52\u51fd\u6570\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u8fc7\u7edd\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\u3001Python\u3001C++\u3001Go\u3001C# \u7b49\uff09\u4e0d\u652f\u6301\u81ea\u52a8\u4f18\u5316\u5c3e\u9012\u5f52\uff0c\u56e0\u6b64\u901a\u5e38\u8ba4\u4e3a\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \u3002
\u51fd\u6570\u548c\u65b9\u6cd5\u8fd9\u4e24\u4e2a\u672f\u8bed\u7684\u533a\u522b\u662f\u4ec0\u4e48\uff1f
\u300c\u51fd\u6570 function\u300d\u53ef\u4ee5\u88ab\u72ec\u7acb\u6267\u884c\uff0c\u6240\u6709\u53c2\u6570\u90fd\u4ee5\u663e\u5f0f\u4f20\u9012\u3002\u300c\u65b9\u6cd5 method\u300d\u4e0e\u4e00\u4e2a\u5bf9\u8c61\u5173\u8054\uff0c\u88ab\u9690\u5f0f\u4f20\u9012\u7ed9\u8c03\u7528\u5b83\u7684\u5bf9\u8c61\uff0c\u80fd\u591f\u5bf9\u7c7b\u7684\u5b9e\u4f8b\u4e2d\u5305\u542b\u7684\u6570\u636e\u8fdb\u884c\u64cd\u4f5c\u3002
\u4e0b\u9762\u4ee5\u51e0\u79cd\u5e38\u89c1\u7684\u7f16\u7a0b\u8bed\u8a00\u4e3a\u4f8b\u6765\u8bf4\u660e\u3002
- C \u8bed\u8a00\u662f\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\u8bed\u8a00\uff0c\u6ca1\u6709\u9762\u5411\u5bf9\u8c61\u7684\u6982\u5ff5\uff0c\u6240\u4ee5\u53ea\u6709\u51fd\u6570\u3002\u4f46\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u521b\u5efa\u7ed3\u6784\u4f53\uff08struct\uff09\u6765\u6a21\u62df\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff0c\u4e0e\u7ed3\u6784\u4f53\u76f8\u5173\u8054\u7684\u51fd\u6570\u5c31\u76f8\u5f53\u4e8e\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u65b9\u6cd5\u3002
- Java \u548c C# \u662f\u9762\u5411\u5bf9\u8c61\u7684\u7f16\u7a0b\u8bed\u8a00\uff0c\u4ee3\u7801\u5757\uff08\u65b9\u6cd5\uff09\u901a\u5e38\u4f5c\u4e3a\u67d0\u4e2a\u7c7b\u7684\u4e00\u90e8\u5206\u3002\u9759\u6001\u65b9\u6cd5\u7684\u884c\u4e3a\u7c7b\u4f3c\u4e8e\u51fd\u6570\uff0c\u56e0\u4e3a\u5b83\u88ab\u7ed1\u5b9a\u5728\u7c7b\u4e0a\uff0c\u4e0d\u80fd\u8bbf\u95ee\u7279\u5b9a\u7684\u5b9e\u4f8b\u53d8\u91cf\u3002
- C++ \u548c Python \u65e2\u652f\u6301\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\uff08\u51fd\u6570\uff09\uff0c\u4e5f\u652f\u6301\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff08\u65b9\u6cd5\uff09\u3002
\u56fe\u89e3\u201c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\uff1f
\u4e0d\u662f\uff0c\u8be5\u56fe\u5c55\u793a\u7684\u662f\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5176\u53cd\u6620\u7684\u662f\u589e\u957f\u8d8b\u52bf\uff0c\u800c\u4e0d\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\u3002
\u5047\u8bbe\u53d6 \\(n = 8\\) \uff0c\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u6bcf\u6761\u66f2\u7ebf\u7684\u503c\u4e0e\u51fd\u6570\u5bf9\u5e94\u4e0d\u4e0a\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6761\u66f2\u7ebf\u90fd\u5305\u542b\u4e00\u4e2a\u5e38\u6570\u9879\uff0c\u7528\u4e8e\u5c06\u53d6\u503c\u8303\u56f4\u538b\u7f29\u5230\u4e00\u4e2a\u89c6\u89c9\u8212\u9002\u7684\u8303\u56f4\u5185\u3002
\u5728\u5b9e\u9645\u4e2d\uff0c\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u4e2a\u65b9\u6cd5\u7684\u201c\u5e38\u6570\u9879\u201d\u590d\u6742\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u65e0\u6cd5\u4ec5\u51ed\u590d\u6742\u5ea6\u6765\u9009\u62e9 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u4f18\u89e3\u6cd5\u3002\u4f46\u5bf9\u4e8e \\(n = 8^5\\) \u5c31\u5f88\u597d\u9009\u4e86\uff0c\u8fd9\u65f6\u589e\u957f\u8d8b\u52bf\u5df2\u7ecf\u5360\u4e3b\u5bfc\u4e86\u3002
"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":"\u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u53cd\u6620\u7b97\u6cd5\u7684\u6548\u7387\u3002\u5982\u679c\u6211\u4eec\u60f3\u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u5e94\u8be5\u5982\u4f55\u64cd\u4f5c\u5462\uff1f
- \u786e\u5b9a\u8fd0\u884c\u5e73\u53f0\uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u56e0\u7d20\u90fd\u4f1a\u5f71\u54cd\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002
- \u8bc4\u4f30\u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c
+\u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c*\u9700\u8981 10 ns \uff0c\u6253\u5370\u64cd\u4f5cprint()\u9700\u8981 5 ns \u7b49\u3002 - \u7edf\u8ba1\u4ee3\u7801\u4e2d\u6240\u6709\u7684\u8ba1\u7b97\u64cd\u4f5c\uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002
\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig# \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int):\n a = 2 # 1 ns\n a = a + 1 # 1 ns\n a = a * 2 # 10 ns\n # \u5faa\u73af n \u6b21\n for _ in range(n): # 1 ns\n print(0) # 5 ns\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n cout << 0 << endl; // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n System.out.println(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid Algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n Console.WriteLine(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\n a := 2 // 1 ns\n a = a + 1 // 1 ns\n a = a * 2 // 10 ns\n // \u5faa\u73af n \u6b21\n for i := 0; i < n; i++ { // 1 ns\n fmt.Println(a) // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\n var a = 2 // 1 ns\n a = a + 1 // 1 ns\n a = a * 2 // 10 ns\n // \u5faa\u73af n \u6b21\n for _ in 0 ..< n { // 1 ns\n print(0) // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\n var a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n console.log(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\n var a: number = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n console.log(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n print(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: i32) {\n let mut a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for _ in 0..n { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n println!(\"{}\", 0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n printf(\"%d\", 0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: usize) void {\n var a: i32 = 2; // 1 ns\n a += 1; // 1 ns\n a *= 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (0..n) |_| { // 1 ns\n std.debug.print(\"{}\\n\", .{0}); // 5 ns\n }\n}\n\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u4e3a \\((6n + 12)\\) ns \uff1a
\\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\]\u4f46\u5b9e\u9645\u4e0a\uff0c\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u5c06\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u56e0\u4e3a\u7b97\u6cd5\u9700\u8981\u5728\u5404\u79cd\u4e0d\u540c\u7684\u5e73\u53f0\u4e0a\u8fd0\u884c\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u7ed9\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#231","title":"2.3.1 \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002
\u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u6bd4\u8f83\u62bd\u8c61\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5
PythonC++JavaC#GoSwiftJSTSDartRustCZigA\u3001B\u548cC\uff1a# \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int):\n print(0)\n# \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int):\n for _ in range(n):\n print(0)\n# \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int):\n for _ in range(1000000):\n print(0)\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n cout << 0 << endl;\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n cout << 0 << endl;\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n cout << 0 << endl;\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n System.out.println(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n System.out.println(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n System.out.println(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmA(int n) {\n Console.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid AlgorithmB(int n) {\n for (int i = 0; i < n; i++) {\n Console.WriteLine(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmC(int n) {\n for (int i = 0; i < 1000000; i++) {\n Console.WriteLine(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\n fmt.Println(0)\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\n for i := 0; i < n; i++ {\n fmt.Println(0)\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\n for i := 0; i < 1000000; i++ {\n fmt.Println(0)\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\n print(0)\n}\n\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\n for _ in 0 ..< n {\n print(0)\n }\n}\n\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\n for _ in 0 ..< 1000000 {\n print(0)\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\n console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\n for (let i = 0; i < n; i++) {\n console.log(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\n for (let i = 0; i < 1000000; i++) {\n console.log(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\n console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\n for (let i = 0; i < n; i++) {\n console.log(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\n for (let i = 0; i < 1000000; i++) {\n console.log(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmA(int n) {\n print(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithmB(int n) {\n for (int i = 0; i < n; i++) {\n print(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmC(int n) {\n for (int i = 0; i < 1000000; i++) {\n print(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: i32) {\n println!(\"{}\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) {\n for _ in 0..n {\n println!(\"{}\", 0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) {\n for _ in 0..1000000 {\n println!(\"{}\", 0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n printf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n printf(\"%d\", 0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n printf(\"%d\", 0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: usize) void {\n _ = n;\n std.debug.print(\"{}\\n\", .{0});\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) void {\n for (0..n) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) void {\n _ = n;\n for (0..1000000) |_| { \n std.debug.print(\"{}\\n\", .{0});\n }\n}\n\u56fe 2-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4e09\u4e2a\u7b97\u6cd5\u51fd\u6570\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002
- \u7b97\u6cd5
A\u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002 - \u7b97\u6cd5
B\u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u5448\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u9636\u201d\u3002 - \u7b97\u6cd5
C\u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u867d\u7136\u8fd0\u884c\u65f6\u95f4\u5f88\u957f\uff0c\u4f46\u5b83\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64C\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548cA\u76f8\u540c\uff0c\u4ecd\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002
\u56fe 2-7 \u00a0 \u7b97\u6cd5 A\u3001B \u548c C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf
\u76f8\u8f83\u4e8e\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u7279\u70b9\u5462\uff1f
- \u65f6\u95f4\u590d\u6742\u5ea6\u80fd\u591f\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5
B\u7684\u8fd0\u884c\u65f6\u95f4\u5448\u7ebf\u6027\u589e\u957f\uff0c\u5728 \\(n > 1\\) \u65f6\u6bd4\u7b97\u6cd5A\u66f4\u6162\uff0c\u5728 \\(n > 1000000\\) \u65f6\u6bd4\u7b97\u6cd5C\u66f4\u6162\u3002\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u201c\u7ebf\u6027\u9636\u201d\u7684\u7b97\u6cd5\uff0c\u8fd9\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u7684\u542b\u4e49\u3002 - \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7b80\u4fbf\u3002\u663e\u7136\uff0c\u8fd0\u884c\u5e73\u53f0\u548c\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u6b64\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u89c6\u4e3a\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u4ece\u800c\u5c06\u201c\u8ba1\u7b97\u64cd\u4f5c\u8fd0\u884c\u65f6\u95f4\u7edf\u8ba1\u201d\u7b80\u5316\u4e3a\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u7edf\u8ba1\u201d\uff0c\u8fd9\u6837\u4e00\u6765\u4f30\u7b97\u96be\u5ea6\u5c31\u5927\u5927\u964d\u4f4e\u4e86\u3002
- \u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5c3d\u7ba1\u7b97\u6cd5
A\u548cC\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\u5dee\u522b\u5f88\u5927\u3002\u540c\u6837\uff0c\u5c3d\u7ba1\u7b97\u6cd5B\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4C\u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5B\u660e\u663e\u4f18\u4e8e\u7b97\u6cd5C\u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u5224\u65ad\u7b97\u6cd5\u6548\u7387\u7684\u9ad8\u4f4e\u3002\u5f53\u7136\uff0c\u5c3d\u7ba1\u5b58\u5728\u4e0a\u8ff0\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002
\u7ed9\u5b9a\u4e00\u4e2a\u8f93\u5165\u5927\u5c0f\u4e3a \\(n\\) \u7684\u51fd\u6570\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef algorithm(n: int):\n a = 1 # +1\n a = a + 1 # +1\n a = a * 2 # +1\n # \u5faa\u73af n \u6b21\n for i in range(n): # +1\n print(0) # +1\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n cout << 0 << endl; // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n System.out.println(0); // +1\n }\n}\nvoid Algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n Console.WriteLine(0); // +1\n }\n}\nfunc algorithm(n int) {\n a := 1 // +1\n a = a + 1 // +1\n a = a * 2 // +1\n // \u5faa\u73af n \u6b21\n for i := 0; i < n; i++ { // +1\n fmt.Println(a) // +1\n }\n}\nfunc algorithm(n: Int) {\n var a = 1 // +1\n a = a + 1 // +1\n a = a * 2 // +1\n // \u5faa\u73af n \u6b21\n for _ in 0 ..< n { // +1\n print(0) // +1\n }\n}\nfunction algorithm(n) {\n var a = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // \u5faa\u73af n \u6b21\n for(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n console.log(0); // +1\n }\n}\nfunction algorithm(n: number): void{\n var a: number = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // \u5faa\u73af n \u6b21\n for(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n console.log(0); // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n print(0); // +1\n }\n}\nfn algorithm(n: i32) {\n let mut a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n\n // \u5faa\u73af n \u6b21\n for _ in 0..n { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n println!(\"{}\", 0); // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n printf(\"%d\", 0); // +1\n }\n} \nfn algorithm(n: usize) void {\n var a: i32 = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // \u5faa\u73af n \u6b21\n for (0..n) |_| { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n std.debug.print(\"{}\\n\", .{0}); // +1\n }\n}\n\u8bbe\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\uff0c\u8bb0\u4e3a \\(T(n)\\) \uff0c\u5219\u4ee5\u4e0a\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a\uff1a
\\[ T(n) = 3 + 2n \\]\\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u5176\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u5b83\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002
\u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u79f0\u4e3a\u300c\u5927 \\(O\\) \u8bb0\u53f7 big-\\(O\\) notation\u300d\uff0c\u8868\u793a\u51fd\u6570 \\(T(n)\\) \u7684\u300c\u6e10\u8fd1\u4e0a\u754c asymptotic upper bound\u300d\u3002
\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u672c\u8d28\u4e0a\u662f\u8ba1\u7b97\u201c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\)\u201d\u7684\u6e10\u8fd1\u4e0a\u754c\uff0c\u5b83\u5177\u6709\u660e\u786e\u7684\u6570\u5b66\u5b9a\u4e49\u3002
\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c
\u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n > n_0\\) \uff0c\u5747\u6709 \\(T(n) \\leq c \\cdot f(n)\\) \uff0c\u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a \\(T(n) = O(f(n))\\) \u3002
\u5982\u56fe 2-8 \u6240\u793a\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5bfb\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5f53 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff0c\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\u3002
\u56fe 2-8 \u00a0 \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c
"},{"location":"chapter_computational_complexity/time_complexity/#233","title":"2.3.3 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u65e0\u987b\u62c5\u5fc3\u3002\u6211\u4eec\u53ef\u4ee5\u5148\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u5728\u4e0d\u65ad\u7684\u5b9e\u8df5\u4e2d\uff0c\u5c31\u53ef\u4ee5\u9010\u6e10\u9886\u609f\u5176\u6570\u5b66\u610f\u4e49\u3002
\u6839\u636e\u5b9a\u4e49\uff0c\u786e\u5b9a \\(f(n)\\) \u4e4b\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\u5982\u4f55\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1. \u00a0 \u7b2c\u4e00\u6b65\uff1a\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"\u9488\u5bf9\u4ee3\u7801\uff0c\u9010\u884c\u4ece\u4e0a\u5230\u4e0b\u8ba1\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u7b80\u5316\u6280\u5de7\u3002
- \u5ffd\u7565 \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\u3002\u56e0\u4e3a\u5b83\u4eec\u90fd\u4e0e \\(n\\) \u65e0\u5173\uff0c\u6240\u4ee5\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002
- \u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7b80\u5316\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u6ca1\u6709\u5f71\u54cd\u3002
- \u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u7b2c
1.\u70b9\u548c\u7b2c2.\u70b9\u7684\u6280\u5de7\u3002
\u7ed9\u5b9a\u4e00\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4e0a\u8ff0\u6280\u5de7\u6765\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef algorithm(n: int):\n a = 1 # +0\uff08\u6280\u5de7 1\uff09\n a = a + n # +0\uff08\u6280\u5de7 1\uff09\n # +n\uff08\u6280\u5de7 2\uff09\n for i in range(5 * n + 1):\n print(0)\n # +n*n\uff08\u6280\u5de7 3\uff09\n for i in range(2 * n):\n for j in range(n + 1):\n print(0)\nvoid algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n cout << 0 << endl;\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n cout << 0 << endl;\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n System.out.println(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n System.out.println(0);\n }\n }\n}\nvoid Algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n Console.WriteLine(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n Console.WriteLine(0);\n }\n }\n}\nfunc algorithm(n int) {\n a := 1 // +0\uff08\u6280\u5de7 1\uff09\n a = a + n // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for i := 0; i < 5 * n + 1; i++ {\n fmt.Println(0)\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for i := 0; i < 2 * n; i++ {\n for j := 0; j < n + 1; j++ {\n fmt.Println(0)\n }\n }\n}\nfunc algorithm(n: Int) {\n var a = 1 // +0\uff08\u6280\u5de7 1\uff09\n a = a + n // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for _ in 0 ..< (5 * n + 1) {\n print(0)\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for _ in 0 ..< (2 * n) {\n for _ in 0 ..< (n + 1) {\n print(0)\n }\n }\n}\nfunction algorithm(n) {\n let a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (let i = 0; i < 5 * n + 1; i++) {\n console.log(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (let i = 0; i < 2 * n; i++) {\n for (let j = 0; j < n + 1; j++) {\n console.log(0);\n }\n }\n}\nfunction algorithm(n: number): void {\n let a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (let i = 0; i < 5 * n + 1; i++) {\n console.log(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (let i = 0; i < 2 * n; i++) {\n for (let j = 0; j < n + 1; j++) {\n console.log(0);\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n print(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n print(0);\n }\n }\n}\nfn algorithm(n: i32) {\n let mut a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n\n // +n\uff08\u6280\u5de7 2\uff09\n for i in 0..(5 * n + 1) {\n println!(\"{}\", 0);\n }\n\n // +n*n\uff08\u6280\u5de7 3\uff09\n for i in 0..(2 * n) {\n for j in 0..(n + 1) {\n println!(\"{}\", 0);\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n printf(\"%d\", 0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n printf(\"%d\", 0);\n }\n }\n}\nfn algorithm(n: usize) void {\n var a: i32 = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + @as(i32, @intCast(n)); // +0\uff08\u6280\u5de7 1\uff09\n\n // +n\uff08\u6280\u5de7 2\uff09\n for(0..(5 * n + 1)) |_| {\n std.debug.print(\"{}\\n\", .{0}); \n }\n\n // +n*n\uff08\u6280\u5de7 3\uff09\n for(0..(2 * n)) |_| {\n for(0..(n + 1)) |_| {\n std.debug.print(\"{}\\n\", .{0}); \n }\n }\n}\n\u4ee5\u4e0b\u516c\u5f0f\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\uff0c\u4e24\u8005\u63a8\u7b97\u51fa\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002
\\[ \\begin{aligned} T(n) & = 2n(n + 1) + (5n + 1) + 2 & \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline & = 2n^2 + 7n + 3 \\newline T(n) & = n^2 + n & \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2. \u00a0 \u7b2c\u4e8c\u6b65\uff1a\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"\u65f6\u95f4\u590d\u6742\u5ea6\u7531 \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u53d1\u6325\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u4ed6\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002
\u8868 2-2 \u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u5938\u5f20\u7684\u503c\u662f\u4e3a\u4e86\u5f3a\u8c03\u201c\u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570\u201d\u8fd9\u4e00\u7ed3\u8bba\u3002\u5f53 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u53d8\u5f97\u65e0\u8db3\u8f7b\u91cd\u3002
\u8868 2-2 \u00a0 \u4e0d\u540c\u64cd\u4f5c\u6570\u91cf\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234","title":"2.3.4 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u5982\u56fe 2-9 \u6240\u793a\uff08\u6309\u7167\u4ece\u4f4e\u5230\u9ad8\u7684\u987a\u5e8f\u6392\u5217\uff09\u3002
\\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n \\log n) < O(n^2) < O(2^n) < O(n!) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u7ebf\u6027\u5bf9\u6570\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} < \\text{\u9636\u4e58\u9636} \\end{aligned} \\]\u56fe 2-9 \u00a0 \u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b
"},{"location":"chapter_computational_complexity/time_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"\u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002
\u5728\u4ee5\u4e0b\u51fd\u6570\u4e2d\uff0c\u5c3d\u7ba1\u64cd\u4f5c\u6570\u91cf
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.pysize\u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u4e8e\u5176\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a
time_complexity.cppdef constant(n: int) -> int:\n \"\"\"\u5e38\u6570\u9636\"\"\"\n count = 0\n size = 100000\n for _ in range(size):\n count += 1\n return count\n
time_complexity.java/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.cs/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.go/* \u5e38\u6570\u9636 */\nint Constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.swift/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\n count := 0\n size := 100000\n for i := 0; i < size; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -> Int {\n var count = 0\n let size = 100_000\n for _ in 0 ..< size {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n let count = 0;\n const size = 100000;\n for (let i = 0; i < size; i++) count++;\n return count;\n}\n
time_complexity.dart/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\n let count = 0;\n const size = 100000;\n for (let i = 0; i < size; i++) count++;\n return count;\n}\n
time_complexity.rs/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (var i = 0; i < size; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -> i32 {\n _ = n;\n let mut count = 0;\n let size = 100_000;\n for _ in 0..size {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n int i = 0;\n for (int i = 0; i < size; i++) {\n count++;\n }\n return count;\n}\n
"},{"location":"chapter_computational_complexity/time_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n _ = n;\n var count: i32 = 0;\n const size: i32 = 100_000;\n var i: i32 = 0;\n while(i<size) : (i += 1) {\n count += 1;\n }\n return count;\n}\n\u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5355\u5c42\u5faa\u73af\u4e2d\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef linear(n: int) -> int:\n \"\"\"\u7ebf\u6027\u9636\"\"\"\n count = 0\n for _ in range(n):\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\n count := 0\n for i := 0; i < n; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -> Int {\n var count = 0\n for _ in 0 ..< n {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n let count = 0;\n for (let i = 0; i < n; i++) count++;\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\n let count = 0;\n for (let i = 0; i < n; i++) count++;\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (var i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -> i32 {\n let mut count = 0;\n for _ in 0..n {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\n var count: i32 = 0;\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n count += 1;\n }\n return count;\n}\n\u904d\u5386\u6570\u7ec4\u548c\u904d\u5386\u94fe\u8868\u7b49\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef array_traversal(nums: list[int]) -> int:\n \"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\n count = 0\n # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for num in nums:\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector<int> &nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int num : nums) {\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int num : nums) {\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n foreach (int num in nums) {\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\n count := 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for range nums {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -> Int {\n var count = 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for _ in nums {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (let i = 0; i < nums.length; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (let i = 0; i < nums.length; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List<int> nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (var _num in nums) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &[i32]) -> i32 {\n let mut count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for _ in nums {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\n var count: i32 = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (nums) |_| {\n count += 1;\n }\n return count;\n}\n\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u9700\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u5177\u4f53\u786e\u5b9a\u3002\u6bd4\u5982\u5728\u7b2c\u4e00\u4e2a\u793a\u4f8b\u4e2d\uff0c\u53d8\u91cf \\(n\\) \u4e3a\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u5728\u7b2c\u4e8c\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u4e3a\u6570\u636e\u5927\u5c0f\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"\u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u56e0\u6b64\u603b\u4f53\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef quadratic(n: int) -> int:\n \"\"\"\u5e73\u65b9\u9636\"\"\"\n count = 0\n # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for i in range(n):\n for j in range(n):\n count += 1\n return count\n
time_complexity.java/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.cs/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.go/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.swift/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\n count := 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for i := 0; i < n; i++ {\n for j := 0; j < n; j++ {\n count++\n }\n }\n return count\n}\n
time_complexity.js/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -> Int {\n var count = 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for _ in 0 ..< n {\n for _ in 0 ..< n {\n count += 1\n }\n }\n return count\n}\n
time_complexity.ts/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.dart/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.rs/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.c/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -> i32 {\n let mut count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for _ in 0..n {\n for _ in 0..n {\n count += 1;\n }\n }\n count\n}\n
time_complexity.zig/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\n var count: i32 = 0;\n var i: i32 = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n while (i < n) : (i += 1) {\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n count += 1;\n }\n }\n return count;\n}\n\u56fe 2-10 \u5bf9\u6bd4\u4e86\u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u4e09\u79cd\u65f6\u95f4\u590d\u6742\u5ea6\u3002
\u56fe 2-10 \u00a0 \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u4ee5\u5192\u6ce1\u6392\u5e8f\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af\u6267\u884c \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af\u6267\u884c \\(n-1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(n / 2\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O((n - 1) n / 2) = O(n^2)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef bubble_sort(nums: list[int]) -> int:\n \"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\n count = 0 # \u8ba1\u6570\u5668\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in range(len(nums) - 1, 0, -1):\n # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j in range(i):\n if nums[j] > nums[j + 1]:\n # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n tmp: int = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n count += 3 # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n return count\n
time_complexity.java/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector<int> &nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.size() - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.cs/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.go/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.Length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.swift/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n count := 0 // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := len(nums) - 1; i > 0; i-- {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j := 0; j < i; j++ {\n if nums[j] > nums[j+1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n tmp := nums[j]\n nums[j] = nums[j+1]\n nums[j+1] = tmp\n count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count\n}\n
time_complexity.js/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -> Int {\n var count = 0 // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0 ..< i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count\n}\n
time_complexity.ts/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n let count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.dart/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n let count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.rs/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List<int> nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (var i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (var j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.c/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &mut [i32]) -> i32 {\n let mut count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in (1..nums.len()).rev() {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0..i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n count\n}\n
time_complexity.zig/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = n - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
"},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n var count: i32 = 0; // \u8ba1\u6570\u5668 \n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n while (i > 0) : (i -= 1) {\n var j: usize = 0;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n while (j < i) : (j += 1) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n var tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n\u751f\u7269\u5b66\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u662f\u6307\u6570\u9636\u589e\u957f\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u53d8\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u53d8\u4e3a \\(4\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002
\u56fe 2-11 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u7ec6\u80de\u5206\u88c2\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef exponential(n: int) -> int:\n \"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n count = 0\n base = 1\n # \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in range(n):\n for _ in range(base):\n count += 1\n base *= 2\n # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n
time_complexity.java/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.cs/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.go/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\n int count = 0, bas = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < bas; j++) {\n count++;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.swift/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\n count, base := 0, 1\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for i := 0; i < n; i++ {\n for j := 0; j < base; j++ {\n count++\n }\n base *= 2\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n}\n
time_complexity.js/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -> Int {\n var count = 0\n var base = 1\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in 0 ..< n {\n for _ in 0 ..< base {\n count += 1\n }\n base *= 2\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n}\n
time_complexity.ts/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\n let count = 0,\n base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.dart/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\n let count = 0,\n base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.rs/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (var i = 0; i < n; i++) {\n for (var j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.c/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -> i32 {\n let mut count = 0;\n let mut base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in 0..n {\n for _ in 0..base {\n count += 1\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n count\n}\n
time_complexity.zig/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0;\n int bas = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < bas; j++) {\n count++;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\n var count: i32 = 0;\n var bas: i32 = 1;\n var i: i32 = 0;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n while (i < n) : (i += 1) {\n var j: i32 = 0;\n while (j < bas) : (j += 1) {\n count += 1;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n\u56fe 2-11 \u00a0 \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u5176\u9012\u5f52\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u7ecf\u8fc7 \\(n\\) \u6b21\u5206\u88c2\u540e\u505c\u6b62\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef exp_recur(n: int) -> int:\n \"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n == 1:\n return 1\n return exp_recur(n - 1) + exp_recur(n - 1) + 1\n
time_complexity.java/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.cs/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.go/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\n if (n == 1) return 1;\n return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n
time_complexity.swift/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\n if n == 1 {\n return 1\n }\n return expRecur(n-1) + expRecur(n-1) + 1\n}\n
time_complexity.js/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -> Int {\n if n == 1 {\n return 1\n }\n return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n
time_complexity.ts/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\n if (n === 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.dart/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\n if (n === 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.rs/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.c/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -> i32 {\n if n == 1 {\n return 1;\n }\n exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n
time_complexity.zig/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\n if (n == 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n\u6307\u6570\u9636\u589e\u957f\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u7a77\u4e3e\u6cd5\uff08\u66b4\u529b\u641c\u7d22\u3001\u56de\u6eaf\u7b49\uff09\u4e2d\u6bd4\u8f83\u5e38\u89c1\u3002\u5bf9\u4e8e\u6570\u636e\u89c4\u6a21\u8f83\u5927\u7684\u95ee\u9898\uff0c\u6307\u6570\u9636\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff0c\u901a\u5e38\u9700\u8981\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u6216\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u6765\u89e3\u51b3\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"\u4e0e\u6307\u6570\u9636\u76f8\u53cd\uff0c\u5bf9\u6570\u9636\u53cd\u6620\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u60c5\u51b5\u3002\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002
\u56fe 2-12 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_2 n)\\) \uff0c\u7b80\u8bb0\u4e3a \\(O(\\log n)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef logarithmic(n: float) -> int:\n \"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n count = 0\n while n > 1:\n n = n / 2\n count += 1\n return count\n
time_complexity.java/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n /= 2;\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\n count := 0\n for n > 1 {\n n = n / 2\n count++\n }\n return count\n}\n
time_complexity.js/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -> Int {\n var count = 0\n var n = n\n while n > 1 {\n n = n / 2\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\n let count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\n let count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(num n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: f32) -> i32 {\n let mut count = 0;\n while n > 1.0 {\n n = n / 2.0;\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\n var count: i32 = 0;\n var n_var = n;\n while (n_var > 1)\n {\n n_var = n_var / 2;\n count +=1;\n }\n return count;\n}\n\u56fe 2-12 \u00a0 \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef log_recur(n: float) -> int:\n \"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n <= 1:\n return 0\n return log_recur(n / 2) + 1\n
time_complexity.java/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.cs/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.go/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(float n) {\n if (n <= 1) return 0;\n return LogRecur(n / 2) + 1;\n}\n
time_complexity.swift/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\n if n <= 1 {\n return 0\n }\n return logRecur(n/2) + 1\n}\n
time_complexity.js/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -> Int {\n if n <= 1 {\n return 0\n }\n return logRecur(n: n / 2) + 1\n}\n
time_complexity.ts/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.dart/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.rs/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(num n) {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.c/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: f32) -> i32 {\n if n <= 1.0 {\n return 0;\n }\n log_recur(n / 2.0) + 1\n}\n
time_complexity.zig/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u7b97\u6cd5\u4e2d\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u591a\u201d\u548c\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002\u5b83\u589e\u957f\u7f13\u6162\uff0c\u662f\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\u7684\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002
\\(O(\\log n)\\) \u7684\u5e95\u6570\u662f\u591a\u5c11\uff1f
\u51c6\u786e\u6765\u8bf4\uff0c\u201c\u4e00\u5206\u4e3a \\(m\\)\u201d\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(\\log_m n)\\) \u3002\u800c\u901a\u8fc7\u5bf9\u6570\u6362\u5e95\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5177\u6709\u4e0d\u540c\u5e95\u6570\u3001\u76f8\u7b49\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a
\\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\]\u4e5f\u5c31\u662f\u8bf4\uff0c\u5e95\u6570 \\(m\\) \u53ef\u4ee5\u5728\u4e0d\u5f71\u54cd\u590d\u6742\u5ea6\u7684\u524d\u63d0\u4e0b\u8f6c\u6362\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u7701\u7565\u5e95\u6570 \\(m\\) \uff0c\u5c06\u5bf9\u6570\u9636\u76f4\u63a5\u8bb0\u4e3a \\(O(\\log n)\\) \u3002
"},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","title":"6. \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"\u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef linear_log_recur(n: float) -> int:\n \"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\n if n <= 1:\n return 1\n count: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n for _ in range(n):\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(float n) {\n if (n <= 1) return 1;\n int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\n if n <= 1 {\n return 1\n }\n count := linearLogRecur(n/2) + linearLogRecur(n/2)\n for i := 0.0; i < n; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -> Int {\n if n <= 1 {\n return 1\n }\n var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n for _ in stride(from: 0, to: n, by: 1) {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\n if (n <= 1) return 1;\n let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (let i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\n if (n <= 1) return 1;\n let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (let i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(num n) {\n if (n <= 1) return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (var i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: f32) -> i32 {\n if n <= 1.0 {\n return 1;\n }\n let mut count = linear_log_recur(n / 2.0) + linear_log_recur(n / 2.0);\n for _ in 0 ..n as i32 {\n count += 1;\n }\n return count\n}\n
time_complexity.zig/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\n if (n <= 1) return 1;\n var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n var i: f32 = 0;\n while (i < n) : (i += 1) {\n count += 1;\n }\n return count;\n}\n\u56fe 2-13 \u5c55\u793a\u4e86\u7ebf\u6027\u5bf9\u6570\u9636\u7684\u751f\u6210\u65b9\u5f0f\u3002\u4e8c\u53c9\u6811\u7684\u6bcf\u4e00\u5c42\u7684\u64cd\u4f5c\u603b\u6570\u90fd\u4e3a \\(n\\) \uff0c\u6811\u5171\u6709 \\(\\log_2 n + 1\\) \u5c42\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
\u56fe 2-13 \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#7-on","title":"7. \u00a0 \u9636\u4e58\u9636 \\(O(n!)\\)","text":"\u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u201c\u5168\u6392\u5217\u201d\u95ee\u9898\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6570\u91cf\u4e3a\uff1a
\\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\]\u9636\u4e58\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u5982\u56fe 2-14 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c42\u65f6\u505c\u6b62\u5206\u88c2\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef factorial_recur(n: int) -> int:\n \"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n == 0:\n return 1\n count = 0\n # \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in range(n):\n count += factorial_recur(n - 1)\n return count\n
time_complexity.java/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.cs/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.go/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\n if (n == 0) return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += FactorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.swift/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\n if n == 0 {\n return 1\n }\n count := 0\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for i := 0; i < n; i++ {\n count += factorialRecur(n - 1)\n }\n return count\n}\n
time_complexity.js/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -> Int {\n if n == 0 {\n return 1\n }\n var count = 0\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in 0 ..< n {\n count += factorialRecur(n: n - 1)\n }\n return count\n}\n
time_complexity.ts/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\n if (n === 0) return 1;\n let count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (let i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.dart/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\n if (n === 0) return 1;\n let count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (let i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.rs/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0) return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (var i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.c/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -> i32 {\n if n == 0 {\n return 1;\n }\n let mut count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in 0..n {\n count += factorial_recur(n - 1);\n }\n count\n}\n
time_complexity.zig/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\n if (n == 0) return 1;\n var count: i32 = 0;\n var i: i32 = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n while (i < n) : (i += 1) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n\u56fe 2-14 \u00a0 \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u5f53 \\(n \\geq 4\\) \u65f6\u6052\u6709 \\(n! > 2^n\\) \uff0c\u6240\u4ee5\u9636\u4e58\u9636\u6bd4\u6307\u6570\u9636\u589e\u957f\u5f97\u66f4\u5feb\uff0c\u5728 \\(n\\) \u8f83\u5927\u65f6\u4e5f\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#235","title":"2.3.5 \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"\u7b97\u6cd5\u7684\u65f6\u95f4\u6548\u7387\u5f80\u5f80\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u5047\u8bbe\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4
nums\uff0c\u5176\u4e2dnums\u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u6bcf\u4e2a\u6570\u5b57\u53ea\u51fa\u73b0\u4e00\u6b21\uff1b\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff0c\u4efb\u52a1\u76ee\u6807\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002- \u5f53
nums = [?, ?, ..., 1]\uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u9700\u8981\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002 - \u5f53
nums = [1, ?, ?, ...]\uff0c\u5373\u5f53\u9996\u4e2a\u5143\u7d20\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\) \u3002
\u201c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\u3002\u76f8\u5e94\u5730\uff0c\u201c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\uff0c\u7528 \\(\\Omega\\) \u8bb0\u53f7\u8868\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig worst_best_time_complexity.py
worst_best_time_complexity.cppdef random_numbers(n: int) -> list[int]:\n \"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n # \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\n nums = [i for i in range(1, n + 1)]\n # \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n random.shuffle(nums)\n return nums\n\ndef find_one(nums: list[int]) -> int:\n \"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n for i in range(len(nums)):\n # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1:\n return i\n return -1\n
worst_best_time_complexity.java/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector<int> randomNumbers(int n) {\n vector<int> nums(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\n unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector<int> &nums) {\n for (int i = 0; i < nums.size(); i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.cs/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\n Integer[] nums = new Integer[n];\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n Collections.shuffle(Arrays.asList(nums));\n // Integer[] -> int[]\n int[] res = new int[n];\n for (int i = 0; i < n; i++) {\n res[i] = nums[i];\n }\n return res;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n for (int i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.go/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\n int[] nums = new int[n];\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (int i = 0; i < nums.Length; i++) {\n int index = new Random().Next(i, nums.Length);\n (nums[i], nums[index]) = (nums[index], nums[i]);\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n for (int i = 0; i < nums.Length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.swift/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\n nums := make([]int, n)\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for i := 0; i < n; i++ {\n nums[i] = i + 1\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n rand.Shuffle(len(nums), func(i, j int) {\n nums[i], nums[j] = nums[j], nums[i]\n })\n return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n for i := 0; i < len(nums); i++ {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return i\n }\n }\n return -1\n}\n
worst_best_time_complexity.js/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -> [Int] {\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n var nums = Array(1 ... n)\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle()\n return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -> Int {\n for i in nums.indices {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return i\n }\n }\n return -1\n}\n
worst_best_time_complexity.ts/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\n const nums = Array(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (let i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (let i = 0; i < n; i++) {\n const r = Math.floor(Math.random() * (i + 1));\n const temp = nums[i];\n nums[i] = nums[r];\n nums[r] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n for (let i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] === 1) {\n return i;\n }\n }\n return -1;\n}\n
worst_best_time_complexity.dart/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\n const nums = Array(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (let i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (let i = 0; i < n; i++) {\n const r = Math.floor(Math.random() * (i + 1));\n const temp = nums[i];\n nums[i] = nums[r];\n nums[r] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n for (let i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] === 1) {\n return i;\n }\n }\n return -1;\n}\n
worst_best_time_complexity.rs/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList<int> randomNumbers(int n) {\n final nums = List.filled(n, 0);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (var i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle();\n\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List<int> nums) {\n for (var i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1) return i;\n }\n\n return -1;\n}\n
worst_best_time_complexity.c/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -> Vec<i32> {\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n let mut nums = (1..=n).collect::<Vec<i32>>();\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle(&mut thread_rng());\n nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &[i32]) -> Option<usize> {\n for i in 0..nums.len() {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return Some(i);\n }\n }\n None\n}\n
worst_best_time_complexity.zig/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n // \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\n int *nums = (int *)malloc(n * sizeof(int));\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (int i = n - 1; i > 0; i--) {\n int j = rand() % (i + 1);\n int temp = nums[i];\n nums[i] = nums[j];\n nums[j] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n for (int i = 0; i < n; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\n var nums: [n]i32 = undefined;\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (&nums, 0..) |*num, i| {\n num.* = @as(i32, @intCast(i)) + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n const rand = std.crypto.random;\n rand.shuffle(i32, &nums);\n return nums;\n}\n\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n for (nums, 0..) |num, i| {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (num == 1) return @intCast(i);\n }\n return -1;\n}\n\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6211\u4eec\u5728\u5b9e\u9645\u4e2d\u5f88\u5c11\u4f7f\u7528\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u56e0\u4e3a\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u53ef\u80fd\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u800c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u6548\u7387\u5b89\u5168\u503c\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002
\u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u4e8e\u201c\u7279\u6b8a\u7684\u6570\u636e\u5206\u5e03\u201d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u5e76\u4e0d\u80fd\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\u6765\u8868\u793a\u3002
\u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5c31\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(n / 2\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(n / 2) = \\Theta(n)\\) \u3002
\u4f46\u5bf9\u4e8e\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5f80\u5f80\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002
\u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f
\u53ef\u80fd\u7531\u4e8e \\(O\\) \u7b26\u53f7\u8fc7\u4e8e\u6717\u6717\u4e0a\u53e3\uff0c\u56e0\u6b64\u6211\u4eec\u5e38\u5e38\u4f7f\u7528\u5b83\u6765\u8868\u793a\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u4f46\u4ece\u4e25\u683c\u610f\u4e49\u4e0a\u8bb2\uff0c\u8fd9\u79cd\u505a\u6cd5\u5e76\u4e0d\u89c4\u8303\u3002\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u7c7b\u4f3c\u201c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8bf7\u5c06\u5176\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u3002
"},{"location":"chapter_data_structure/","title":"\u7b2c 3 \u7ae0 \u00a0 \u6570\u636e\u7ed3\u6784","text":"Abstract
\u6570\u636e\u7ed3\u6784\u5982\u540c\u4e00\u526f\u7a33\u56fa\u800c\u591a\u6837\u7684\u6846\u67b6\u3002
\u5b83\u4e3a\u6570\u636e\u7684\u6709\u5e8f\u7ec4\u7ec7\u63d0\u4f9b\u4e86\u84dd\u56fe\uff0c\u7b97\u6cd5\u5f97\u4ee5\u5728\u6b64\u57fa\u7840\u4e0a\u751f\u52a8\u8d77\u6765\u3002
"},{"location":"chapter_data_structure/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b
- 3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b
- 3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *
- 3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *
- 3.5 \u00a0 \u5c0f\u7ed3
\u5f53\u8c08\u53ca\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\u65f6\uff0c\u6211\u4eec\u4f1a\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u79cd\u5f62\u5f0f\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u6570\u636e\u7684\u7ec4\u7ec7\u5f62\u5f0f\u5404\u5f02\uff0c\u4f46\u5b83\u4eec\u90fd\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u3002
\u57fa\u672c\u6570\u636e\u7c7b\u578b\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\uff0c\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002
- \u6574\u6570\u7c7b\u578b
byte\u3001short\u3001int\u3001long\u3002 - \u6d6e\u70b9\u6570\u7c7b\u578b
float\u3001double\uff0c\u7528\u4e8e\u8868\u793a\u5c0f\u6570\u3002 - \u5b57\u7b26\u7c7b\u578b
char\uff0c\u7528\u4e8e\u8868\u793a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u6bcd\u3001\u6807\u70b9\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002 - \u5e03\u5c14\u7c7b\u578b
bool\uff0c\u7528\u4e8e\u8868\u793a\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\u5224\u65ad\u3002
\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4ee5\u4e8c\u8fdb\u5236\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u3002\u4e00\u4e2a\u4e8c\u8fdb\u5236\u4f4d\u5373\u4e3a \\(1\\) \u6bd4\u7279\u3002\u5728\u7edd\u5927\u591a\u6570\u73b0\u4ee3\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\\(1\\) \u5b57\u8282\uff08byte\uff09\u7531 \\(8\\) \u6bd4\u7279\uff08bit\uff09\u7ec4\u6210\u3002
\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5176\u5360\u7528\u7684\u7a7a\u95f4\u5927\u5c0f\u3002\u4e0b\u9762\u4ee5 Java \u4e3a\u4f8b\u3002
- \u6574\u6570\u7c7b\u578b
byte\u5360\u7528 \\(1\\) byte = \\(8\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u4e2a\u6570\u5b57\u3002 - \u6574\u6570\u7c7b\u578b
int\u5360\u7528 \\(4\\) bytes = \\(32\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u6570\u5b57\u3002
\u8868 3-1 \u5217\u4e3e\u4e86 Java \u4e2d\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u3002\u6b64\u8868\u683c\u65e0\u987b\u6b7b\u8bb0\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002
\u8868 3-1 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u548c\u53d6\u503c\u8303\u56f4
\u7c7b\u578b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9ed8\u8ba4\u503c \u6574\u6570byte1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\)short2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\)int4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\)long8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570float4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\)double8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26char2 bytes \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14bool1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\)\u8bf7\u6ce8\u610f\uff0c\u8868 3-1 \u9488\u5bf9\u7684\u662f Java \u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u60c5\u51b5\u3002\u6bcf\u79cd\u7f16\u7a0b\u8bed\u8a00\u90fd\u6709\u5404\u81ea\u7684\u6570\u636e\u7c7b\u578b\u5b9a\u4e49\uff0c\u5b83\u4eec\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002
- \u5728 Python \u4e2d\uff0c\u6574\u6570\u7c7b\u578b
int\u53ef\u4ee5\u662f\u4efb\u610f\u5927\u5c0f\uff0c\u53ea\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\uff1b\u6d6e\u70b9\u6570float\u662f\u53cc\u7cbe\u5ea6 64 \u4f4d\uff1b\u6ca1\u6709char\u7c7b\u578b\uff0c\u5355\u4e2a\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32str\u3002 - C \u548c C++ \u672a\u660e\u786e\u89c4\u5b9a\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5927\u5c0f\uff0c\u800c\u56e0\u5b9e\u73b0\u548c\u5e73\u53f0\u5404\u5f02\u3002\u8868 3-1 \u9075\u5faa LP64 \u6570\u636e\u6a21\u578b\uff0c\u5176\u7528\u4e8e\u5305\u62ec Linux \u548c macOS \u5728\u5185\u7684 Unix 64 \u4f4d\u64cd\u4f5c\u7cfb\u7edf\u3002
- \u5b57\u7b26
char\u7684\u5927\u5c0f\u5728 C \u548c C++ \u4e2d\u4e3a 1 \u5b57\u8282\uff0c\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\u53d6\u51b3\u4e8e\u7279\u5b9a\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\uff0c\u8be6\u89c1\u201c\u5b57\u7b26\u7f16\u7801\u201d\u7ae0\u8282\u3002 - \u5373\u4f7f\u8868\u793a\u5e03\u5c14\u91cf\u4ec5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u5185\u5b58\u4e2d\u901a\u5e38\u5b58\u50a8\u4e3a 1 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u673a CPU \u901a\u5e38\u5c06 1 \u5b57\u8282\u4f5c\u4e3a\u6700\u5c0f\u5bfb\u5740\u5185\u5b58\u5355\u5143\u3002
\u90a3\u4e48\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u95f4\u6709\u4ec0\u4e48\u8054\u7cfb\u5462\uff1f\u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u8fd9\u53e5\u8bdd\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\u800c\u975e\u201c\u6570\u636e\u201d\u3002
\u5982\u679c\u60f3\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u6211\u4eec\u81ea\u7136\u4f1a\u60f3\u5230\u4f7f\u7528\u6570\u7ec4\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u7684\u7ebf\u6027\u7ed3\u6784\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u548c\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5b58\u50a8\u7684\u5185\u5bb9\u662f\u6574\u6570
int\u3001\u5c0f\u6570float\u6216\u662f\u5b57\u7b26char\uff0c\u5219\u4e0e\u201c\u6570\u636e\u7ed3\u6784\u201d\u65e0\u5173\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u7528\u76f8\u540c\u7684\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\uff09\u6765\u5b58\u50a8\u4e0e\u8868\u793a\u4e0d\u540c\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u5305\u62ec
PythonC++JavaC#GoSwiftJSTSDartRustCZigint\u3001float\u3001char\u3001bool\u7b49\u3002# \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python \u7684\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python \u7684\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\u5f15\u7528\ndata = [0, 0.0, 'a', False, ListNode(0)]\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers = Array(repeating: Int(), count: 5)\nlet decimals = Array(repeating: Double(), count: 5)\nlet characters = Array(repeating: Character(\"a\"), count: 5)\nlet bools = Array(repeating: Bool(), count: 5)\n// JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\nconst array = [0, 0.0, 'a', false];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nList<int> numbers = List.filled(5, 0);\nList<double> decimals = List.filled(5, 0.0);\nList<String> characters = List.filled(5, 'a');\nList<bool> bools = List.filled(5, false);\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers: Vec<i32> = vec![0; 5];\nlet decimals: Vec<f32> = vec![0.0, 5];\nlet characters: Vec<char> = vec!['0'; 5];\nlet bools: Vec<bool> = vec![false; 5];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n
"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *","text":"\n\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6240\u6709\u6570\u636e\u90fd\u662f\u4ee5\u4e8c\u8fdb\u5236\u6570\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\uff0c\u5b57\u7b26
"},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1 \u00a0 ASCII \u5b57\u7b26\u96c6","text":"char\u4e5f\u4e0d\u4f8b\u5916\u3002\u4e3a\u4e86\u8868\u793a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u5efa\u7acb\u4e00\u5957\u201c\u5b57\u7b26\u96c6\u201d\uff0c\u89c4\u5b9a\u6bcf\u4e2a\u5b57\u7b26\u548c\u4e8c\u8fdb\u5236\u6570\u4e4b\u95f4\u7684\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u540e\uff0c\u8ba1\u7b97\u673a\u5c31\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5b8c\u6210\u4e8c\u8fdb\u5236\u6570\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002\u300cASCII \u7801\u300d\u662f\u6700\u65e9\u51fa\u73b0\u7684\u5b57\u7b26\u96c6\uff0c\u5176\u5168\u79f0\u4e3a American Standard Code for Information Interchange\uff08\u7f8e\u56fd\u6807\u51c6\u4fe1\u606f\u4ea4\u6362\u4ee3\u7801\uff09\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u8fdb\u5236\u6570\uff08\u4e00\u4e2a\u5b57\u8282\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6700\u591a\u80fd\u591f\u8868\u793a 128 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u5982\u56fe 3-6 \u6240\u793a\uff0cASCII \u7801\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5199\u3001\u6570\u5b57 0 ~ 9\u3001\u4e00\u4e9b\u6807\u70b9\u7b26\u53f7\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u7b26\uff08\u5982\u6362\u884c\u7b26\u548c\u5236\u8868\u7b26\uff09\u3002
\u56fe 3-6 \u00a0 ASCII \u7801
\u7136\u800c\uff0cASCII \u7801\u4ec5\u80fd\u591f\u8868\u793a\u82f1\u6587\u3002\u968f\u7740\u8ba1\u7b97\u673a\u7684\u5168\u7403\u5316\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u80fd\u591f\u8868\u793a\u66f4\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u96c6\u300cEASCII\u300d\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u7840\u4e0a\u6269\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u591f\u8868\u793a 256 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002
\u5728\u4e16\u754c\u8303\u56f4\u5185\uff0c\u9646\u7eed\u51fa\u73b0\u4e86\u4e00\u6279\u9002\u7528\u4e8e\u4e0d\u540c\u5730\u533a\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u8fd9\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u4e2a\u5b57\u7b26\u7edf\u4e00\u4e3a ASCII \u7801\uff0c\u540e 128 \u4e2a\u5b57\u7b26\u5b9a\u4e49\u4e0d\u540c\uff0c\u4ee5\u9002\u5e94\u4e0d\u540c\u8bed\u8a00\u7684\u9700\u6c42\u3002
"},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2 \u00a0 GBK \u5b57\u7b26\u96c6","text":"\u540e\u6765\u4eba\u4eec\u53d1\u73b0\uff0cEASCII \u7801\u4ecd\u7136\u65e0\u6cd5\u6ee1\u8db3\u8bb8\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u6570\u91cf\u8981\u6c42\u3002\u6bd4\u5982\u6c49\u5b57\u6709\u8fd1\u5341\u4e07\u4e2a\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u51e0\u5343\u4e2a\u3002\u4e2d\u56fd\u56fd\u5bb6\u6807\u51c6\u603b\u5c40\u4e8e 1980 \u5e74\u53d1\u5e03\u4e86\u300cGB2312\u300d\u5b57\u7b26\u96c6\uff0c\u5176\u6536\u5f55\u4e86 6763 \u4e2a\u6c49\u5b57\uff0c\u57fa\u672c\u6ee1\u8db3\u4e86\u6c49\u5b57\u7684\u8ba1\u7b97\u673a\u5904\u7406\u9700\u8981\u3002
\u7136\u800c\uff0cGB2312 \u65e0\u6cd5\u5904\u7406\u90e8\u5206\u7f55\u89c1\u5b57\u548c\u7e41\u4f53\u5b57\u3002\u300cGBK\u300d\u5b57\u7b26\u96c6\u662f\u5728 GB2312 \u7684\u57fa\u7840\u4e0a\u6269\u5c55\u5f97\u5230\u7684\uff0c\u5b83\u5171\u6536\u5f55\u4e86 21886 \u4e2a\u6c49\u5b57\u3002\u5728 GBK \u7684\u7f16\u7801\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u7b26\u4f7f\u7528\u4e00\u4e2a\u5b57\u8282\u8868\u793a\uff0c\u6c49\u5b57\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u8868\u793a\u3002
"},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3 \u00a0 Unicode \u5b57\u7b26\u96c6","text":"\u968f\u7740\u8ba1\u7b97\u673a\u6280\u672f\u7684\u84ec\u52c3\u53d1\u5c55\uff0c\u5b57\u7b26\u96c6\u4e0e\u7f16\u7801\u6807\u51c6\u767e\u82b1\u9f50\u653e\uff0c\u800c\u8fd9\u5e26\u6765\u4e86\u8bb8\u591a\u95ee\u9898\u3002\u4e00\u65b9\u9762\uff0c\u8fd9\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u4e49\u4e86\u7279\u5b9a\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u65e0\u6cd5\u5728\u591a\u8bed\u8a00\u73af\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u79cd\u8bed\u8a00\u5b58\u5728\u591a\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u5982\u679c\u4e24\u53f0\u8ba1\u7b97\u673a\u4f7f\u7528\u7684\u662f\u4e0d\u540c\u7684\u7f16\u7801\u6807\u51c6\uff0c\u5219\u5728\u4fe1\u606f\u4f20\u9012\u65f6\u5c31\u4f1a\u51fa\u73b0\u4e71\u7801\u3002
\u90a3\u4e2a\u65f6\u4ee3\u7684\u7814\u7a76\u4eba\u5458\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u4e2a\u8db3\u591f\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c06\u4e16\u754c\u8303\u56f4\u5185\u7684\u6240\u6709\u8bed\u8a00\u548c\u7b26\u53f7\u90fd\u6536\u5f55\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u51b3\u8de8\u8bed\u8a00\u73af\u5883\u548c\u4e71\u7801\u95ee\u9898\u4e86\u5417\uff1f\u5728\u8fd9\u79cd\u60f3\u6cd5\u7684\u9a71\u52a8\u4e0b\uff0c\u4e00\u4e2a\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u5e94\u8fd0\u800c\u751f\u3002
\u300cUnicode\u300d\u7684\u4e2d\u6587\u540d\u79f0\u4e3a\u201c\u7edf\u4e00\u7801\u201d\uff0c\u7406\u8bba\u4e0a\u80fd\u5bb9\u7eb3 100 \u591a\u4e07\u4e2a\u5b57\u7b26\u3002\u5b83\u81f4\u529b\u4e8e\u5c06\u5168\u7403\u8303\u56f4\u5185\u7684\u5b57\u7b26\u7eb3\u5165\u7edf\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u79cd\u901a\u7528\u7684\u5b57\u7b26\u96c6\u6765\u5904\u7406\u548c\u663e\u793a\u5404\u79cd\u8bed\u8a00\u6587\u5b57\uff0c\u51cf\u5c11\u56e0\u4e3a\u7f16\u7801\u6807\u51c6\u4e0d\u540c\u800c\u4ea7\u751f\u7684\u4e71\u7801\u95ee\u9898\u3002
\u81ea 1991 \u5e74\u53d1\u5e03\u4ee5\u6765\uff0cUnicode \u4e0d\u65ad\u6269\u5145\u65b0\u7684\u8bed\u8a00\u4e0e\u5b57\u7b26\u3002\u622a\u81f3 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7ecf\u5305\u542b 149186 \u4e2a\u5b57\u7b26\uff0c\u5305\u62ec\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\u3001\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002\u5728\u5e9e\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u7b26\u5360\u7528 3 \u5b57\u8282\u751a\u81f3 4 \u5b57\u8282\u3002
Unicode \u662f\u4e00\u79cd\u901a\u7528\u5b57\u7b26\u96c6\uff0c\u672c\u8d28\u4e0a\u662f\u7ed9\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u4e00\u4e2a\u7f16\u53f7\uff08\u79f0\u4e3a\u201c\u7801\u70b9\u201d\uff09\uff0c\u4f46\u5b83\u5e76\u6ca1\u6709\u89c4\u5b9a\u5728\u8ba1\u7b97\u673a\u4e2d\u5982\u4f55\u5b58\u50a8\u8fd9\u4e9b\u5b57\u7b26\u7801\u70b9\u3002\u6211\u4eec\u4e0d\u7981\u4f1a\u95ee\uff1a\u5f53\u591a\u79cd\u957f\u5ea6\u7684 Unicode \u7801\u70b9\u540c\u65f6\u51fa\u73b0\u5728\u4e00\u4e2a\u6587\u672c\u4e2d\u65f6\uff0c\u7cfb\u7edf\u5982\u4f55\u89e3\u6790\u5b57\u7b26\uff1f\u4f8b\u5982\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u7684\u7f16\u7801\uff0c\u7cfb\u7edf\u5982\u4f55\u786e\u8ba4\u5b83\u662f\u4e00\u4e2a 2 \u5b57\u8282\u7684\u5b57\u7b26\u8fd8\u662f\u4e24\u4e2a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff1f
\u5bf9\u4e8e\u4ee5\u4e0a\u95ee\u9898\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u662f\u5c06\u6240\u6709\u5b57\u7b26\u5b58\u50a8\u4e3a\u7b49\u957f\u7684\u7f16\u7801\u3002\u5982\u56fe 3-7 \u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff0c\u201c\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9ad8\u4f4d\u586b 0 \u5c06\u201cHello \u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u7b26\u90fd\u7f16\u7801\u4e3a 2 \u5b57\u8282\u957f\u5ea6\u3002\u8fd9\u6837\u7cfb\u7edf\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u5b57\u8282\u89e3\u6790\u4e00\u4e2a\u5b57\u7b26\uff0c\u6062\u590d\u8fd9\u4e2a\u77ed\u8bed\u7684\u5185\u5bb9\u4e86\u3002
\u56fe 3-7 \u00a0 Unicode \u7f16\u7801\u793a\u4f8b
\u7136\u800c ASCII \u7801\u5df2\u7ecf\u5411\u6211\u4eec\u8bc1\u660e\uff0c\u7f16\u7801\u82f1\u6587\u53ea\u9700 1 \u5b57\u8282\u3002\u82e5\u91c7\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u672c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u5c06\u4f1a\u662f ASCII \u7f16\u7801\u4e0b\u7684\u4e24\u500d\uff0c\u975e\u5e38\u6d6a\u8d39\u5185\u5b58\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002
"},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4 \u00a0 UTF-8 \u7f16\u7801","text":"\u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u4e3a\u56fd\u9645\u4e0a\u4f7f\u7528\u6700\u5e7f\u6cdb\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u79cd\u53ef\u53d8\u957f\u5ea6\u7684\u7f16\u7801\uff0c\u4f7f\u7528 1 \u5230 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6839\u636e\u5b57\u7b26\u7684\u590d\u6742\u6027\u800c\u53d8\u3002ASCII \u5b57\u7b26\u53ea\u9700 1 \u5b57\u8282\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u814a\u5b57\u6bcd\u9700\u8981 2 \u5b57\u8282\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u9700\u8981 3 \u5b57\u8282\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u7b26\u9700\u8981 4 \u5b57\u8282\u3002
UTF-8 \u7684\u7f16\u7801\u89c4\u5219\u5e76\u4e0d\u590d\u6742\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002
- \u5bf9\u4e8e\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff0c\u5c06\u6700\u9ad8\u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff0c\u5176\u4f59 7 \u4f4d\u8bbe\u7f6e\u4e3a Unicode \u7801\u70b9\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u7b26\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u5360\u636e\u4e86\u524d 128 \u4e2a\u7801\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0cUTF-8 \u7f16\u7801\u53ef\u4ee5\u5411\u4e0b\u517c\u5bb9 ASCII \u7801\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u6765\u89e3\u6790\u5e74\u4ee3\u4e45\u8fdc\u7684 ASCII \u7801\u6587\u672c\u3002
- \u5bf9\u4e8e\u957f\u5ea6\u4e3a \\(n\\) \u5b57\u8282\u7684\u5b57\u7b26\uff08\u5176\u4e2d \\(n > 1\\)\uff09\uff0c\u5c06\u9996\u4e2a\u5b57\u8282\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u7b2c \\(n + 1\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff1b\u4ece\u7b2c\u4e8c\u4e2a\u5b57\u8282\u5f00\u59cb\uff0c\u5c06\u6bcf\u4e2a\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \uff1b\u5176\u4f59\u6240\u6709\u4f4d\u7528\u4e8e\u586b\u5145\u5b57\u7b26\u7684 Unicode \u7801\u70b9\u3002
\u56fe 3-8 \u5c55\u793a\u4e86\u201cHello\u7b97\u6cd5\u201d\u5bf9\u5e94\u7684 UTF-8 \u7f16\u7801\u3002\u89c2\u5bdf\u53d1\u73b0\uff0c\u7531\u4e8e\u6700\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u56e0\u6b64\u7cfb\u7edf\u53ef\u4ee5\u901a\u8fc7\u8bfb\u53d6\u6700\u9ad8\u4f4d \\(1\\) \u7684\u4e2a\u6570\u6765\u89e3\u6790\u51fa\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(n\\) \u3002
\u4f46\u4e3a\u4ec0\u4e48\u8981\u5c06\u5176\u4f59\u6240\u6709\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \u5462\uff1f\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a \\(10\\) \u80fd\u591f\u8d77\u5230\u6821\u9a8c\u7b26\u7684\u4f5c\u7528\u3002\u5047\u8bbe\u7cfb\u7edf\u4ece\u4e00\u4e2a\u9519\u8bef\u7684\u5b57\u8282\u5f00\u59cb\u89e3\u6790\u6587\u672c\uff0c\u5b57\u8282\u5934\u90e8\u7684 \\(10\\) \u80fd\u591f\u5e2e\u52a9\u7cfb\u7edf\u5feb\u901f\u5224\u65ad\u51fa\u5f02\u5e38\u3002
\u4e4b\u6240\u4ee5\u5c06 \\(10\\) \u5f53\u4f5c\u6821\u9a8c\u7b26\uff0c\u662f\u56e0\u4e3a\u5728 UTF-8 \u7f16\u7801\u89c4\u5219\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \u3002\u8fd9\u4e2a\u7ed3\u8bba\u53ef\u4ee5\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\uff1a\u5047\u8bbe\u4e00\u4e2a\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \uff0c\u8bf4\u660e\u8be5\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(1\\) \uff0c\u5bf9\u5e94 ASCII \u7801\u3002\u800c ASCII \u7801\u7684\u6700\u9ad8\u4f4d\u5e94\u8be5\u662f \\(0\\) \uff0c\u4e0e\u5047\u8bbe\u77db\u76fe\u3002
\u56fe 3-8 \u00a0 UTF-8 \u7f16\u7801\u793a\u4f8b
\u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u89c1\u7684\u7f16\u7801\u65b9\u5f0f\u8fd8\u5305\u62ec\u4ee5\u4e0b\u4e24\u79cd\u3002
- UTF-16 \u7f16\u7801\uff1a\u4f7f\u7528 2 \u6216 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u3002\u6240\u6709\u7684 ASCII \u5b57\u7b26\u548c\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u7b26\uff0c\u90fd\u7528 2 \u5b57\u8282\u8868\u793a\uff1b\u5c11\u6570\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u5b57\u8282\u8868\u793a\u3002\u5bf9\u4e8e 2 \u5b57\u8282\u7684\u5b57\u7b26\uff0cUTF-16 \u7f16\u7801\u4e0e Unicode \u7801\u70b9\u76f8\u7b49\u3002
- UTF-32 \u7f16\u7801\uff1a\u6bcf\u4e2a\u5b57\u7b26\u90fd\u4f7f\u7528 4 \u5b57\u8282\u3002\u8fd9\u610f\u5473\u7740 UTF-32 \u6bd4 UTF-8 \u548c UTF-16 \u66f4\u5360\u7528\u7a7a\u95f4\uff0c\u7279\u522b\u662f\u5bf9\u4e8e ASCII \u5b57\u7b26\u5360\u6bd4\u8f83\u9ad8\u7684\u6587\u672c\u3002
\u4ece\u5b58\u50a8\u7a7a\u95f4\u5360\u7528\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u7b26\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 1 \u5b57\u8282\uff1b\u4f7f\u7528 UTF-16 \u7f16\u7801\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u7b26\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u4f1a\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 2 \u5b57\u8282\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u5b57\u8282\u3002
\u4ece\u517c\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8bb8\u591a\u5de5\u5177\u548c\u5e93\u4f18\u5148\u652f\u6301 UTF-8 \u3002
"},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801","text":"\u5bf9\u4e8e\u4ee5\u5f80\u7684\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff0c\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u7684\u5b57\u7b26\u4e32\u90fd\u91c7\u7528 UTF-16 \u6216 UTF-32 \u8fd9\u7c7b\u7b49\u957f\u7684\u7f16\u7801\u3002\u5728\u7b49\u957f\u7f16\u7801\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u7b26\u4e32\u770b\u4f5c\u6570\u7ec4\u6765\u5904\u7406\uff0c\u8fd9\u79cd\u505a\u6cd5\u5177\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002
- \u968f\u673a\u8bbf\u95ee\uff1aUTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\u3002UTF-8 \u662f\u4e00\u79cd\u53d8\u957f\u7f16\u7801\uff0c\u8981\u60f3\u627e\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u4ece\u5b57\u7b26\u4e32\u7684\u5f00\u59cb\u5904\u904d\u5386\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u8fd9\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u3002
- \u5b57\u7b26\u8ba1\u6570\uff1a\u4e0e\u968f\u673a\u8bbf\u95ee\u7c7b\u4f3c\uff0c\u8ba1\u7b97 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8ba1\u7b97 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u9700\u8981\u904d\u5386\u6574\u4e2a\u5b57\u7b26\u4e32\u3002
- \u5b57\u7b26\u4e32\u64cd\u4f5c\uff1a\u5728 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u5f88\u591a\u5b57\u7b26\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u8fde\u63a5\u3001\u63d2\u5165\u3001\u5220\u9664\u7b49\uff09\u66f4\u5bb9\u6613\u8fdb\u884c\u3002\u5728 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u8fdb\u884c\u8fd9\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u989d\u5916\u7684\u8ba1\u7b97\uff0c\u4ee5\u786e\u4fdd\u4e0d\u4f1a\u4ea7\u751f\u65e0\u6548\u7684 UTF-8 \u7f16\u7801\u3002
\u5b9e\u9645\u4e0a\uff0c\u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6848\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5f88\u6709\u8da3\u7684\u8bdd\u9898\uff0c\u6d89\u53ca\u8bb8\u591a\u56e0\u7d20\u3002
- Java \u7684
String\u7c7b\u578b\u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a Java \u8bed\u8a00\u8bbe\u8ba1\u4e4b\u521d\uff0c\u4eba\u4eec\u8ba4\u4e3a 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u7b26\u3002\u7136\u800c\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0d\u6b63\u786e\u7684\u5224\u65ad\u3002\u540e\u6765 Unicode \u89c4\u8303\u6269\u5c55\u5230\u4e86\u8d85\u8fc7 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u7b26\u73b0\u5728\u53ef\u80fd\u7531\u4e00\u5bf9 16 \u4f4d\u7684\u503c\uff08\u79f0\u4e3a\u201c\u4ee3\u7406\u5bf9\u201d\uff09\u8868\u793a\u3002 - JavaScript \u548c TypeScript \u7684\u5b57\u7b26\u4e32\u4f7f\u7528 UTF-16 \u7f16\u7801\u7684\u539f\u56e0\u4e0e Java \u7c7b\u4f3c\u3002\u5f53 1995 \u5e74 Netscape \u516c\u53f8\u9996\u6b21\u63a8\u51fa JavaScript \u8bed\u8a00\u65f6\uff0cUnicode \u8fd8\u5904\u4e8e\u53d1\u5c55\u65e9\u671f\uff0c\u90a3\u65f6\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7f16\u7801\u5c31\u8db3\u4ee5\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u7b26\u4e86\u3002
- C# \u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u4e3b\u8981\u662f\u56e0\u4e3a .NET \u5e73\u53f0\u662f\u7531 Microsoft \u8bbe\u8ba1\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u672f\uff08\u5305\u62ec Windows \u64cd\u4f5c\u7cfb\u7edf\uff09\u90fd\u5e7f\u6cdb\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002
\u7531\u4e8e\u4ee5\u4e0a\u7f16\u7a0b\u8bed\u8a00\u5bf9\u5b57\u7b26\u6570\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u4eec\u4e0d\u5f97\u4e0d\u91c7\u53d6\u201c\u4ee3\u7406\u5bf9\u201d\u7684\u65b9\u5f0f\u6765\u8868\u793a\u8d85\u8fc7 16 \u4f4d\u957f\u5ea6\u7684 Unicode \u5b57\u7b26\u3002\u8fd9\u662f\u4e00\u4e2a\u4e0d\u5f97\u5df2\u4e3a\u4e4b\u7684\u65e0\u5948\u4e4b\u4e3e\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5bf9\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u4e00\u4e2a\u5b57\u7b26\u53ef\u80fd\u5360\u7528 2 \u5b57\u8282\u6216 4 \u5b57\u8282\uff0c\u4ece\u800c\u4e27\u5931\u4e86\u7b49\u957f\u7f16\u7801\u7684\u4f18\u52bf\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5904\u7406\u4ee3\u7406\u5bf9\u9700\u8981\u589e\u52a0\u989d\u5916\u4ee3\u7801\uff0c\u8fd9\u63d0\u9ad8\u4e86\u7f16\u7a0b\u7684\u590d\u6742\u6027\u548c\u8c03\u8bd5\u96be\u5ea6\u3002
\u51fa\u4e8e\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7f16\u7a0b\u8bed\u8a00\u63d0\u51fa\u4e86\u4e00\u4e9b\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6848\u3002
- Python \u4e2d\u7684
str\u4f7f\u7528 Unicode \u7f16\u7801\uff0c\u5e76\u91c7\u7528\u4e00\u79cd\u7075\u6d3b\u7684\u5b57\u7b26\u4e32\u8868\u793a\uff0c\u5b58\u50a8\u7684\u5b57\u7b26\u957f\u5ea6\u53d6\u51b3\u4e8e\u5b57\u7b26\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u7801\u70b9\u3002\u82e5\u5b57\u7b26\u4e32\u4e2d\u5168\u90e8\u662f ASCII \u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff1b\u5982\u679c\u6709\u5b57\u7b26\u8d85\u51fa\u4e86 ASCII \u8303\u56f4\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8bed\u8a00\u5e73\u9762\uff08BMP\uff09\u5185\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff1b\u5982\u679c\u6709\u8d85\u51fa BMP \u7684\u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 4 \u5b57\u8282\u3002 - Go \u8bed\u8a00\u7684
string\u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Go \u8bed\u8a00\u8fd8\u63d0\u4f9b\u4e86rune\u7c7b\u578b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002 - Rust \u8bed\u8a00\u7684 str \u548c String \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Rust \u4e5f\u63d0\u4f9b\u4e86
char\u7c7b\u578b\uff0c\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002
\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8ba8\u8bba\u7684\u90fd\u662f\u5b57\u7b26\u4e32\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u8fd9\u548c\u5b57\u7b26\u4e32\u5982\u4f55\u5728\u6587\u4ef6\u4e2d\u5b58\u50a8\u6216\u5728\u7f51\u7edc\u4e2d\u4f20\u8f93\u662f\u4e0d\u540c\u7684\u95ee\u9898\u3002\u5728\u6587\u4ef6\u5b58\u50a8\u6216\u7f51\u7edc\u4f20\u8f93\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u5b57\u7b26\u4e32\u7f16\u7801\u4e3a UTF-8 \u683c\u5f0f\uff0c\u4ee5\u8fbe\u5230\u6700\u4f18\u7684\u517c\u5bb9\u6027\u548c\u7a7a\u95f4\u6548\u7387\u3002
"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\uff0c\u5b83\u4eec\u53ef\u4ee5\u4ece\u201c\u903b\u8f91\u7ed3\u6784\u201d\u548c\u201c\u7269\u7406\u7ed3\u6784\u201d\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002
"},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1 \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"\u903b\u8f91\u7ed3\u6784\u63ed\u793a\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u5728\u6570\u7ec4\u548c\u94fe\u8868\u4e2d\uff0c\u6570\u636e\u6309\u7167\u4e00\u5b9a\u987a\u5e8f\u6392\u5217\uff0c\u4f53\u73b0\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u800c\u5728\u6811\u4e2d\uff0c\u6570\u636e\u4ece\u9876\u90e8\u5411\u4e0b\u6309\u5c42\u6b21\u6392\u5217\uff0c\u8868\u73b0\u51fa\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u5219\u7531\u8282\u70b9\u548c\u8fb9\u6784\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7684\u7f51\u7edc\u5173\u7cfb\u3002
\u5982\u56fe 3-1 \u6240\u793a\uff0c\u903b\u8f91\u7ed3\u6784\u53ef\u5206\u4e3a\u201c\u7ebf\u6027\u201d\u548c\u201c\u975e\u7ebf\u6027\u201d\u4e24\u5927\u7c7b\u3002\u7ebf\u6027\u7ed3\u6784\u6bd4\u8f83\u76f4\u89c2\uff0c\u6307\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u5448\u7ebf\u6027\u6392\u5217\uff1b\u975e\u7ebf\u6027\u7ed3\u6784\u5219\u76f8\u53cd\uff0c\u5448\u975e\u7ebf\u6027\u6392\u5217\u3002
- \u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3002
- \u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u56fe\u3001\u54c8\u5e0c\u8868\u3002
\u56fe 3-1 \u00a0 \u7ebf\u6027\u6570\u636e\u7ed3\u6784\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784
\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u6811\u5f62\u7ed3\u6784\u548c\u7f51\u72b6\u7ed3\u6784\u3002
- \u7ebf\u6027\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u961f\u5217\u3001\u6808\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u4e00\u7684\u987a\u5e8f\u5173\u7cfb\u3002
- \u6811\u5f62\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u591a\u7684\u5173\u7cfb\u3002
- \u7f51\u72b6\u7ed3\u6784\uff1a\u56fe\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u591a\u5bf9\u591a\u7684\u5173\u7cfb\u3002
\u5f53\u7b97\u6cd5\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u56fe 3-2 \u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8\u4e00\u5b9a\u5927\u5c0f\u7684\u6570\u636e\u3002
\u7cfb\u7edf\u901a\u8fc7\u5185\u5b58\u5730\u5740\u6765\u8bbf\u95ee\u76ee\u6807\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u5982\u56fe 3-2 \u6240\u793a\uff0c\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u4e3a\u8868\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5206\u914d\u7f16\u53f7\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u552f\u4e00\u7684\u5185\u5b58\u5730\u5740\u3002\u6709\u4e86\u8fd9\u4e9b\u5730\u5740\uff0c\u7a0b\u5e8f\u4fbf\u53ef\u4ee5\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002
\u56fe 3-2 \u00a0 \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740
Tip
\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5c06\u5185\u5b58\u6bd4\u4f5c Excel \u8868\u683c\u662f\u4e00\u4e2a\u7b80\u5316\u7684\u7c7b\u6bd4\uff0c\u5b9e\u9645\u5185\u5b58\u7684\u5de5\u4f5c\u673a\u5236\u6bd4\u8f83\u590d\u6742\uff0c\u6d89\u53ca\u5730\u5740\u7a7a\u95f4\u3001\u5185\u5b58\u7ba1\u7406\u3001\u7f13\u5b58\u673a\u5236\u3001\u865a\u62df\u5185\u5b58\u548c\u7269\u7406\u5185\u5b58\u7b49\u6982\u5ff5\u3002
\u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u5171\u4eab\u8d44\u6e90\uff0c\u5f53\u67d0\u5757\u5185\u5b58\u88ab\u67d0\u4e2a\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u5219\u65e0\u6cd5\u88ab\u5176\u4ed6\u7a0b\u5e8f\u540c\u65f6\u4f7f\u7528\u4e86\u3002\u56e0\u6b64\u5728\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bbe\u8ba1\u4e2d\uff0c\u5185\u5b58\u8d44\u6e90\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u8003\u8651\u56e0\u7d20\u3002\u6bd4\u5982\uff0c\u7b97\u6cd5\u6240\u5360\u7528\u7684\u5185\u5b58\u5cf0\u503c\u4e0d\u5e94\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u7a7a\u95f2\u5185\u5b58\uff1b\u5982\u679c\u7f3a\u5c11\u8fde\u7eed\u5927\u5757\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u90a3\u4e48\u6240\u9009\u7528\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u5206\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002
\u5982\u56fe 3-3 \u6240\u793a\uff0c\u7269\u7406\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u53ef\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u70b9\u3002
\u56fe 3-3 \u00a0 \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u5206\u6563\u7a7a\u95f4\u5b58\u50a8
\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e8c\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\uff0c\u6808\u548c\u961f\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff1b\u800c\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u6570\u7ec4\u548c\u94fe\u8868\u3002
- \u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\u3002
- \u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u3002
\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u610f\u5473\u7740\u6b64\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u5bf9\u5e94\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u79f0\u201c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u5bf9\u5176\u957f\u5ea6\u8fdb\u884c\u8c03\u6574\u3002
Tip
\u5982\u679c\u4f60\u611f\u89c9\u7269\u7406\u7ed3\u6784\u7406\u89e3\u8d77\u6765\u6709\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u9605\u8bfb\u4e0b\u4e00\u7ae0\uff0c\u7136\u540e\u518d\u56de\u987e\u672c\u8282\u5185\u5bb9\u3002
"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *","text":"Note
\u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u5e26\u6709 * \u7b26\u53f7\u7684\u662f\u9009\u8bfb\u7ae0\u8282\u3002\u5982\u679c\u4f60\u65f6\u95f4\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u518d\u5355\u72ec\u653b\u514b\u3002
"},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1 \u00a0 \u6574\u6570\u7f16\u7801","text":"\u5728\u4e0a\u4e00\u8282\u7684\u8868\u683c\u4e2d\u6211\u4eec\u53d1\u73b0\uff0c\u6240\u6709\u6574\u6570\u7c7b\u578b\u80fd\u591f\u8868\u793a\u7684\u8d1f\u6570\u90fd\u6bd4\u6b63\u6570\u591a\u4e00\u4e2a\uff0c\u4f8b\u5982
byte\u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \u3002\u8fd9\u4e2a\u73b0\u8c61\u6bd4\u8f83\u53cd\u76f4\u89c9\uff0c\u5b83\u7684\u5185\u5728\u539f\u56e0\u6d89\u53ca\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u76f8\u5173\u77e5\u8bc6\u3002\u9996\u5148\u9700\u8981\u6307\u51fa\uff0c\u6570\u5b57\u662f\u4ee5\u201c\u8865\u7801\u201d\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u7684\u3002\u5728\u5206\u6790\u8fd9\u6837\u505a\u7684\u539f\u56e0\u4e4b\u524d\uff0c\u9996\u5148\u7ed9\u51fa\u4e09\u8005\u7684\u5b9a\u4e49\u3002
- \u539f\u7801\uff1a\u6211\u4eec\u5c06\u6570\u5b57\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u89c6\u4e3a\u7b26\u53f7\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6570\uff0c\\(1\\) \u8868\u793a\u8d1f\u6570\uff0c\u5176\u4f59\u4f4d\u8868\u793a\u6570\u5b57\u7684\u503c\u3002
- \u53cd\u7801\uff1a\u6b63\u6570\u7684\u53cd\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u53cd\u7801\u662f\u5bf9\u5176\u539f\u7801\u9664\u7b26\u53f7\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002
- \u8865\u7801\uff1a\u6b63\u6570\u7684\u8865\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u8865\u7801\u662f\u5728\u5176\u53cd\u7801\u7684\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u3002
\u56fe 3-4 \u5c55\u793a\u4e86\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u4e4b\u95f4\u7684\u8f6c\u6362\u65b9\u6cd5\u3002
\u56fe 3-4 \u00a0 \u539f\u7801\u3001\u53cd\u7801\u4e0e\u8865\u7801\u4e4b\u95f4\u7684\u76f8\u4e92\u8f6c\u6362
\u300c\u539f\u7801 sign-magnitude\u300d\u867d\u7136\u6700\u76f4\u89c2\uff0c\u4f46\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u8d1f\u6570\u7684\u539f\u7801\u4e0d\u80fd\u76f4\u63a5\u7528\u4e8e\u8fd0\u7b97\u3002\u4f8b\u5982\u5728\u539f\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7ed3\u679c\u662f \\(-3\\) \uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002
\\[ \\begin{aligned} & 1 + (-2) \\newline & \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline & = 1000 \\; 0011 \\newline & \\rightarrow -3 \\end{aligned} \\]\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8ba1\u7b97\u673a\u5f15\u5165\u4e86\u300c\u53cd\u7801 1's complement\u300d\u3002\u5982\u679c\u6211\u4eec\u5148\u5c06\u539f\u7801\u8f6c\u6362\u4e3a\u53cd\u7801\uff0c\u5e76\u5728\u53cd\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u6700\u540e\u5c06\u7ed3\u679c\u4ece\u53cd\u7801\u8f6c\u6362\u56de\u539f\u7801\uff0c\u5219\u53ef\u5f97\u5230\u6b63\u786e\u7ed3\u679c \\(-1\\) \u3002
\\[ \\begin{aligned} & 1 + (-2) \\newline & \\rightarrow 0000 \\; 0001 \\; \\text{(\u539f\u7801)} + 1000 \\; 0010 \\; \\text{(\u539f\u7801)} \\newline & = 0000 \\; 0001 \\; \\text{(\u53cd\u7801)} + 1111 \\; 1101 \\; \\text{(\u53cd\u7801)} \\newline & = 1111 \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline & = 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline & \\rightarrow -1 \\end{aligned} \\]\u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b57\u96f6\u7684\u539f\u7801\u6709 \\(+0\\) \u548c \\(-0\\) \u4e24\u79cd\u8868\u793a\u65b9\u5f0f\u3002\u8fd9\u610f\u5473\u7740\u6570\u5b57\u96f6\u5bf9\u5e94\u4e24\u4e2a\u4e0d\u540c\u7684\u4e8c\u8fdb\u5236\u7f16\u7801\uff0c\u8fd9\u53ef\u80fd\u4f1a\u5e26\u6765\u6b67\u4e49\u3002\u6bd4\u5982\u5728\u6761\u4ef6\u5224\u65ad\u4e2d\uff0c\u5982\u679c\u6ca1\u6709\u533a\u5206\u6b63\u96f6\u548c\u8d1f\u96f6\uff0c\u5219\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5224\u65ad\u7ed3\u679c\u51fa\u9519\u3002\u800c\u5982\u679c\u6211\u4eec\u60f3\u5904\u7406\u6b63\u96f6\u548c\u8d1f\u96f6\u6b67\u4e49\uff0c\u5219\u9700\u8981\u5f15\u5165\u989d\u5916\u7684\u5224\u65ad\u64cd\u4f5c\uff0c\u8fd9\u53ef\u80fd\u4f1a\u964d\u4f4e\u8ba1\u7b97\u673a\u7684\u8fd0\u7b97\u6548\u7387\u3002
\\[ \\begin{aligned} +0 & \\rightarrow 0000 \\; 0000 \\newline -0 & \\rightarrow 1000 \\; 0000 \\end{aligned} \\]\u4e0e\u539f\u7801\u4e00\u6837\uff0c\u53cd\u7801\u4e5f\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\uff0c\u56e0\u6b64\u8ba1\u7b97\u673a\u8fdb\u4e00\u6b65\u5f15\u5165\u4e86\u300c\u8865\u7801 2's complement\u300d\u3002\u6211\u4eec\u5148\u6765\u89c2\u5bdf\u4e00\u4e0b\u8d1f\u96f6\u7684\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u8f6c\u6362\u8fc7\u7a0b\uff1a
\\[ \\begin{aligned} -0 \\rightarrow \\; & 1000 \\; 0000 \\; \\text{(\u539f\u7801)} \\newline = \\; & 1111 \\; 1111 \\; \\text{(\u53cd\u7801)} \\newline = 1 \\; & 0000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline \\end{aligned} \\]\u5728\u8d1f\u96f6\u7684\u53cd\u7801\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u4f1a\u4ea7\u751f\u8fdb\u4f4d\uff0c\u4f46
byte\u7c7b\u578b\u7684\u957f\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u51fa\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u4f1a\u88ab\u820d\u5f03\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8d1f\u96f6\u7684\u8865\u7801\u4e3a \\(0000 \\; 0000\\) \uff0c\u4e0e\u6b63\u96f6\u7684\u8865\u7801\u76f8\u540c\u3002\u8fd9\u610f\u5473\u7740\u5728\u8865\u7801\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u4e2a\u96f6\uff0c\u6b63\u8d1f\u96f6\u6b67\u4e49\u4ece\u800c\u5f97\u5230\u89e3\u51b3\u3002\u8fd8\u5269\u6700\u540e\u4e00\u4e2a\u7591\u60d1\uff1a
byte\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u6765\u7684\u4e00\u4e2a\u8d1f\u6570 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u4eec\u6ce8\u610f\u5230\uff0c\u533a\u95f4 \\([-127, +127]\\) \u5185\u7684\u6240\u6709\u6574\u6570\u90fd\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\uff0c\u5e76\u4e14\u539f\u7801\u548c\u8865\u7801\u4e4b\u95f4\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\u3002\u7136\u800c\uff0c\u8865\u7801 \\(1000 \\; 0000\\) \u662f\u4e00\u4e2a\u4f8b\u5916\uff0c\u5b83\u5e76\u6ca1\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3002\u6839\u636e\u8f6c\u6362\u65b9\u6cd5\uff0c\u6211\u4eec\u5f97\u5230\u8be5\u8865\u7801\u7684\u539f\u7801\u4e3a \\(0000 \\; 0000\\) \u3002\u8fd9\u663e\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u4e3a\u8be5\u539f\u7801\u8868\u793a\u6570\u5b57 \\(0\\) \uff0c\u5b83\u7684\u8865\u7801\u5e94\u8be5\u662f\u81ea\u8eab\u3002\u8ba1\u7b97\u673a\u89c4\u5b9a\u8fd9\u4e2a\u7279\u6b8a\u7684\u8865\u7801 \\(1000 \\; 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5b9e\u9645\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u8865\u7801\u4e0b\u7684\u8ba1\u7b97\u7ed3\u679c\u5c31\u662f \\(-128\\) \u3002
\\[ \\begin{aligned} & (-127) + (-1) \\newline & \\rightarrow 1111 \\; 1111 \\; \\text{(\u539f\u7801)} + 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline & = 1000 \\; 0000 \\; \\text{(\u53cd\u7801)} + 1111 \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline & = 1000 \\; 0001 \\; \\text{(\u8865\u7801)} + 1111 \\; 1111 \\; \\text{(\u8865\u7801)} \\newline & = 1000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline & \\rightarrow -128 \\end{aligned} \\]\u4f60\u53ef\u80fd\u5df2\u7ecf\u53d1\u73b0\u4e86\uff0c\u4e0a\u8ff0\u6240\u6709\u8ba1\u7b97\u90fd\u662f\u52a0\u6cd5\u8fd0\u7b97\u3002\u8fd9\u6697\u793a\u7740\u4e00\u4e2a\u91cd\u8981\u4e8b\u5b9e\uff1a\u8ba1\u7b97\u673a\u5185\u90e8\u7684\u786c\u4ef6\u7535\u8def\u4e3b\u8981\u662f\u57fa\u4e8e\u52a0\u6cd5\u8fd0\u7b97\u8bbe\u8ba1\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u52a0\u6cd5\u8fd0\u7b97\u76f8\u5bf9\u4e8e\u5176\u4ed6\u8fd0\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u51cf\u6cd5\uff09\u6765\u8bf4\uff0c\u786c\u4ef6\u5b9e\u73b0\u8d77\u6765\u66f4\u7b80\u5355\uff0c\u66f4\u5bb9\u6613\u8fdb\u884c\u5e76\u884c\u5316\u5904\u7406\uff0c\u8fd0\u7b97\u901f\u5ea6\u66f4\u5feb\u3002
\u8bf7\u6ce8\u610f\uff0c\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u8ba1\u7b97\u673a\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u901a\u8fc7\u5c06\u52a0\u6cd5\u4e0e\u4e00\u4e9b\u57fa\u672c\u903b\u8f91\u8fd0\u7b97\u7ed3\u5408\uff0c\u8ba1\u7b97\u673a\u80fd\u591f\u5b9e\u73b0\u5404\u79cd\u5176\u4ed6\u7684\u6570\u5b66\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u51cf\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8ba1\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u51cf\u6cd5\u3002
\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u8ba1\u7b97\u673a\u4f7f\u7528\u8865\u7801\u7684\u539f\u56e0\uff1a\u57fa\u4e8e\u8865\u7801\u8868\u793a\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u7528\u540c\u6837\u7684\u7535\u8def\u548c\u64cd\u4f5c\u6765\u5904\u7406\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\u6765\u5904\u7406\u51cf\u6cd5\uff0c\u5e76\u4e14\u65e0\u987b\u7279\u522b\u5904\u7406\u6b63\u8d1f\u96f6\u7684\u6b67\u4e49\u95ee\u9898\u3002\u8fd9\u5927\u5927\u7b80\u5316\u4e86\u786c\u4ef6\u8bbe\u8ba1\uff0c\u63d0\u9ad8\u4e86\u8fd0\u7b97\u6548\u7387\u3002
\u8865\u7801\u7684\u8bbe\u8ba1\u975e\u5e38\u7cbe\u5999\uff0c\u56e0\u7bc7\u5e45\u5173\u7cfb\u6211\u4eec\u5c31\u5148\u4ecb\u7ecd\u5230\u8fd9\u91cc\uff0c\u5efa\u8bae\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8fdb\u4e00\u6b65\u6df1\u5165\u4e86\u89e3\u3002
"},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2 \u00a0 \u6d6e\u70b9\u6570\u7f16\u7801","text":"\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\uff1a
int\u548cfloat\u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 bytes \uff0c\u4f46\u4e3a\u4ec0\u4e48float\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8eint\uff1f\u8fd9\u975e\u5e38\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u6309\u7406\u8bf4float\u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570
\\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\]float\u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u8bb0\u4e00\u4e2a 32-bit \u957f\u5ea6\u7684\u4e8c\u8fdb\u5236\u6570\u4e3a\uff1a\u6839\u636e IEEE 754 \u6807\u51c6\uff0c32-bit \u957f\u5ea6\u7684
float\u7531\u4ee5\u4e0b\u4e09\u4e2a\u90e8\u5206\u6784\u6210\u3002- \u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 bit \uff0c\u5bf9\u5e94 \\(b_{31}\\) \u3002
- \u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 bits \uff0c\u5bf9\u5e94 \\(b_{30} b_{29} \\ldots b_{23}\\) \u3002
- \u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 23 bits \uff0c\u5bf9\u5e94 \\(b_{22} b_{21} \\ldots b_0\\) \u3002
\u4e8c\u8fdb\u5236\u6570
\\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\]float\u5bf9\u5e94\u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u4e3a\uff1a\u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a
\\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\]\u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a\uff1a
\\[ \\begin{aligned} \\mathrm{S} \\in & \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = & (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\]\u56fe 3-5 \u00a0 IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u7684\u8ba1\u7b97\u793a\u4f8b
\u89c2\u5bdf\u56fe 3-5 \uff0c\u7ed9\u5b9a\u4e00\u4e2a\u793a\u4f8b\u6570\u636e \\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u5219\u6709\uff1a
\\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\]\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u95ee\u9898\uff1a
float\u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8eint\u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0cfloat\u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002\u5c3d\u7ba1\u6d6e\u70b9\u6570
float\u6269\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578bint\u5c06\u5168\u90e8 32 \u4f4d\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570float\u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002\u5982\u8868 3-2 \u6240\u793a\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002
\u8868 3-2 \u00a0 \u6307\u6570\u4f4d\u542b\u4e49
\u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\)\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u6d6e\u70b9\u6570\u7684\u7cbe\u5ea6\u3002\u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126}\\) \uff0c\u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23}\\) \u3002
\u53cc\u7cbe\u5ea6
"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_data_structure/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"double\u4e5f\u91c7\u7528\u7c7b\u4f3c\u4e8efloat\u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u505a\u8d58\u8ff0\u3002- \u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u800c\u7269\u7406\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002
- \u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u5305\u62ec\u7ebf\u6027\u3001\u6811\u72b6\u548c\u7f51\u72b6\u7b49\u3002\u901a\u5e38\u6211\u4eec\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002
- \u5f53\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u88ab\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u3002\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u62e5\u6709\u5bf9\u5e94\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u901a\u8fc7\u8fd9\u4e9b\u5185\u5b58\u5730\u5740\u8bbf\u95ee\u6570\u636e\u3002
- \u7269\u7406\u7ed3\u6784\u4e3b\u8981\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e24\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002
- \u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5305\u62ec\u6574\u6570
byte\u3001short\u3001int\u3001long\uff0c\u6d6e\u70b9\u6570float\u3001double\uff0c\u5b57\u7b26char\u548c\u5e03\u5c14boolean\u3002\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002 - \u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7f16\u7801\u6570\u5b57\u7684\u4e09\u79cd\u65b9\u6cd5\uff0c\u5b83\u4eec\u4e4b\u95f4\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\u3002\u6574\u6570\u7684\u539f\u7801\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u53f7\u4f4d\uff0c\u5176\u4f59\u4f4d\u662f\u6570\u5b57\u7684\u503c\u3002
- \u6574\u6570\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u8865\u7801\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\u3002\u5728\u8865\u7801\u8868\u793a\u4e0b\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u5bf9\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\u4e00\u89c6\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u4e3a\u51cf\u6cd5\u64cd\u4f5c\u5355\u72ec\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\uff0c\u5e76\u4e14\u4e0d\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u7684\u95ee\u9898\u3002
- \u6d6e\u70b9\u6570\u7684\u7f16\u7801\u7531 1 \u4f4d\u7b26\u53f7\u4f4d\u30018 \u4f4d\u6307\u6570\u4f4d\u548c 23 \u4f4d\u5206\u6570\u4f4d\u6784\u6210\u3002\u7531\u4e8e\u5b58\u5728\u6307\u6570\u4f4d\uff0c\u56e0\u6b64\u6d6e\u70b9\u6570\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e\u6574\u6570\uff0c\u4ee3\u4ef7\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002
- ASCII \u7801\u662f\u6700\u65e9\u51fa\u73b0\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u957f\u5ea6\u4e3a 1 \u5b57\u8282\uff0c\u5171\u6536\u5f55 127 \u4e2a\u5b57\u7b26\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u5f55\u4e24\u4e07\u591a\u4e2a\u6c49\u5b57\u3002Unicode \u81f4\u529b\u4e8e\u63d0\u4f9b\u4e00\u4e2a\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u6536\u5f55\u4e16\u754c\u5185\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u4ece\u800c\u89e3\u51b3\u7531\u4e8e\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5bfc\u81f4\u7684\u4e71\u7801\u95ee\u9898\u3002
- UTF-8 \u662f\u6700\u53d7\u6b22\u8fce\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u79cd\u53d8\u957f\u7684\u7f16\u7801\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u6269\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5b58\u50a8\u7a7a\u95f4\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u957f\u7684\u7f16\u7801\u65b9\u6cd5\u3002\u5728\u7f16\u7801\u4e2d\u6587\u65f6\uff0cUTF-16 \u6bd4 UTF-8 \u7684\u5360\u7528\u7a7a\u95f4\u66f4\u5c0f\u3002Java \u548c C# \u7b49\u7f16\u7a0b\u8bed\u8a00\u9ed8\u8ba4\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002
\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1f
\u54c8\u5e0c\u8868\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u800c\u4e3a\u4e86\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\uff0c\u6211\u4eec\u53ef\u80fd\u4f1a\u4f7f\u7528\u201c\u94fe\u5f0f\u5730\u5740\u201d\uff08\u540e\u7eed\u201c\u54c8\u5e0c\u51b2\u7a81\u201d\u7ae0\u8282\u4f1a\u8bb2\uff09\uff1a\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u6876\u6307\u5411\u4e00\u4e2a\u94fe\u8868\uff0c\u5f53\u94fe\u8868\u957f\u5ea6\u8d85\u8fc7\u4e00\u5b9a\u9608\u503c\u65f6\uff0c\u53c8\u53ef\u80fd\u88ab\u8f6c\u5316\u4e3a\u6811\uff08\u901a\u5e38\u4e3a\u7ea2\u9ed1\u6811\uff09\u3002 \u4ece\u5b58\u50a8\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u54c8\u5e0c\u8868\u7684\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u6876\u69fd\u4f4d\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u503c\uff0c\u4e5f\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u94fe\u8868\u6216\u4e00\u68f5\u6811\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\u3001\u94fe\u8868\uff09\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff08\u6811\uff09\u3002
char\u7c7b\u578b\u7684\u957f\u5ea6\u662f 1 byte \u5417\uff1fchar\u7c7b\u578b\u7684\u957f\u5ea6\u7531\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u7f16\u7801\u65b9\u6cd5\u51b3\u5b9a\u3002\u4f8b\u5982\uff0cJava\u3001JavaScript\u3001TypeScript\u3001C# \u90fd\u91c7\u7528 UTF-16 \u7f16\u7801\uff08\u4fdd\u5b58 Unicode \u7801\u70b9\uff09\uff0c\u56e0\u6b64 char \u7c7b\u578b\u7684\u957f\u5ea6\u4e3a 2 bytes\u3002\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d \u662f\u5426\u6709\u6b67\u4e49\uff1f\u56e0\u4e3a\u6808\u4e5f\u53ef\u4ee5\u8fdb\u884c\u51fa\u6808\u548c\u5165\u6808\u7b49\u64cd\u4f5c\uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u662f\u201c\u52a8\u6001\u201d\u7684\u3002
\u6808\u786e\u5b9e\u53ef\u4ee5\u5b9e\u73b0\u52a8\u6001\u7684\u6570\u636e\u64cd\u4f5c\uff0c\u4f46\u6570\u636e\u7ed3\u6784\u4ecd\u7136\u662f\u201c\u9759\u6001\u201d\uff08\u957f\u5ea6\u4e0d\u53ef\u53d8\uff09\u7684\u3002\u5c3d\u7ba1\u57fa\u4e8e\u6570\u7ec4\u7684\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u52a8\u6001\u5730\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\uff0c\u4f46\u5b83\u4eec\u7684\u5bb9\u91cf\u662f\u56fa\u5b9a\u7684\u3002\u5982\u679c\u6570\u636e\u91cf\u8d85\u51fa\u4e86\u9884\u5206\u914d\u7684\u5927\u5c0f\uff0c\u5c31\u9700\u8981\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u65e7\u6570\u7ec4\u7684\u5185\u5bb9\u590d\u5236\u5230\u65b0\u6570\u7ec4\u4e2d\u3002
\u5728\u6784\u5efa\u6808\uff08\u961f\u5217\uff09\u7684\u65f6\u5019\uff0c\u672a\u6307\u5b9a\u5b83\u7684\u5927\u5c0f\uff0c\u4e3a\u4ec0\u4e48\u5b83\u4eec\u662f\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\u5462\uff1f
\u5728\u9ad8\u7ea7\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e0\u987b\u4eba\u5de5\u6307\u5b9a\u6808\uff08\u961f\u5217\uff09\u7684\u521d\u59cb\u5bb9\u91cf\uff0c\u8fd9\u4e2a\u5de5\u4f5c\u7531\u7c7b\u5185\u90e8\u81ea\u52a8\u5b8c\u6210\u3002\u4f8b\u5982\uff0cJava \u7684 ArrayList \u7684\u521d\u59cb\u5bb9\u91cf\u901a\u5e38\u4e3a 10\u3002\u53e6\u5916\uff0c\u6269\u5bb9\u64cd\u4f5c\u4e5f\u662f\u81ea\u52a8\u5b9e\u73b0\u7684\u3002\u8be6\u89c1\u540e\u7eed\u7684\u201c\u5217\u8868\u201d\u7ae0\u8282\u3002
"},{"location":"chapter_divide_and_conquer/","title":"\u7b2c 12 \u7ae0 \u00a0 \u5206\u6cbb","text":"Abstract
\u96be\u9898\u88ab\u9010\u5c42\u62c6\u89e3\uff0c\u6bcf\u4e00\u6b21\u7684\u62c6\u89e3\u90fd\u4f7f\u5b83\u53d8\u5f97\u66f4\u4e3a\u7b80\u5355\u3002
\u5206\u800c\u6cbb\u4e4b\u63ed\u793a\u4e86\u4e00\u4e2a\u91cd\u8981\u7684\u4e8b\u5b9e\uff1a\u4ece\u7b80\u5355\u505a\u8d77\uff0c\u4e00\u5207\u90fd\u4e0d\u518d\u590d\u6742\u3002
"},{"location":"chapter_divide_and_conquer/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5
- 12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565
- 12.3 \u00a0 \u6784\u5efa\u6811\u95ee\u9898
- 12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898
- 12.5 \u00a0 \u5c0f\u7ed3
\u6211\u4eec\u5df2\u7ecf\u5b66\u8fc7\uff0c\u641c\u7d22\u7b97\u6cd5\u5206\u4e3a\u4e24\u5927\u7c7b\u3002
- \u66b4\u529b\u641c\u7d22\uff1a\u5b83\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
- \u81ea\u9002\u5e94\u641c\u7d22\uff1a\u5b83\u5229\u7528\u7279\u6709\u7684\u6570\u636e\u7ec4\u7ec7\u5f62\u5f0f\u6216\u5148\u9a8c\u4fe1\u606f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002
\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u548c\u6811\u3002
- \u4e8c\u5206\u67e5\u627e\u7684\u6bcf\u4e00\u6b65\u90fd\u5c06\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\u5206\u89e3\u4e3a\u4e00\u4e2a\u5c0f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u4e00\u76f4\u6301\u7eed\u5230\u6570\u7ec4\u4e3a\u7a7a\u6216\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002
- \u6811\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4ee3\u8868\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u5806\u7b49\u6570\u636e\u7ed3\u6784\u4e2d\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(\\log n)\\) \u3002
\u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u7b56\u7565\u5982\u4e0b\u6240\u793a\u3002
- \u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4e8c\u5206\u67e5\u627e\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\u5206\u89e3\u4e3a\u5b50\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\uff0c\u8fd9\u662f\u901a\u8fc7\u6bd4\u8f83\u4e2d\u95f4\u5143\u7d20\u548c\u76ee\u6807\u5143\u7d20\u6765\u5b9e\u73b0\u7684\u3002
- \u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u5904\u7406\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5b83\u4e0d\u53d7\u5176\u4ed6\u5b50\u95ee\u9898\u7684\u5f71\u54cd\u3002
- \u5b50\u95ee\u9898\u7684\u89e3\u65e0\u987b\u5408\u5e76\uff1a\u4e8c\u5206\u67e5\u627e\u65e8\u5728\u67e5\u627e\u4e00\u4e2a\u7279\u5b9a\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u3002\u5f53\u5b50\u95ee\u9898\u5f97\u5230\u89e3\u51b3\u65f6\uff0c\u539f\u95ee\u9898\u4e5f\u4f1a\u540c\u65f6\u5f97\u5230\u89e3\u51b3\u3002
\u5206\u6cbb\u80fd\u591f\u63d0\u5347\u641c\u7d22\u6548\u7387\uff0c\u672c\u8d28\u4e0a\u662f\u56e0\u4e3a\u66b4\u529b\u641c\u7d22\u6bcf\u8f6e\u53ea\u80fd\u6392\u9664\u4e00\u4e2a\u9009\u9879\uff0c\u800c\u5206\u6cbb\u641c\u7d22\u6bcf\u8f6e\u53ef\u4ee5\u6392\u9664\u4e00\u534a\u9009\u9879\u3002
"},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","title":"1. \u00a0 \u57fa\u4e8e\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e","text":"\u5728\u4e4b\u524d\u7684\u7ae0\u8282\u4e2d\uff0c\u4e8c\u5206\u67e5\u627e\u662f\u57fa\u4e8e\u9012\u63a8\uff08\u8fed\u4ee3\uff09\u5b9e\u73b0\u7684\u3002\u73b0\u5728\u6211\u4eec\u57fa\u4e8e\u5206\u6cbb\uff08\u9012\u5f52\uff09\u6765\u5b9e\u73b0\u5b83\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4
nums\uff0c\u5176\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\uff0c\u8bf7\u67e5\u627e\u5143\u7d20target\u3002\u4ece\u5206\u6cbb\u89d2\u5ea6\uff0c\u6211\u4eec\u5c06\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u8bb0\u4e3a \\(f(i, j)\\) \u3002
\u4ee5\u539f\u95ee\u9898 \\(f(0, n-1)\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u8fdb\u884c\u4e8c\u5206\u67e5\u627e\u3002
- \u8ba1\u7b97\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u7684\u4e2d\u70b9 \\(m\\) \uff0c\u6839\u636e\u5b83\u6392\u9664\u4e00\u534a\u641c\u7d22\u533a\u95f4\u3002
- \u9012\u5f52\u6c42\u89e3\u89c4\u6a21\u51cf\u5c0f\u4e00\u534a\u7684\u5b50\u95ee\u9898\uff0c\u53ef\u80fd\u4e3a \\(f(i, m-1)\\) \u6216 \\(f(m+1, j)\\) \u3002
- \u5faa\u73af\u7b2c
1.\u6b65\u548c\u7b2c2.\u6b65\uff0c\u76f4\u81f3\u627e\u5230target\u6216\u533a\u95f4\u4e3a\u7a7a\u65f6\u8fd4\u56de\u3002
\u56fe 12-4 \u5c55\u793a\u4e86\u5728\u6570\u7ec4\u4e2d\u4e8c\u5206\u67e5\u627e\u5143\u7d20 \\(6\\) \u7684\u5206\u6cbb\u8fc7\u7a0b\u3002
\u56fe 12-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u8fc7\u7a0b
\u5728\u5b9e\u73b0\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_recur.pydfs()\u6765\u6c42\u89e3\u95ee\u9898 \\(f(i, j)\\) \uff1a
binary_search_recur.cppdef dfs(nums: list[int], target: int, i: int, j: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j)\"\"\"\n # \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if i > j:\n return -1\n # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n m = (i + j) // 2\n if nums[m] < target:\n # \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j)\n elif nums[m] > target:\n # \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1)\n else:\n # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n\ndef binary_search(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\"\"\"\n n = len(nums)\n # \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1)\n
binary_search_recur.java/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(vector<int> &nums, int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) / 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(vector<int> &nums, int target) {\n int n = nums.size();\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.cs/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) / 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int[] nums, int target) {\n int n = nums.length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.go/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) / 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return DFS(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return DFS(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint BinarySearch(int[] nums, int target) {\n int n = nums.Length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return DFS(nums, target, 0, n - 1);\n}\n
binary_search_recur.swift/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n // \u5982\u679c\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u6ca1\u6709\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if i > j {\n return -1\n }\n // \u8ba1\u7b97\u7d22\u5f15\u4e2d\u70b9\n m := i + ((j - i) >> 1)\n //\u5224\u65ad\u4e2d\u70b9\u4e0e\u76ee\u6807\u5143\u7d20\u5927\u5c0f\n if nums[m] < target {\n // \u5c0f\u4e8e\u5219\u9012\u5f52\u53f3\u534a\u6570\u7ec4\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m+1, j)\n } else if nums[m] > target {\n // \u5c0f\u4e8e\u5219\u9012\u5f52\u5de6\u534a\u6570\u7ec4\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m-1)\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums []int, target int) int {\n n := len(nums)\n return dfs(nums, target, 0, n-1)\n}\n
binary_search_recur.js/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -> Int {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if i > j {\n return -1\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n let m = (i + j) / 2\n if nums[m] < target {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums: nums, target: target, i: m + 1, j: j)\n } else if nums[m] > target {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums: nums, target: target, i: i, j: m - 1)\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums: [Int], target: Int) -> Int {\n let n = nums.count\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums: nums, target: target, i: 0, j: n - 1)\n}\n
binary_search_recur.ts/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums, target, i, j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n const m = i + ((j - i) >> 1);\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums, target) {\n const n = nums.length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.dart/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n const m = i + ((j - i) >> 1);\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums: number[], target: number): number {\n const n = nums.length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.rs/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(List<int> nums, int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) ~/ 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(List<int> nums, int target) {\n int n = nums.length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.c/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfn dfs(nums: &[i32], target: i32, i: i32, j: i32) -> i32 {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if i > j { return -1; }\n let m: i32 = (i + j) / 2;\n if nums[m as usize] < target {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if nums[m as usize] > target {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfn binary_search(nums: &[i32], target: i32) -> i32 {\n let n = nums.len() as i32;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n dfs(nums, target, 0, n - 1)\n}\n
binary_search_recur.zig/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) / 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int nums[], int target, int numsSize) {\n int n = numsSize;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","title":"12.3 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u95ee\u9898","text":"[class]{}-[func]{dfs}\n\n[class]{}-[func]{binarySearch}\nQuestion
\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\u7684\u524d\u5e8f\u904d\u5386
preorder\u548c\u4e2d\u5e8f\u904d\u5386inorder\uff0c\u8bf7\u4ece\u4e2d\u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u8fd4\u56de\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u3002\u5047\u8bbe\u4e8c\u53c9\u6811\u4e2d\u6ca1\u6709\u503c\u91cd\u590d\u7684\u8282\u70b9\u3002\u56fe 12-5 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u793a\u4f8b\u6570\u636e
"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","title":"1. \u00a0 \u5224\u65ad\u662f\u5426\u4e3a\u5206\u6cbb\u95ee\u9898","text":"\u539f\u95ee\u9898\u5b9a\u4e49\u4e3a\u4ece
preorder\u548cinorder\u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u662f\u4e00\u4e2a\u5178\u578b\u7684\u5206\u6cbb\u95ee\u9898\u3002- \u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4ece\u5206\u6cbb\u7684\u89d2\u5ea6\u5207\u5165\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\u3001\u6784\u5efa\u53f3\u5b50\u6811\uff0c\u52a0\u4e0a\u4e00\u6b65\u64cd\u4f5c\uff1a\u521d\u59cb\u5316\u6839\u8282\u70b9\u3002\u800c\u5bf9\u4e8e\u6bcf\u68f5\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u590d\u7528\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u5c06\u5176\u5212\u5206\u4e3a\u66f4\u5c0f\u7684\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u7a7a\u5b50\u6811\uff09\u65f6\u7ec8\u6b62\u3002
- \u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u4ea4\u96c6\u3002\u5728\u6784\u5efa\u5de6\u5b50\u6811\u65f6\uff0c\u6211\u4eec\u53ea\u9700\u5173\u6ce8\u4e2d\u5e8f\u904d\u5386\u548c\u524d\u5e8f\u904d\u5386\u4e2d\u4e0e\u5de6\u5b50\u6811\u5bf9\u5e94\u7684\u90e8\u5206\u3002\u53f3\u5b50\u6811\u540c\u7406\u3002
- \u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e00\u65e6\u5f97\u5230\u4e86\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u5b83\u4eec\u94fe\u63a5\u5230\u6839\u8282\u70b9\u4e0a\uff0c\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u8fd9\u9053\u9898\u53ef\u4ee5\u4f7f\u7528\u5206\u6cbb\u6765\u6c42\u89e3\uff0c\u4f46\u5982\u4f55\u901a\u8fc7\u524d\u5e8f\u904d\u5386
preorder\u548c\u4e2d\u5e8f\u904d\u5386inorder\u6765\u5212\u5206\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u5462\uff1f\u6839\u636e\u5b9a\u4e49\uff0c
preorder\u548cinorder\u90fd\u53ef\u4ee5\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002- \u524d\u5e8f\u904d\u5386\uff1a
[ \u6839\u8282\u70b9 | \u5de6\u5b50\u6811 | \u53f3\u5b50\u6811 ]\uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94[ 3 | 9 | 2 1 7 ]\u3002 - \u4e2d\u5e8f\u904d\u5386\uff1a
[ \u5de6\u5b50\u6811 | \u6839\u8282\u70b9 \uff5c \u53f3\u5b50\u6811 ]\uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94[ 9 | 3 | 1 2 7 ]\u3002
\u4ee5\u4e0a\u56fe\u6570\u636e\u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u56fe 12-6 \u6240\u793a\u7684\u6b65\u9aa4\u5f97\u5230\u5212\u5206\u7ed3\u679c\u3002
- \u524d\u5e8f\u904d\u5386\u7684\u9996\u5143\u7d20 3 \u662f\u6839\u8282\u70b9\u7684\u503c\u3002
- \u67e5\u627e\u6839\u8282\u70b9 3 \u5728
inorder\u4e2d\u7684\u7d22\u5f15\uff0c\u5229\u7528\u8be5\u7d22\u5f15\u53ef\u5c06inorder\u5212\u5206\u4e3a[ 9 | 3 \uff5c 1 2 7 ]\u3002 - \u6839\u636e
inorder\u5212\u5206\u7ed3\u679c\uff0c\u6613\u5f97\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u5206\u522b\u4e3a 1 \u548c 3 \uff0c\u4ece\u800c\u53ef\u5c06preorder\u5212\u5206\u4e3a[ 3 | 9 | 2 1 7 ]\u3002
\u56fe 12-6 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u5212\u5206\u5b50\u6811
"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","title":"3. \u00a0 \u57fa\u4e8e\u53d8\u91cf\u63cf\u8ff0\u5b50\u6811\u533a\u95f4","text":"\u6839\u636e\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u6211\u4eec\u5df2\u7ecf\u5f97\u5230\u6839\u8282\u70b9\u3001\u5de6\u5b50\u6811\u3001\u53f3\u5b50\u6811\u5728
preorder\u548cinorder\u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002\u800c\u4e3a\u4e86\u63cf\u8ff0\u8fd9\u4e9b\u7d22\u5f15\u533a\u95f4\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u3002- \u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728
preorder\u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(i\\) \u3002 - \u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728
inorder\u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(m\\) \u3002 - \u5c06\u5f53\u524d\u6811\u5728
inorder\u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u8bb0\u4e3a \\([l, r]\\) \u3002
\u5982\u8868 12-1 \u6240\u793a\uff0c\u901a\u8fc7\u4ee5\u4e0a\u53d8\u91cf\u5373\u53ef\u8868\u793a\u6839\u8282\u70b9\u5728
preorder\u4e2d\u7684\u7d22\u5f15\uff0c\u4ee5\u53ca\u5b50\u6811\u5728inorder\u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002\u8868 12-1 \u00a0 \u6839\u8282\u70b9\u548c\u5b50\u6811\u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u7d22\u5f15
\u6839\u8282\u70b9\u5728preorder\u4e2d\u7684\u7d22\u5f15 \u5b50\u6811\u5728inorder\u4e2d\u7684\u7d22\u5f15\u533a\u95f4 \u5f53\u524d\u6811 \\(i\\) \\([l, r]\\) \u5de6\u5b50\u6811 \\(i + 1\\) \\([l, m-1]\\) \u53f3\u5b50\u6811 \\(i + 1 + (m - l)\\) \\([m+1, r]\\)\u8bf7\u6ce8\u610f\uff0c\u53f3\u5b50\u6811\u6839\u8282\u70b9\u7d22\u5f15\u4e2d\u7684 \\((m-l)\\) \u7684\u542b\u4e49\u662f\u201c\u5de6\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u201d\uff0c\u5efa\u8bae\u914d\u5408\u56fe 12-7 \u7406\u89e3\u3002
\u56fe 12-7 \u00a0 \u6839\u8282\u70b9\u548c\u5de6\u53f3\u5b50\u6811\u7684\u7d22\u5f15\u533a\u95f4\u8868\u793a
"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","title":"4. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u4e3a\u4e86\u63d0\u5347\u67e5\u8be2 \\(m\\) \u7684\u6548\u7387\uff0c\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig build_tree.pyhmap\u6765\u5b58\u50a8\u6570\u7ec4inorder\u4e2d\u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff1a
build_tree.cppdef dfs(\n preorder: list[int],\n inorder_map: dict[int, int],\n i: int,\n l: int,\n r: int,\n) -> TreeNode | None:\n \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb\"\"\"\n # \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if r - l < 0:\n return None\n # \u521d\u59cb\u5316\u6839\u8282\u70b9\n root = TreeNode(preorder[i])\n # \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n m = inorder_map[preorder[i]]\n # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n # \u8fd4\u56de\u6839\u8282\u70b9\n return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -> TreeNode | None:\n \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\"\"\"\n # \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n inorder_map = {val: i for i, val in enumerate(inorder)}\n root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n return root\n
build_tree.java/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(vector<int> &preorder, unordered_map<int, int> &inorderMap, int i, int l, int r) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0)\n return NULL;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode *root = new TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap[preorder[i]];\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root->left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root->right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(vector<int> &preorder, vector<int> &inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n unordered_map<int, int> inorderMap;\n for (int i = 0; i < inorder.size(); i++) {\n inorderMap[inorder[i]] = i;\n }\n TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n return root;\n}\n
build_tree.cs/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode dfs(int[] preorder, Map<Integer, Integer> inorderMap, int i, int l, int r) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0)\n return null;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode root = new TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap.get(preorder[i]);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n Map<Integer, Integer> inorderMap = new HashMap<>();\n for (int i = 0; i < inorder.length; i++) {\n inorderMap.put(inorder[i], i);\n }\n TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n return root;\n}\n
build_tree.go/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? DFS(int[] preorder, Dictionary<int, int> inorderMap, int i, int l, int r) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0)\n return null;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode root = new(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap[preorder[i]];\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n Dictionary<int, int> inorderMap = [];\n for (int i = 0; i < inorder.Length; i++) {\n inorderMap.TryAdd(inorder[i], i);\n }\n TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n return root;\n}\n
build_tree.swift/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if r-l < 0 {\n return nil\n }\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n root := NewTreeNode(preorder[i])\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n m := inorderMap[preorder[i]]\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n inorderMap := make(map[int]int, len(inorder))\n for i := 0; i < len(inorder); i++ {\n inorderMap[inorder[i]] = i\n }\n\n root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n return root\n}\n
build_tree.js/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -> TreeNode? {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if r - l < 0 {\n return nil\n }\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n let root = TreeNode(x: preorder[i])\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n let m = inorderMap[preorder[i]]!\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -> TreeNode? {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n return dfs(preorder: preorder, inorderMap: inorderMap, i: 0, l: 0, r: inorder.count - 1)\n}\n
build_tree.ts/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(preorder, inorderMap, i, l, r) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0) return null;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n const root = new TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n const m = inorderMap.get(preorder[i]);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder, inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n let inorderMap = new Map();\n for (let i = 0; i < inorder.length; i++) {\n inorderMap.set(inorder[i], i);\n }\n const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n return root;\n}\n
build_tree.dart/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(\n preorder: number[],\n inorderMap: Map<number, number>,\n i: number,\n l: number,\n r: number\n): TreeNode | null {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0) return null;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n const root: TreeNode = new TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n const m = inorderMap.get(preorder[i]);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n let inorderMap = new Map<number, number>();\n for (let i = 0; i < inorder.length; i++) {\n inorderMap.set(inorder[i], i);\n }\n const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n return root;\n}\n
build_tree.rs/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? dfs(\n List<int> preorder,\n Map<int, int> inorderMap,\n int i,\n int l,\n int r,\n) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0) {\n return null;\n }\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode? root = TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap[preorder[i]]!;\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? buildTree(List<int> preorder, List<int> inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n Map<int, int> inorderMap = {};\n for (int i = 0; i < inorder.length; i++) {\n inorderMap[inorder[i]] = i;\n }\n TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n return root;\n}\n
build_tree.c/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfn dfs(preorder: &[i32], inorder_map: &HashMap<i32, i32>, i: i32, l: i32, r: i32) -> Option<Rc<RefCell<TreeNode>>> {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if r - l < 0 { return None; }\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n let root = TreeNode::new(preorder[i as usize]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n let m = inorder_map.get(&preorder[i as usize]).unwrap();\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n Some(root)\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfn build_tree(preorder: &[i32], inorder: &[i32]) -> Option<Rc<RefCell<TreeNode>>> {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n let mut inorder_map: HashMap<i32, i32> = HashMap::new();\n for i in 0..inorder.len() {\n inorder_map.insert(inorder[i], i as i32);\n }\n let root = dfs(preorder, &inorder_map, 0, 0, inorder.len() as i32 - 1);\n root\n}\n
build_tree.zig/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0)\n return NULL;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n root->val = preorder[i];\n root->left = NULL;\n root->right = NULL;\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap[preorder[i]];\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root->left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root->right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n for (int i = 0; i < inorderSize; i++) {\n inorderMap[inorder[i]] = i;\n }\n TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n free(inorderMap);\n return root;\n}\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{buildTree}\n\u56fe 12-8 \u5c55\u793a\u4e86\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5404\u4e2a\u8282\u70b9\u662f\u5728\u5411\u4e0b\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\uff0c\u800c\u5404\u6761\u8fb9\uff08\u5f15\u7528\uff09\u662f\u5728\u5411\u4e0a\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\u3002
<1><2><3><4><5><6><7><8><9>\u56fe 12-8 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b
\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u5185\u7684\u524d\u5e8f\u904d\u5386
preorder\u548c\u4e2d\u5e8f\u904d\u5386inorder\u7684\u5212\u5206\u7ed3\u679c\u5982\u56fe 12-9 \u6240\u793a\u3002\u56fe 12-9 \u00a0 \u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u7684\u5212\u5206\u7ed3\u679c
\u8bbe\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u521d\u59cb\u5316\u6bcf\u4e00\u4e2a\u8282\u70b9\uff08\u6267\u884c\u4e00\u4e2a\u9012\u5f52\u51fd\u6570
dfs()\uff09\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u54c8\u5e0c\u8868\u5b58\u50a8
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/","title":"12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5","text":"inorder\u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u300c\u5206\u6cbb divide and conquer\u300d\uff0c\u5168\u79f0\u5206\u800c\u6cbb\u4e4b\uff0c\u662f\u4e00\u79cd\u975e\u5e38\u91cd\u8981\u4e14\u5e38\u89c1\u7684\u7b97\u6cd5\u7b56\u7565\u3002\u5206\u6cbb\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff0c\u5305\u62ec\u201c\u5206\u201d\u548c\u201c\u6cbb\u201d\u4e24\u4e2a\u6b65\u9aa4\u3002
- \u5206\uff08\u5212\u5206\u9636\u6bb5\uff09\uff1a\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u6216\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898\u65f6\u7ec8\u6b62\u3002
- \u6cbb\uff08\u5408\u5e76\u9636\u6bb5\uff09\uff1a\u4ece\u5df2\u77e5\u89e3\u7684\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002
\u5982\u56fe 12-1 \u6240\u793a\uff0c\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\u4e4b\u4e00\u3002
- \u5206\uff1a\u9012\u5f52\u5730\u5c06\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u5230\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\uff08\u6700\u5c0f\u5b50\u95ee\u9898\uff09\u3002
- \u6cbb\uff1a\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u6709\u5e8f\u7684\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u7684\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002
\u56fe 12-1 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","title":"12.1.1 \u00a0 \u5982\u4f55\u5224\u65ad\u5206\u6cbb\u95ee\u9898","text":"\u4e00\u4e2a\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u5206\u6cbb\u89e3\u51b3\uff0c\u901a\u5e38\u53ef\u4ee5\u53c2\u8003\u4ee5\u4e0b\u51e0\u4e2a\u5224\u65ad\u4f9d\u636e\u3002
- \u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u539f\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\u6210\u89c4\u6a21\u66f4\u5c0f\u3001\u7c7b\u4f3c\u7684\u5b50\u95ee\u9898\uff0c\u4ee5\u53ca\u80fd\u591f\u4ee5\u76f8\u540c\u65b9\u5f0f\u9012\u5f52\u5730\u8fdb\u884c\u5212\u5206\u3002
- \u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5b50\u95ee\u9898\u4e4b\u95f4\u6ca1\u6709\u91cd\u53e0\uff0c\u4e92\u4e0d\u4f9d\u8d56\uff0c\u53ef\u4ee5\u72ec\u7acb\u89e3\u51b3\u3002
- \u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u539f\u95ee\u9898\u7684\u89e3\u901a\u8fc7\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\u5f97\u6765\u3002
\u663e\u7136\uff0c\u5f52\u5e76\u6392\u5e8f\u6ee1\u8db3\u4ee5\u4e0a\u4e09\u6761\u5224\u65ad\u4f9d\u636e\u3002
- \u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u9012\u5f52\u5730\u5c06\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\u3002
- \u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u53ef\u4ee5\u72ec\u7acb\u5730\u8fdb\u884c\u6392\u5e8f\uff08\u5b50\u95ee\u9898\u53ef\u4ee5\u72ec\u7acb\u8fdb\u884c\u6c42\u89e3\uff09\u3002
- \u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u53ef\u4ee5\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002
\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\uff0c\u5f80\u5f80\u8fd8\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u5728\u6392\u5e8f\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u76f8\u8f83\u4e8e\u9009\u62e9\u3001\u5192\u6ce1\u3001\u63d2\u5165\u6392\u5e8f\u66f4\u5feb\uff0c\u5c31\u662f\u56e0\u4e3a\u5b83\u4eec\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\u3002
\u90a3\u4e48\uff0c\u6211\u4eec\u4e0d\u7981\u53d1\u95ee\uff1a\u4e3a\u4ec0\u4e48\u5206\u6cbb\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u5176\u5e95\u5c42\u903b\u8f91\u662f\u4ec0\u4e48\uff1f\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c06\u5927\u95ee\u9898\u5206\u89e3\u4e3a\u591a\u4e2a\u5b50\u95ee\u9898\u3001\u89e3\u51b3\u5b50\u95ee\u9898\u3001\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u5408\u5e76\u4e3a\u539f\u95ee\u9898\u7684\u89e3\uff0c\u8fd9\u51e0\u6b65\u7684\u6548\u7387\u4e3a\u4ec0\u4e48\u6bd4\u76f4\u63a5\u89e3\u51b3\u539f\u95ee\u9898\u7684\u6548\u7387\u66f4\u9ad8\uff1f\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u4ece\u64cd\u4f5c\u6570\u91cf\u548c\u5e76\u884c\u8ba1\u7b97\u4e24\u65b9\u9762\u6765\u8ba8\u8bba\u3002
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","title":"1. \u00a0 \u64cd\u4f5c\u6570\u91cf\u4f18\u5316","text":"\u4ee5\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u4e3a\u4f8b\uff0c\u5176\u5904\u7406\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u9700\u8981 \\(O(n^2)\\) \u65f6\u95f4\u3002\u5047\u8bbe\u6211\u4eec\u6309\u7167\u56fe 12-2 \u6240\u793a\u7684\u65b9\u5f0f\uff0c\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u5219\u5212\u5206\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6bcf\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O((n / 2)^2)\\) \u65f6\u95f4\uff0c\u5408\u5e76\u4e24\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a
\\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\]\u56fe 12-2 \u00a0 \u5212\u5206\u6570\u7ec4\u524d\u540e\u7684\u5192\u6ce1\u6392\u5e8f
\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u8ba1\u7b97\u4ee5\u4e0b\u4e0d\u7b49\u5f0f\uff0c\u5176\u5de6\u8fb9\u548c\u53f3\u8fb9\u5206\u522b\u4e3a\u5212\u5206\u524d\u548c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u603b\u6570\uff1a
\\[ \\begin{aligned} n^2 & > \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n & > 0 \\newline n(n - 4) & > 0 \\end{aligned} \\]\u8fd9\u610f\u5473\u7740\u5f53 \\(n > 4\\) \u65f6\uff0c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u6570\u91cf\u66f4\u5c11\uff0c\u6392\u5e8f\u6548\u7387\u5e94\u8be5\u66f4\u9ad8\u3002\u8bf7\u6ce8\u610f\uff0c\u5212\u5206\u540e\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u7136\u662f\u5e73\u65b9\u9636 \\(O(n^2)\\) \uff0c\u53ea\u662f\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\u4e86\u3002
\u8fdb\u4e00\u6b65\u60f3\uff0c\u5982\u679c\u6211\u4eec\u628a\u5b50\u6570\u7ec4\u4e0d\u65ad\u5730\u518d\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u505c\u6b62\u5212\u5206\u5462\uff1f\u8fd9\u79cd\u601d\u8def\u5b9e\u9645\u4e0a\u5c31\u662f\u201c\u5f52\u5e76\u6392\u5e8f\u201d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
\u518d\u601d\u8003\uff0c\u5982\u679c\u6211\u4eec\u591a\u8bbe\u7f6e\u51e0\u4e2a\u5212\u5206\u70b9\uff0c\u5c06\u539f\u6570\u7ec4\u5e73\u5747\u5212\u5206\u4e3a \\(k\\) \u4e2a\u5b50\u6570\u7ec4\u5462\uff1f\u8fd9\u79cd\u60c5\u51b5\u4e0e\u201c\u6876\u6392\u5e8f\u201d\u975e\u5e38\u7c7b\u4f3c\uff0c\u5b83\u975e\u5e38\u9002\u5408\u6392\u5e8f\u6d77\u91cf\u6570\u636e\uff0c\u7406\u8bba\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n + k)\\) \u3002
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","title":"2. \u00a0 \u5e76\u884c\u8ba1\u7b97\u4f18\u5316","text":"\u6211\u4eec\u77e5\u9053\uff0c\u5206\u6cbb\u751f\u6210\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u56e0\u6b64\u901a\u5e38\u53ef\u4ee5\u5e76\u884c\u89e3\u51b3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8fd8\u6709\u5229\u4e8e\u64cd\u4f5c\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002
\u5e76\u884c\u4f18\u5316\u5728\u591a\u6838\u6216\u591a\u5904\u7406\u5668\u7684\u73af\u5883\u4e2d\u5c24\u5176\u6709\u6548\uff0c\u56e0\u4e3a\u7cfb\u7edf\u53ef\u4ee5\u540c\u65f6\u5904\u7406\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u66f4\u52a0\u5145\u5206\u5730\u5229\u7528\u8ba1\u7b97\u8d44\u6e90\uff0c\u4ece\u800c\u663e\u8457\u51cf\u5c11\u603b\u4f53\u7684\u8fd0\u884c\u65f6\u95f4\u3002
\u6bd4\u5982\u5728\u56fe 12-3 \u6240\u793a\u7684\u201c\u6876\u6392\u5e8f\u201d\u4e2d\uff0c\u6211\u4eec\u5c06\u6d77\u91cf\u7684\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u5219\u53ef\u6240\u6709\u6876\u7684\u6392\u5e8f\u4efb\u52a1\u5206\u6563\u5230\u5404\u4e2a\u8ba1\u7b97\u5355\u5143\uff0c\u5b8c\u6210\u540e\u518d\u5408\u5e76\u7ed3\u679c\u3002
\u56fe 12-3 \u00a0 \u6876\u6392\u5e8f\u7684\u5e76\u884c\u8ba1\u7b97
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","title":"12.1.3 \u00a0 \u5206\u6cbb\u5e38\u89c1\u5e94\u7528","text":"\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u8bb8\u591a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\u3002
- \u5bfb\u627e\u6700\u8fd1\u70b9\u5bf9\uff1a\u8be5\u7b97\u6cd5\u9996\u5148\u5c06\u70b9\u96c6\u5206\u6210\u4e24\u90e8\u5206\uff0c\u7136\u540e\u5206\u522b\u627e\u51fa\u4e24\u90e8\u5206\u4e2d\u7684\u6700\u8fd1\u70b9\u5bf9\uff0c\u6700\u540e\u627e\u51fa\u8de8\u8d8a\u4e24\u90e8\u5206\u7684\u6700\u8fd1\u70b9\u5bf9\u3002
- \u5927\u6574\u6570\u4e58\u6cd5\uff1a\u4f8b\u5982 Karatsuba \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u6574\u6570\u4e58\u6cd5\u5206\u89e3\u4e3a\u51e0\u4e2a\u8f83\u5c0f\u7684\u6574\u6570\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002
- \u77e9\u9635\u4e58\u6cd5\uff1a\u4f8b\u5982 Strassen \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u77e9\u9635\u4e58\u6cd5\u5206\u89e3\u4e3a\u591a\u4e2a\u5c0f\u77e9\u9635\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002
- \u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u6c49\u8bfa\u5854\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u89e3\u51b3\uff0c\u8fd9\u662f\u5178\u578b\u7684\u5206\u6cbb\u7b56\u7565\u5e94\u7528\u3002
- \u6c42\u89e3\u9006\u5e8f\u5bf9\uff1a\u5728\u4e00\u4e2a\u5e8f\u5217\u4e2d\uff0c\u5982\u679c\u524d\u9762\u7684\u6570\u5b57\u5927\u4e8e\u540e\u9762\u7684\u6570\u5b57\uff0c\u90a3\u4e48\u8fd9\u4e24\u4e2a\u6570\u5b57\u6784\u6210\u4e00\u4e2a\u9006\u5e8f\u5bf9\u3002\u6c42\u89e3\u9006\u5e8f\u5bf9\u95ee\u9898\u53ef\u4ee5\u5229\u7528\u5206\u6cbb\u7684\u601d\u60f3\uff0c\u501f\u52a9\u5f52\u5e76\u6392\u5e8f\u8fdb\u884c\u6c42\u89e3\u3002
\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5728\u7b97\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u4e2d\u5e94\u7528\u975e\u5e38\u5e7f\u6cdb\u3002
- \u4e8c\u5206\u67e5\u627e\uff1a\u4e8c\u5206\u67e5\u627e\u662f\u5c06\u6709\u5e8f\u6570\u7ec4\u4ece\u4e2d\u70b9\u7d22\u5f15\u5904\u5206\u4e3a\u4e24\u90e8\u5206\uff0c\u7136\u540e\u6839\u636e\u76ee\u6807\u503c\u4e0e\u4e2d\u95f4\u5143\u7d20\u503c\u6bd4\u8f83\u7ed3\u679c\uff0c\u51b3\u5b9a\u6392\u9664\u54ea\u4e00\u534a\u533a\u95f4\uff0c\u5e76\u5728\u5269\u4f59\u533a\u95f4\u6267\u884c\u76f8\u540c\u7684\u4e8c\u5206\u64cd\u4f5c\u3002
- \u5f52\u5e76\u6392\u5e8f\uff1a\u672c\u8282\u5f00\u5934\u5df2\u4ecb\u7ecd\uff0c\u4e0d\u518d\u8d58\u8ff0\u3002
- \u5feb\u901f\u6392\u5e8f\uff1a\u5feb\u901f\u6392\u5e8f\u662f\u9009\u53d6\u4e00\u4e2a\u57fa\u51c6\u503c\uff0c\u7136\u540e\u628a\u6570\u7ec4\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u4e00\u4e2a\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5c0f\uff0c\u53e6\u4e00\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5927\uff0c\u518d\u5bf9\u8fd9\u4e24\u90e8\u5206\u8fdb\u884c\u76f8\u540c\u7684\u5212\u5206\u64cd\u4f5c\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e0b\u4e00\u4e2a\u5143\u7d20\u3002
- \u6876\u6392\u5e8f\uff1a\u6876\u6392\u5e8f\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c06\u6570\u636e\u5206\u6563\u5230\u591a\u4e2a\u6876\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u5404\u4e2a\u6876\u7684\u5143\u7d20\u4f9d\u6b21\u53d6\u51fa\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002
- \u6811\uff1a\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u7ea2\u9ed1\u6811\u3001B \u6811\u3001B+ \u6811\u7b49\uff0c\u5b83\u4eec\u7684\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u7b49\u64cd\u4f5c\u90fd\u53ef\u4ee5\u89c6\u4e3a\u5206\u6cbb\u7b56\u7565\u7684\u5e94\u7528\u3002
- \u5806\uff1a\u5806\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u5176\u5404\u79cd\u64cd\u4f5c\uff0c\u5982\u63d2\u5165\u3001\u5220\u9664\u548c\u5806\u5316\uff0c\u5b9e\u9645\u4e0a\u90fd\u9690\u542b\u4e86\u5206\u6cbb\u7684\u601d\u60f3\u3002
- \u54c8\u5e0c\u8868\uff1a\u867d\u7136\u54c8\u5e0c\u8868\u6765\u5e76\u4e0d\u76f4\u63a5\u5e94\u7528\u5206\u6cbb\uff0c\u4f46\u67d0\u4e9b\u54c8\u5e0c\u51b2\u7a81\u89e3\u51b3\u65b9\u6848\u95f4\u63a5\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u4f8b\u5982\uff0c\u94fe\u5f0f\u5730\u5740\u4e2d\u7684\u957f\u94fe\u8868\u4f1a\u88ab\u8f6c\u5316\u4e3a\u7ea2\u9ed1\u6811\uff0c\u4ee5\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002
\u53ef\u4ee5\u770b\u51fa\uff0c\u5206\u6cbb\u662f\u4e00\u79cd\u201c\u6da6\u7269\u7ec6\u65e0\u58f0\u201d\u7684\u7b97\u6cd5\u601d\u60f3\uff0c\u9690\u542b\u5728\u5404\u79cd\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u4e2d\u3002
"},{"location":"chapter_divide_and_conquer/hanota_problem/","title":"12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898","text":"\u5728\u5f52\u5e76\u6392\u5e8f\u548c\u6784\u5efa\u4e8c\u53c9\u6811\u4e2d\uff0c\u6211\u4eec\u90fd\u662f\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a\u539f\u95ee\u9898\u4e00\u534a\u7684\u5b50\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u6c49\u8bfa\u5854\u95ee\u9898\uff0c\u6211\u4eec\u91c7\u7528\u4e0d\u540c\u7684\u5206\u89e3\u7b56\u7565\u3002
Question
\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\uff0c\u8bb0\u4e3a
A\u3001B\u548cC\u3002\u8d77\u59cb\u72b6\u6001\u4e0b\uff0c\u67f1\u5b50A\u4e0a\u5957\u7740 \\(n\\) \u4e2a\u5706\u76d8\uff0c\u5b83\u4eec\u4ece\u4e0a\u5230\u4e0b\u6309\u7167\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u3002\u6211\u4eec\u7684\u4efb\u52a1\u662f\u8981\u628a\u8fd9 \\(n\\) \u4e2a\u5706\u76d8\u79fb\u5230\u67f1\u5b50C\u4e0a\uff0c\u5e76\u4fdd\u6301\u5b83\u4eec\u7684\u539f\u6709\u987a\u5e8f\u4e0d\u53d8\uff08\u5982\u56fe 12-10 \u6240\u793a\uff09\u3002\u5728\u79fb\u52a8\u5706\u76d8\u7684\u8fc7\u7a0b\u4e2d\uff0c\u9700\u8981\u9075\u5b88\u4ee5\u4e0b\u89c4\u5219\u3002- \u5706\u76d8\u53ea\u80fd\u4ece\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u62ff\u51fa\uff0c\u4ece\u53e6\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u653e\u5165\u3002
- \u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\u3002
- \u5c0f\u5706\u76d8\u5fc5\u987b\u65f6\u523b\u4f4d\u4e8e\u5927\u5706\u76d8\u4e4b\u4e0a\u3002
\u56fe 12-10 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u793a\u4f8b
\u6211\u4eec\u5c06\u89c4\u6a21\u4e3a \\(i\\) \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u8bb0\u4f5c \\(f(i)\\) \u3002\u4f8b\u5982 \\(f(3)\\) \u4ee3\u8868\u5c06 \\(3\\) \u4e2a\u5706\u76d8\u4ece
"},{"location":"chapter_divide_and_conquer/hanota_problem/#1","title":"1. \u00a0 \u8003\u8651\u57fa\u672c\u60c5\u51b5","text":"A\u79fb\u52a8\u81f3C\u7684\u6c49\u8bfa\u5854\u95ee\u9898\u3002\u5982\u56fe 12-11 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(1)\\) \uff0c\u5373\u5f53\u53ea\u6709\u4e00\u4e2a\u5706\u76d8\u65f6\uff0c\u6211\u4eec\u5c06\u5b83\u76f4\u63a5\u4ece
<1><2>A\u79fb\u52a8\u81f3C\u5373\u53ef\u3002\u56fe 12-11 \u00a0 \u89c4\u6a21\u4e3a 1 \u95ee\u9898\u7684\u89e3
\u5982\u56fe 12-12 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(2)\\) \uff0c\u5373\u5f53\u6709\u4e24\u4e2a\u5706\u76d8\u65f6\uff0c\u7531\u4e8e\u8981\u65f6\u523b\u6ee1\u8db3\u5c0f\u5706\u76d8\u5728\u5927\u5706\u76d8\u4e4b\u4e0a\uff0c\u56e0\u6b64\u9700\u8981\u501f\u52a9
B\u6765\u5b8c\u6210\u79fb\u52a8\u3002- \u5148\u5c06\u4e0a\u9762\u7684\u5c0f\u5706\u76d8\u4ece
A\u79fb\u81f3B\u3002 - \u518d\u5c06\u5927\u5706\u76d8\u4ece
A\u79fb\u81f3C\u3002 - \u6700\u540e\u5c06\u5c0f\u5706\u76d8\u4ece
B\u79fb\u81f3C\u3002
\u56fe 12-12 \u00a0 \u89c4\u6a21\u4e3a 2 \u95ee\u9898\u7684\u89e3
\u89e3\u51b3\u95ee\u9898 \\(f(2)\\) \u7684\u8fc7\u7a0b\u53ef\u603b\u7ed3\u4e3a\uff1a\u5c06\u4e24\u4e2a\u5706\u76d8\u501f\u52a9
"},{"location":"chapter_divide_and_conquer/hanota_problem/#2","title":"2. \u00a0 \u5b50\u95ee\u9898\u5206\u89e3","text":"B\u4eceA\u79fb\u81f3C\u3002\u5176\u4e2d\uff0cC\u79f0\u4e3a\u76ee\u6807\u67f1\u3001B\u79f0\u4e3a\u7f13\u51b2\u67f1\u3002\u5bf9\u4e8e\u95ee\u9898 \\(f(3)\\) \uff0c\u5373\u5f53\u6709\u4e09\u4e2a\u5706\u76d8\u65f6\uff0c\u60c5\u51b5\u53d8\u5f97\u7a0d\u5fae\u590d\u6742\u4e86\u4e00\u4e9b\u3002
\u56e0\u4e3a\u5df2\u77e5 \\(f(1)\\) \u548c \\(f(2)\\) \u7684\u89e3\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ece\u5206\u6cbb\u89d2\u5ea6\u601d\u8003\uff0c\u5c06
A\u9876\u90e8\u7684\u4e24\u4e2a\u5706\u76d8\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u6267\u884c\u56fe 12-13 \u6240\u793a\u7684\u6b65\u9aa4\u3002\u8fd9\u6837\u4e09\u4e2a\u5706\u76d8\u5c31\u88ab\u987a\u5229\u5730\u4eceA\u79fb\u81f3C\u4e86\u3002- \u4ee4
B\u4e3a\u76ee\u6807\u67f1\u3001C\u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4eceA\u79fb\u81f3B\u3002 - \u5c06
A\u4e2d\u5269\u4f59\u7684\u4e00\u4e2a\u5706\u76d8\u4eceA\u76f4\u63a5\u79fb\u52a8\u81f3C\u3002 - \u4ee4
C\u4e3a\u76ee\u6807\u67f1\u3001A\u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4eceB\u79fb\u81f3C\u3002
\u56fe 12-13 \u00a0 \u89c4\u6a21\u4e3a 3 \u95ee\u9898\u7684\u89e3
\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u5c06\u95ee\u9898 \\(f(3)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(2)\\) \u548c\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u4e4b\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002\u8fd9\u8bf4\u660e\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff0c\u800c\u4e14\u89e3\u53ef\u4ee5\u5408\u5e76\u3002
\u81f3\u6b64\uff0c\u6211\u4eec\u53ef\u603b\u7ed3\u51fa\u56fe 12-14 \u6240\u793a\u7684\u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565\uff1a\u5c06\u539f\u95ee\u9898 \\(f(n)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \u548c\u4e00\u4e2a\u5b50\u95ee\u9898 \\(f(1)\\) \uff0c\u5e76\u6309\u7167\u4ee5\u4e0b\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u3002
- \u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9
C\u4eceA\u79fb\u81f3B\u3002 - \u5c06\u5269\u4f59 \\(1\\) \u4e2a\u5706\u76d8\u4ece
A\u76f4\u63a5\u79fb\u81f3C\u3002 - \u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9
A\u4eceB\u79fb\u81f3C\u3002
\u5bf9\u4e8e\u8fd9\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \uff0c\u53ef\u4ee5\u901a\u8fc7\u76f8\u540c\u7684\u65b9\u5f0f\u8fdb\u884c\u9012\u5f52\u5212\u5206\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u800c \\(f(1)\\) \u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u53ea\u9700\u4e00\u6b21\u79fb\u52a8\u64cd\u4f5c\u5373\u53ef\u3002
\u56fe 12-14 \u00a0 \u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565
"},{"location":"chapter_divide_and_conquer/hanota_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570
PythonC++JavaC#GoSwiftJSTSDartRustCZig hanota.pydfs(i, src, buf, tar)\uff0c\u5b83\u7684\u4f5c\u7528\u662f\u5c06\u67f1src\u9876\u90e8\u7684 \\(i\\) \u4e2a\u5706\u76d8\u501f\u52a9\u7f13\u51b2\u67f1buf\u79fb\u52a8\u81f3\u76ee\u6807\u67f1tar\uff1a
hanota.cppdef move(src: list[int], tar: list[int]):\n \"\"\"\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\"\"\"\n # \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n pan = src.pop()\n # \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i)\"\"\"\n # \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if i == 1:\n move(src, tar)\n return\n # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf)\n # \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar)\n # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898\"\"\"\n n = len(A)\n # \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C)\n
hanota.java/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(vector<int> &src, vector<int> &tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n int pan = src.back();\n src.pop_back();\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.push_back(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, vector<int> &src, vector<int> &buf, vector<int> &tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(vector<int> &A, vector<int> &B, vector<int> &C) {\n int n = A.size();\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.cs/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List<Integer> src, List<Integer> tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n Integer pan = src.remove(src.size() - 1);\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List<Integer> src, List<Integer> buf, List<Integer> tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List<Integer> A, List<Integer> B, List<Integer> C) {\n int n = A.size();\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.go/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid Move(List<int> src, List<int> tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n int pan = src[^1];\n src.RemoveAt(src.Count - 1);\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.Add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid DFS(int i, List<int> src, List<int> buf, List<int> tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n Move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n DFS(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n Move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n DFS(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid SolveHanota(List<int> A, List<int> B, List<int> C) {\n int n = A.Count;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n DFS(n, A, B, C);\n}\n
hanota.swift/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src, tar *list.List) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n pan := src.Back()\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.PushBack(pan.Value)\n // \u79fb\u9664 src \u9876\u90e8\u5706\u76d8\n src.Remove(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if i == 1 {\n move(src, tar)\n return\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfsHanota(i-1, src, tar, buf)\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar)\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfsHanota(i-1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A, B, C *list.List) {\n n := A.Len()\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfsHanota(n, A, B, C)\n}\n
hanota.js/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n let pan = src.popLast()!\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.append(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if i == 1 {\n move(src: &src, tar: &tar)\n return\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i: i - 1, src: &src, buf: &tar, tar: &buf)\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src: &src, tar: &tar)\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i: i - 1, src: &buf, buf: &src, tar: &tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n let n = A.count\n // \u5217\u8868\u5c3e\u90e8\u662f\u67f1\u5b50\u9876\u90e8\n // \u5c06 src \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(i: n, src: &A, buf: &B, tar: &C)\n}\n
hanota.ts/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src, tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n const pan = src.pop();\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i, src, buf, tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i === 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A, B, C) {\n const n = A.length;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.dart/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src: number[], tar: number[]): void {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n const pan = src.pop();\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i === 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n const n = A.length;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.rs/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List<int> src, List<int> tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n int pan = src.removeLast();\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List<int> src, List<int> buf, List<int> tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List<int> A, List<int> B, List<int> C) {\n int n = A.length;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.c/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfn move_pan(src: &mut Vec<i32>, tar: &mut Vec<i32>) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n let pan = src.remove(src.len() - 1);\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfn dfs(i: i32, src: &mut Vec<i32>, buf: &mut Vec<i32>, tar: &mut Vec<i32>) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if i == 1 {\n move_pan(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move_pan(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfn solve_hanota(A: &mut Vec<i32>, B: &mut Vec<i32>, C: &mut Vec<i32>) {\n let n = A.len() as i32;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.zig/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n int pan = src[*srcSize - 1];\n src[*srcSize - 1] = 0;\n (*srcSize)--;\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar[*tarSize] = pan;\n (*tarSize)++;\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n move(src, srcSize, tar, tarSize);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, srcSize, tar, tarSize);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n[class]{}-[func]{move}\n\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{solveHanota}\n\u5982\u56fe 12-15 \u6240\u793a\uff0c\u6c49\u8bfa\u5854\u95ee\u9898\u5f62\u6210\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5bf9\u5e94\u4e00\u4e2a\u5f00\u542f\u7684
dfs()\u51fd\u6570\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u56fe 12-15 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u7684\u9012\u5f52\u6811
Quote
\u6c49\u8bfa\u5854\u95ee\u9898\u6e90\u81ea\u4e00\u4e2a\u53e4\u8001\u7684\u4f20\u8bf4\u3002\u5728\u53e4\u5370\u5ea6\u7684\u4e00\u4e2a\u5bfa\u5e99\u91cc\uff0c\u50e7\u4fa3\u4eec\u6709\u4e09\u6839\u9ad8\u5927\u7684\u94bb\u77f3\u67f1\u5b50\uff0c\u4ee5\u53ca \\(64\\) \u4e2a\u5927\u5c0f\u4e0d\u4e00\u7684\u91d1\u5706\u76d8\u3002\u50e7\u4fa3\u4eec\u4e0d\u65ad\u5730\u79fb\u52a8\u5706\u76d8\uff0c\u4ed6\u4eec\u76f8\u4fe1\u5728\u6700\u540e\u4e00\u4e2a\u5706\u76d8\u88ab\u6b63\u786e\u653e\u7f6e\u7684\u90a3\u4e00\u523b\uff0c\u8fd9\u4e2a\u4e16\u754c\u5c31\u4f1a\u7ed3\u675f\u3002
\u7136\u800c\uff0c\u5373\u4f7f\u50e7\u4fa3\u4eec\u6bcf\u79d2\u949f\u79fb\u52a8\u4e00\u6b21\uff0c\u603b\u5171\u9700\u8981\u5927\u7ea6 \\(2^{64} \\approx 1.84\u00d710^{19}\\) \u79d2\uff0c\u5408\u7ea6 \\(5850\\) \u4ebf\u5e74\uff0c\u8fdc\u8fdc\u8d85\u8fc7\u4e86\u73b0\u5728\u5bf9\u5b87\u5b99\u5e74\u9f84\u7684\u4f30\u8ba1\u3002\u6240\u4ee5\uff0c\u5018\u82e5\u8fd9\u4e2a\u4f20\u8bf4\u662f\u771f\u7684\uff0c\u6211\u4eec\u5e94\u8be5\u4e0d\u9700\u8981\u62c5\u5fc3\u4e16\u754c\u672b\u65e5\u7684\u5230\u6765\u3002
"},{"location":"chapter_divide_and_conquer/summary/","title":"12.5 \u00a0 \u5c0f\u7ed3","text":"- \u5206\u6cbb\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u7b97\u6cd5\u8bbe\u8ba1\u7b56\u7565\uff0c\u5305\u62ec\u5206\uff08\u5212\u5206\uff09\u548c\u6cbb\uff08\u5408\u5e76\uff09\u4e24\u4e2a\u9636\u6bb5\uff0c\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002
- \u5224\u65ad\u662f\u5426\u662f\u5206\u6cbb\u7b97\u6cd5\u95ee\u9898\u7684\u4f9d\u636e\u5305\u62ec\uff1a\u95ee\u9898\u80fd\u5426\u5206\u89e3\u3001\u5b50\u95ee\u9898\u662f\u5426\u72ec\u7acb\u3001\u5b50\u95ee\u9898\u80fd\u5426\u5408\u5e76\u3002
- \u5f52\u5e76\u6392\u5e8f\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\uff0c\u5176\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u7b49\u957f\u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u5230\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u5f00\u59cb\u9010\u5c42\u5408\u5e76\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002
- \u5f15\u5165\u5206\u6cbb\u7b56\u7565\u5f80\u5f80\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u7b56\u7565\u51cf\u5c11\u4e86\u64cd\u4f5c\u6570\u91cf\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u540e\u6709\u5229\u4e8e\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002
- \u5206\u6cbb\u65e2\u53ef\u4ee5\u89e3\u51b3\u8bb8\u591a\u7b97\u6cd5\u95ee\u9898\uff0c\u4e5f\u5e7f\u6cdb\u5e94\u7528\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u5904\u5904\u53ef\u89c1\u5176\u8eab\u5f71\u3002
- \u76f8\u8f83\u4e8e\u66b4\u529b\u641c\u7d22\uff0c\u81ea\u9002\u5e94\u641c\u7d22\u6548\u7387\u66f4\u9ad8\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\u3002
- \u4e8c\u5206\u67e5\u627e\u662f\u5206\u6cbb\u7b56\u7565\u7684\u53e6\u4e00\u4e2a\u5178\u578b\u5e94\u7528\uff0c\u5b83\u4e0d\u5305\u542b\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u7684\u6b65\u9aa4\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e\u3002
- \u5728\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u95ee\u9898\u4e2d\uff0c\u6784\u5efa\u6811\uff08\u539f\u95ee\u9898\uff09\u53ef\u4ee5\u5212\u5206\u4e3a\u6784\u5efa\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5212\u5206\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u7684\u7d22\u5f15\u533a\u95f4\u6765\u5b9e\u73b0\u3002
- \u5728\u6c49\u8bfa\u5854\u95ee\u9898\u4e2d\uff0c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(n\\) \u7684\u95ee\u9898\u53ef\u4ee5\u5212\u5206\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a \\(n-1\\) \u7684\u5b50\u95ee\u9898\u548c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(1\\) \u7684\u5b50\u95ee\u9898\u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002
Abstract
\u5c0f\u6eaa\u6c47\u5165\u6cb3\u6d41\uff0c\u6c5f\u6cb3\u6c47\u5165\u5927\u6d77\u3002
\u52a8\u6001\u89c4\u5212\u5c06\u5c0f\u95ee\u9898\u7684\u89e3\u6c47\u96c6\u6210\u5927\u95ee\u9898\u7684\u7b54\u6848\uff0c\u4e00\u6b65\u6b65\u5f15\u9886\u6211\u4eec\u8d70\u5411\u89e3\u51b3\u95ee\u9898\u7684\u5f7c\u5cb8\u3002
"},{"location":"chapter_dynamic_programming/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212
- 14.2 \u00a0 DP \u95ee\u9898\u7279\u6027
- 14.3 \u00a0 DP \u89e3\u9898\u601d\u8def
- 14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898
- 14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898
- 14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898
- 14.7 \u00a0 \u5c0f\u7ed3
\u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u52a8\u6001\u89c4\u5212\u662f\u5982\u4f55\u901a\u8fc7\u5b50\u95ee\u9898\u5206\u89e3\u6765\u6c42\u89e3\u539f\u95ee\u9898\u7684\u3002\u5b9e\u9645\u4e0a\uff0c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u7684\u4fa7\u91cd\u70b9\u4e0d\u540c\u3002
- \u5206\u6cbb\u7b97\u6cd5\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u591a\u4e2a\u76f8\u4e92\u72ec\u7acb\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u6700\u5c0f\u5b50\u95ee\u9898\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6700\u7ec8\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002
- \u52a8\u6001\u89c4\u5212\u4e5f\u5bf9\u95ee\u9898\u8fdb\u884c\u9012\u5f52\u5206\u89e3\uff0c\u4f46\u4e0e\u5206\u6cbb\u7b97\u6cd5\u7684\u4e3b\u8981\u533a\u522b\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u4f9d\u8d56\u7684\uff0c\u5728\u5206\u89e3\u8fc7\u7a0b\u4e2d\u4f1a\u51fa\u73b0\u8bb8\u591a\u91cd\u53e0\u5b50\u95ee\u9898\u3002
- \u56de\u6eaf\u7b97\u6cd5\u5728\u5c1d\u8bd5\u548c\u56de\u9000\u4e2d\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u5e76\u901a\u8fc7\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\u3002\u539f\u95ee\u9898\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\u6784\u6210\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u51b3\u7b56\u6b65\u9aa4\u4e4b\u524d\u7684\u5b50\u5e8f\u5217\u770b\u4f5c\u4e00\u4e2a\u5b50\u95ee\u9898\u3002
\u5b9e\u9645\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5b83\u4eec\u4e0d\u4ec5\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u8fd8\u5177\u6709\u53e6\u5916\u4e24\u5927\u7279\u6027\uff1a\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002
"},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","title":"14.2.1 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784","text":"\u6211\u4eec\u5bf9\u722c\u697c\u68af\u95ee\u9898\u7a0d\u4f5c\u6539\u52a8\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9002\u5408\u5c55\u793a\u6700\u4f18\u5b50\u7ed3\u6784\u6982\u5ff5\u3002
\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7
\u7ed9\u5b9a\u4e00\u4e2a\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u6bcf\u4e00\u9636\u697c\u68af\u4e0a\u90fd\u8d34\u6709\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u4f60\u5728\u8be5\u53f0\u9636\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\u3002\u7ed9\u5b9a\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\u6570\u7ec4 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u4e2a\u53f0\u9636\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\\(cost[0]\\) \u4e3a\u5730\u9762\uff08\u8d77\u59cb\u70b9\uff09\u3002\u8bf7\u8ba1\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u4ef7\u624d\u80fd\u5230\u8fbe\u9876\u90e8\uff1f
\u5982\u56fe 14-6 \u6240\u793a\uff0c\u82e5\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9636\u7684\u4ee3\u4ef7\u5206\u522b\u4e3a \\(1\\)\u3001\\(10\\)\u3001\\(1\\) \uff0c\u5219\u4ece\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7\u4e3a \\(2\\) \u3002
\u56fe 14-6 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7
\u8bbe \\(dp[i]\\) \u4e3a\u722c\u5230\u7b2c \\(i\\) \u9636\u7d2f\u8ba1\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\u7531\u4e8e\u7b2c \\(i\\) \u9636\u53ea\u53ef\u80fd\u4ece \\(i - 1\\) \u9636\u6216 \\(i - 2\\) \u9636\u8d70\u6765\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u4e8e \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u4e3a\u4e86\u5c3d\u53ef\u80fd\u51cf\u5c11\u4ee3\u4ef7\uff0c\u6211\u4eec\u5e94\u8be5\u9009\u62e9\u4e24\u8005\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\uff1a
\\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\]\u8fd9\u4fbf\u53ef\u4ee5\u5f15\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u542b\u4e49\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u662f\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\u7684\u3002
\u672c\u9898\u663e\u7136\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6211\u4eec\u4ece\u4e24\u4e2a\u5b50\u95ee\u9898\u6700\u4f18\u89e3 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u4e2d\u6311\u9009\u51fa\u8f83\u4f18\u7684\u90a3\u4e00\u4e2a\uff0c\u5e76\u7528\u5b83\u6784\u5efa\u51fa\u539f\u95ee\u9898 \\(dp[i]\\) \u7684\u6700\u4f18\u89e3\u3002
\u90a3\u4e48\uff0c\u4e0a\u4e00\u8282\u7684\u722c\u697c\u68af\u9898\u76ee\u6709\u6ca1\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u5462\uff1f\u5b83\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u4e2a\u8ba1\u6570\u95ee\u9898\uff0c\u4f46\u5982\u679c\u6362\u4e00\u79cd\u95ee\u6cd5\uff1a\u201c\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6570\u91cf\u201d\u3002\u6211\u4eec\u610f\u5916\u5730\u53d1\u73b0\uff0c\u867d\u7136\u9898\u76ee\u4fee\u6539\u524d\u540e\u662f\u7b49\u4ef7\u7684\uff0c\u4f46\u6700\u4f18\u5b50\u7ed3\u6784\u6d6e\u73b0\u51fa\u6765\u4e86\uff1a\u7b2c \\(n\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u7b49\u4e8e\u7b2c \\(n-1\\) \u9636\u548c\u7b2c \\(n-2\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8bf4\uff0c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u89e3\u91ca\u65b9\u5f0f\u6bd4\u8f83\u7075\u6d3b\uff0c\u5728\u4e0d\u540c\u95ee\u9898\u4e2d\u4f1a\u6709\u4e0d\u540c\u7684\u542b\u4e49\u3002
\u6839\u636e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72b6\u6001 \\(dp[1] = cost[1]\\) \u548c \\(dp[2] = cost[2]\\) \uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u52a8\u6001\u89c4\u5212\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_cost_climbing_stairs_dp.py
min_cost_climbing_stairs_dp.cppdef min_cost_climbing_stairs_dp(cost: list[int]) -> int:\n \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(cost) - 1\n if n == 1 or n == 2:\n return cost[n]\n # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp = [0] * (n + 1)\n # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1], dp[2] = cost[1], cost[2]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in range(3, n + 1):\n dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n return dp[n]\n
min_cost_climbing_stairs_dp.java/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(vector<int> &cost) {\n int n = cost.size() - 1;\n if (n == 1 || n == 2)\n return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n vector<int> dp(n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.cs/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\n int n = cost.length - 1;\n if (n == 1 || n == 2)\n return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[] dp = new int[n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.go/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDP(int[] cost) {\n int n = cost.Length - 1;\n if (n == 1 || n == 2)\n return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[] dp = new int[n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.swift/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost []int) int {\n n := len(cost) - 1\n if n == 1 || n == 2 {\n return cost[n]\n }\n min := func(a, b int) int {\n if a < b {\n return a\n }\n return b\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp := make([]int, n+1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1]\n dp[2] = cost[2]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n }\n return dp[n]\n}\n
min_cost_climbing_stairs_dp.js/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost: [Int]) -> Int {\n let n = cost.count - 1\n if n == 1 || n == 2 {\n return cost[n]\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = Array(repeating: 0, count: n + 1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1]\n dp[2] = cost[2]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in stride(from: 3, through: n, by: 1) {\n dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n }\n return dp[n]\n}\n
min_cost_climbing_stairs_dp.ts/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost) {\n const n = cost.length - 1;\n if (n === 1 || n === 2) {\n return cost[n];\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = new Array(n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.dart/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost: Array<number>): number {\n const n = cost.length - 1;\n if (n === 1 || n === 2) {\n return cost[n];\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = new Array(n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.rs/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(List<int> cost) {\n int n = cost.length - 1;\n if (n == 1 || n == 2) return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n List<int> dp = List.filled(n + 1, 0);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.c/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp(cost: &[i32]) -> i32 {\n let n = cost.len() - 1;\n if n == 1 || n == 2 { return cost[n]; }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n let mut dp = vec![-1; n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in 3..=n {\n dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n dp[n]\n}\n
min_cost_climbing_stairs_dp.zig/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n int n = costSize - 1;\n if (n == 1 || n == 2)\n return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int *dp = calloc(n + 1, sizeof(int));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n }\n int res = dp[n];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDP(comptime cost: []i32) i32 {\n comptime var n = cost.len - 1;\n if (n == 1 or n == 2) {\n return cost[n];\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = [_]i32{-1} ** (n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (3..n + 1) |i| {\n dp[i] = @min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n\u56fe 14-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\u3002
\u56fe 14-7 \u00a0 \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
\u672c\u9898\u4e5f\u53ef\u4ee5\u8fdb\u884c\u7a7a\u95f4\u4f18\u5316\uff0c\u5c06\u4e00\u7ef4\u538b\u7f29\u81f3\u96f6\u7ef4\uff0c\u4f7f\u5f97\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_cost_climbing_stairs_dp.py
min_cost_climbing_stairs_dp.cppdef min_cost_climbing_stairs_dp_comp(cost: list[int]) -> int:\n \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(cost) - 1\n if n == 1 or n == 2:\n return cost[n]\n a, b = cost[1], cost[2]\n for i in range(3, n + 1):\n a, b = b, min(a, b) + cost[i]\n return b\n
min_cost_climbing_stairs_dp.java/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(vector<int> &cost) {\n int n = cost.size() - 1;\n if (n == 1 || n == 2)\n return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.cs/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\n int n = cost.length - 1;\n if (n == 1 || n == 2)\n return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = Math.min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.go/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDPComp(int[] cost) {\n int n = cost.Length - 1;\n if (n == 1 || n == 2)\n return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = Math.Min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.swift/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n n := len(cost) - 1\n if n == 1 || n == 2 {\n return cost[n]\n }\n min := func(a, b int) int {\n if a < b {\n return a\n }\n return b\n }\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n a, b := cost[1], cost[2]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n tmp := b\n b = min(a, tmp) + cost[i]\n a = tmp\n }\n return b\n}\n
min_cost_climbing_stairs_dp.js/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -> Int {\n let n = cost.count - 1\n if n == 1 || n == 2 {\n return cost[n]\n }\n var (a, b) = (cost[1], cost[2])\n for i in stride(from: 3, through: n, by: 1) {\n (a, b) = (b, min(a, b) + cost[i])\n }\n return b\n}\n
min_cost_climbing_stairs_dp.ts/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost) {\n const n = cost.length - 1;\n if (n === 1 || n === 2) {\n return cost[n];\n }\n let a = cost[1],\n b = cost[2];\n for (let i = 3; i <= n; i++) {\n const tmp = b;\n b = Math.min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.dart/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost: Array<number>): number {\n const n = cost.length - 1;\n if (n === 1 || n === 2) {\n return cost[n];\n }\n let a = cost[1],\n b = cost[2];\n for (let i = 3; i <= n; i++) {\n const tmp = b;\n b = Math.min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.rs/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(List<int> cost) {\n int n = cost.length - 1;\n if (n == 1 || n == 2) return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.c/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp_comp(cost: &[i32]) -> i32 {\n let n = cost.len() - 1;\n if n == 1 || n == 2 { return cost[n] };\n let (mut a, mut b) = (cost[1], cost[2]);\n for i in 3..=n {\n let tmp = b;\n b = cmp::min(a, tmp) + cost[i];\n a = tmp;\n }\n b\n}\n
min_cost_climbing_stairs_dp.zig/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n int n = costSize - 1;\n if (n == 1 || n == 2)\n return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = myMin(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
"},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","title":"14.2.2 \u00a0 \u65e0\u540e\u6548\u6027","text":"// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDPComp(cost: []i32) i32 {\n var n = cost.len - 1;\n if (n == 1 or n == 2) {\n return cost[n];\n }\n var a = cost[1];\n var b = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (3..n + 1) |i| {\n var tmp = b;\n b = @min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n\u65e0\u540e\u6548\u6027\u662f\u52a8\u6001\u89c4\u5212\u80fd\u591f\u6709\u6548\u89e3\u51b3\u95ee\u9898\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5176\u5b9a\u4e49\u4e3a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u786e\u5b9a\u7684\u72b6\u6001\uff0c\u5b83\u7684\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u5f53\u524d\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002
\u4ee5\u722c\u697c\u68af\u95ee\u9898\u4e3a\u4f8b\uff0c\u7ed9\u5b9a\u72b6\u6001 \\(i\\) \uff0c\u5b83\u4f1a\u53d1\u5c55\u51fa\u72b6\u6001 \\(i+1\\) \u548c\u72b6\u6001 \\(i+2\\) \uff0c\u5206\u522b\u5bf9\u5e94\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u8fd9\u4e24\u79cd\u9009\u62e9\u65f6\uff0c\u6211\u4eec\u65e0\u987b\u8003\u8651\u72b6\u6001 \\(i\\) \u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5b83\u4eec\u5bf9\u72b6\u6001 \\(i\\) \u7684\u672a\u6765\u6ca1\u6709\u5f71\u54cd\u3002
\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u7ed9\u722c\u697c\u68af\u95ee\u9898\u6dfb\u52a0\u4e00\u4e2a\u7ea6\u675f\uff0c\u60c5\u51b5\u5c31\u4e0d\u4e00\u6837\u4e86\u3002
\u5e26\u7ea6\u675f\u722c\u697c\u68af
\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u4f46\u4e0d\u80fd\u8fde\u7eed\u4e24\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f
\u5982\u56fe 14-8 \u6240\u793a\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u9636\u4ec5\u5269 \\(2\\) \u79cd\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u8fde\u7eed\u4e09\u6b21\u8df3 \\(1\\) \u9636\u7684\u65b9\u6848\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\uff0c\u56e0\u6b64\u88ab\u820d\u5f03\u3002
\u56fe 14-8 \u00a0 \u5e26\u7ea6\u675f\u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf
\u5728\u8be5\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\uff0c\u90a3\u4e48\u4e0b\u4e00\u8f6e\u5c31\u5fc5\u987b\u8df3 \\(2\\) \u9636\u3002\u8fd9\u610f\u5473\u7740\uff0c\u4e0b\u4e00\u6b65\u9009\u62e9\u4e0d\u80fd\u7531\u5f53\u524d\u72b6\u6001\uff08\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u72ec\u7acb\u51b3\u5b9a\uff0c\u8fd8\u548c\u524d\u4e00\u4e2a\u72b6\u6001\uff08\u4e0a\u8f6e\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u6709\u5173\u3002
\u4e0d\u96be\u53d1\u73b0\uff0c\u6b64\u95ee\u9898\u5df2\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u4e3a \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8bb8\u591a\u201c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\u201d\u65b9\u6848\uff0c\u800c\u4e3a\u4e86\u6ee1\u8db3\u7ea6\u675f\uff0c\u6211\u4eec\u5c31\u4e0d\u80fd\u5c06 \\(dp[i-1]\\) \u76f4\u63a5\u8ba1\u5165 \\(dp[i]\\) \u4e2d\u3002
\u4e3a\u6b64\uff0c\u6211\u4eec\u9700\u8981\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff1a\u72b6\u6001 \\([i, j]\\) \u8868\u793a\u5904\u5728\u7b2c \\(i\\) \u9636\u5e76\u4e14\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(j\\) \u9636\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72b6\u6001\u5b9a\u4e49\u6709\u6548\u5730\u533a\u5206\u4e86\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u8fd8\u662f \\(2\\) \u9636\uff0c\u6211\u4eec\u53ef\u4ee5\u636e\u6b64\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u4ece\u4f55\u800c\u6765\u7684\u3002
- \u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ea\u80fd\u9009\u62e9\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 1]\\) \u53ea\u80fd\u4ece \\(dp[i-1, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002
- \u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(2\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ef\u9009\u62e9\u8df3 \\(1\\) \u9636\u6216\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 2]\\) \u53ef\u4ee5\u4ece \\(dp[i-2, 1]\\) \u6216 \\(dp[i-2, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002
\u5982\u56fe 14-9 \u6240\u793a\uff0c\u5728\u8be5\u5b9a\u4e49\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u65b9\u6848\u6570\u3002\u6b64\u65f6\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a
\\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\]\u56fe 14-9 \u00a0 \u8003\u8651\u7ea6\u675f\u4e0b\u7684\u9012\u63a8\u5173\u7cfb
\u6700\u7ec8\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u4e24\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u9636\u7684\u65b9\u6848\u603b\u6570\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_constraint_dp.py
climbing_stairs_constraint_dp.cppdef climbing_stairs_constraint_dp(n: int) -> int:\n \"\"\"\u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n if n == 1 or n == 2:\n return 1\n # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp = [[0] * 3 for _ in range(n + 1)]\n # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1], dp[1][2] = 1, 0\n dp[2][1], dp[2][2] = 0, 1\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in range(3, n + 1):\n dp[i][1] = dp[i - 1][2]\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n return dp[n][1] + dp[n][2]\n
climbing_stairs_constraint_dp.java/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n vector<vector<int>> dp(n + 1, vector<int>(3, 0));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.cs/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[][] dp = new int[n + 1][3];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.go/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[,] dp = new int[n + 1, 3];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1, 1] = 1;\n dp[1, 2] = 0;\n dp[2, 1] = 0;\n dp[2, 2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i, 1] = dp[i - 1, 2];\n dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n }\n return dp[n, 1] + dp[n, 2];\n}\n
climbing_stairs_constraint_dp.swift/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n int) int {\n if n == 1 || n == 2 {\n return 1\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp := make([][3]int, n+1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1\n dp[1][2] = 0\n dp[2][1] = 0\n dp[2][2] = 1\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n dp[i][1] = dp[i-1][2]\n dp[i][2] = dp[i-2][1] + dp[i-2][2]\n }\n return dp[n][1] + dp[n][2]\n}\n
climbing_stairs_constraint_dp.js/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n: Int) -> Int {\n if n == 1 || n == 2 {\n return 1\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1\n dp[1][2] = 0\n dp[2][1] = 0\n dp[2][2] = 1\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in stride(from: 3, through: n, by: 1) {\n dp[i][1] = dp[i - 1][2]\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n }\n return dp[n][1] + dp[n][2]\n}\n
climbing_stairs_constraint_dp.ts/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n) {\n if (n === 1 || n === 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = Array.from(new Array(n + 1), () => new Array(3));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.dart/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n: number): number {\n if (n === 1 || n === 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = Array.from({ length: n + 1 }, () => new Array(3));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.rs/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(3, 0));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.c/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_constraint_dp(n: usize) -> i32 {\n if n == 1 || n == 2 { return 1 };\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n let mut dp = vec![vec![-1; 3]; n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in 3..=n {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n dp[n][1] + dp[n][2]\n}\n
climbing_stairs_constraint_dp.zig/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(3, sizeof(int));\n }\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n int res = dp[n][1] + dp[n][2];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n free(dp);\n return res;\n}\n// \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsConstraintDP(comptime n: usize) i32 {\n if (n == 1 or n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = [_][3]i32{ [_]i32{ -1, -1, -1 } } ** (n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (3..n + 1) |i| {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n\u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u4e8e\u4ec5\u9700\u591a\u8003\u8651\u524d\u9762\u4e00\u4e2a\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u901a\u8fc7\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff0c\u4f7f\u5f97\u95ee\u9898\u91cd\u65b0\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u95ee\u9898\u5177\u6709\u975e\u5e38\u4e25\u91cd\u7684\u201c\u6709\u540e\u6548\u6027\u201d\u3002
\u722c\u697c\u68af\u4e0e\u969c\u788d\u751f\u6210
\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\u3002\u89c4\u5b9a\u5f53\u722c\u5230\u7b2c \\(i\\) \u9636\u65f6\uff0c\u7cfb\u7edf\u81ea\u52a8\u4f1a\u5728\u7b2c \\(2i\\) \u9636\u4e0a\u653e\u4e0a\u969c\u788d\u7269\uff0c\u4e4b\u540e\u6240\u6709\u8f6e\u90fd\u4e0d\u5141\u8bb8\u8df3\u5230\u7b2c \\(2i\\) \u9636\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u4e24\u8f6e\u5206\u522b\u8df3\u5230\u4e86\u7b2c \\(2\\)\u3001\\(3\\) \u9636\u4e0a\uff0c\u5219\u4e4b\u540e\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4\\)\u3001\\(6\\) \u9636\u4e0a\u3002\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f
\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8dc3\u4f9d\u8d56\u8fc7\u53bb\u6240\u6709\u7684\u72b6\u6001\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b21\u8df3\u8dc3\u90fd\u4f1a\u5728\u66f4\u9ad8\u7684\u9636\u68af\u4e0a\u8bbe\u7f6e\u969c\u788d\uff0c\u5e76\u5f71\u54cd\u672a\u6765\u7684\u8df3\u8dc3\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u5f80\u5f80\u96be\u4ee5\u89e3\u51b3\u3002
\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u590d\u6742\u7684\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff08\u4f8b\u5982\u65c5\u884c\u5546\u95ee\u9898\uff09\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u542f\u53d1\u5f0f\u641c\u7d22\u3001\u9057\u4f20\u7b97\u6cd5\u3001\u5f3a\u5316\u5b66\u4e60\u7b49\uff0c\u4ece\u800c\u5728\u6709\u9650\u65f6\u95f4\u5185\u5f97\u5230\u53ef\u7528\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","title":"14.3 \u00a0 \u52a8\u6001\u89c4\u5212\u89e3\u9898\u601d\u8def","text":"\u4e0a\u4e24\u8282\u4ecb\u7ecd\u4e86\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4e3b\u8981\u7279\u5f81\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u4e00\u8d77\u63a2\u7a76\u4e24\u4e2a\u66f4\u52a0\u5b9e\u7528\u7684\u95ee\u9898\u3002
- \u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u95ee\u9898\u662f\u4e0d\u662f\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff1f
- \u6c42\u89e3\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8be5\u4ece\u4f55\u5904\u5165\u624b\uff0c\u5b8c\u6574\u6b65\u9aa4\u662f\u4ec0\u4e48\uff1f
\u603b\u7684\u6765\u8bf4\uff0c\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u90a3\u4e48\u5b83\u901a\u5e38\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u6c42\u89e3\u3002\u7136\u800c\uff0c\u6211\u4eec\u5f88\u96be\u4ece\u95ee\u9898\u63cf\u8ff0\u4e2d\u76f4\u63a5\u63d0\u53d6\u51fa\u8fd9\u4e9b\u7279\u6027\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u653e\u5bbd\u6761\u4ef6\uff0c\u5148\u89c2\u5bdf\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u56de\u6eaf\uff08\u7a77\u4e3e\uff09\u89e3\u51b3\u3002
\u9002\u5408\u7528\u56de\u6eaf\u89e3\u51b3\u7684\u95ee\u9898\u901a\u5e38\u6ee1\u8db3\u201c\u51b3\u7b56\u6811\u6a21\u578b\u201d\uff0c\u8fd9\u79cd\u95ee\u9898\u53ef\u4ee5\u4f7f\u7528\u6811\u5f62\u7ed3\u6784\u6765\u63cf\u8ff0\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\uff0c\u6bcf\u4e00\u6761\u8def\u5f84\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\u3002
\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u95ee\u9898\u5305\u542b\u660e\u786e\u7684\u51b3\u7b56\u6982\u5ff5\uff0c\u5e76\u4e14\u89e3\u662f\u901a\u8fc7\u4e00\u7cfb\u5217\u51b3\u7b56\u4ea7\u751f\u7684\uff0c\u90a3\u4e48\u5b83\u5c31\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u901a\u5e38\u53ef\u4ee5\u4f7f\u7528\u56de\u6eaf\u6765\u89e3\u51b3\u3002
\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8fd8\u6709\u4e00\u4e9b\u5224\u65ad\u7684\u201c\u52a0\u5206\u9879\u201d\u3002
- \u95ee\u9898\u5305\u542b\u6700\u5927\uff08\u5c0f\uff09\u6216\u6700\u591a\uff08\u5c11\uff09\u7b49\u6700\u4f18\u5316\u63cf\u8ff0\u3002
- \u95ee\u9898\u7684\u72b6\u6001\u80fd\u591f\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u3001\u591a\u7ef4\u77e9\u9635\u6216\u6811\u6765\u8868\u793a\uff0c\u5e76\u4e14\u4e00\u4e2a\u72b6\u6001\u4e0e\u5176\u5468\u56f4\u7684\u72b6\u6001\u5b58\u5728\u9012\u63a8\u5173\u7cfb\u3002
\u76f8\u5e94\u5730\uff0c\u4e5f\u5b58\u5728\u4e00\u4e9b\u201c\u51cf\u5206\u9879\u201d\u3002
- \u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e0d\u662f\u627e\u51fa\u6700\u4f18\u89e3\u3002
- \u95ee\u9898\u63cf\u8ff0\u4e2d\u6709\u660e\u663e\u7684\u6392\u5217\u7ec4\u5408\u7684\u7279\u5f81\uff0c\u9700\u8981\u8fd4\u56de\u5177\u4f53\u7684\u591a\u4e2a\u65b9\u6848\u3002
\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u5e76\u5177\u6709\u8f83\u4e3a\u660e\u663e\u7684\u201c\u52a0\u5206\u9879\u201c\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5047\u8bbe\u5b83\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u5e76\u5728\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u9a8c\u8bc1\u5b83\u3002
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","title":"14.3.2 \u00a0 \u95ee\u9898\u6c42\u89e3\u6b65\u9aa4","text":"\u52a8\u6001\u89c4\u5212\u7684\u89e3\u9898\u6d41\u7a0b\u4f1a\u56e0\u95ee\u9898\u7684\u6027\u8d28\u548c\u96be\u5ea6\u800c\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u901a\u5e38\u9075\u5faa\u4ee5\u4e0b\u6b65\u9aa4\uff1a\u63cf\u8ff0\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u5efa\u7acb \\(dp\\) \u8868\uff0c\u63a8\u5bfc\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u7b49\u3002
\u4e3a\u4e86\u66f4\u5f62\u8c61\u5730\u5c55\u793a\u89e3\u9898\u6b65\u9aa4\uff0c\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\u201c\u6700\u5c0f\u8def\u5f84\u548c\u201d\u6765\u4e3e\u4f8b\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a \\(n \\times m\\) \u7684\u4e8c\u7ef4\u7f51\u683c
grid\uff0c\u7f51\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5305\u542b\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u8be5\u5355\u5143\u683c\u7684\u4ee3\u4ef7\u3002\u673a\u5668\u4eba\u4ee5\u5de6\u4e0a\u89d2\u5355\u5143\u683c\u4e3a\u8d77\u59cb\u70b9\uff0c\u6bcf\u6b21\u53ea\u80fd\u5411\u4e0b\u6216\u8005\u5411\u53f3\u79fb\u52a8\u4e00\u6b65\uff0c\u76f4\u81f3\u5230\u8fbe\u53f3\u4e0b\u89d2\u5355\u5143\u683c\u3002\u8bf7\u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230\u53f3\u4e0b\u89d2\u7684\u6700\u5c0f\u8def\u5f84\u548c\u3002\u56fe 14-10 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u7f51\u683c\u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e3a \\(13\\) \u3002
\u56fe 14-10 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u793a\u4f8b\u6570\u636e
\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868
\u672c\u9898\u7684\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u5c31\u662f\u4ece\u5f53\u524d\u683c\u5b50\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u3002\u8bbe\u5f53\u524d\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\([i, j]\\) \uff0c\u5219\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u540e\uff0c\u7d22\u5f15\u53d8\u4e3a \\([i+1, j]\\) \u6216 \\([i, j+1]\\) \u3002\u56e0\u6b64\uff0c\u72b6\u6001\u5e94\u5305\u542b\u884c\u7d22\u5f15\u548c\u5217\u7d22\u5f15\u4e24\u4e2a\u53d8\u91cf\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002
\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u4ece\u8d77\u59cb\u70b9 \\([0, 0]\\) \u8d70\u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\uff0c\u89e3\u8bb0\u4e3a \\(dp[i, j]\\) \u3002
\u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u56fe 14-11 \u6240\u793a\u7684\u4e8c\u7ef4 \\(dp\\) \u77e9\u9635\uff0c\u5176\u5c3a\u5bf8\u4e0e\u8f93\u5165\u7f51\u683c \\(grid\\) \u76f8\u540c\u3002
\u56fe 14-11 \u00a0 \u72b6\u6001\u5b9a\u4e49\u4e0e dp \u8868
Note
\u52a8\u6001\u89c4\u5212\u548c\u56de\u6eaf\u8fc7\u7a0b\u53ef\u4ee5\u63cf\u8ff0\u4e3a\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\uff0c\u800c\u72b6\u6001\u7531\u6240\u6709\u51b3\u7b56\u53d8\u91cf\u6784\u6210\u3002\u5b83\u5e94\u5f53\u5305\u542b\u63cf\u8ff0\u89e3\u9898\u8fdb\u5ea6\u7684\u6240\u6709\u53d8\u91cf\uff0c\u5176\u5305\u542b\u4e86\u8db3\u591f\u7684\u4fe1\u606f\uff0c\u80fd\u591f\u7528\u6765\u63a8\u5bfc\u51fa\u4e0b\u4e00\u4e2a\u72b6\u6001\u3002
\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5b9a\u4e49\u4e00\u4e2a \\(dp\\) \u8868\u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u72b6\u6001\u7684\u6bcf\u4e2a\u72ec\u7acb\u53d8\u91cf\u90fd\u662f \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\\(dp\\) \u8868\u662f\u72b6\u6001\u548c\u5b50\u95ee\u9898\u7684\u89e3\u4e4b\u95f4\u7684\u6620\u5c04\u3002
\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
\u5bf9\u4e8e\u72b6\u6001 \\([i, j]\\) \uff0c\u5b83\u53ea\u80fd\u4ece\u4e0a\u8fb9\u683c\u5b50 \\([i-1, j]\\) \u548c\u5de6\u8fb9\u683c\u5b50 \\([i, j-1]\\) \u8f6c\u79fb\u800c\u6765\u3002\u56e0\u6b64\u6700\u4f18\u5b50\u7ed3\u6784\u4e3a\uff1a\u5230\u8fbe \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u7531 \\([i, j-1]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e0e \\([i-1, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\u51b3\u5b9a\u3002
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u63a8\u51fa\u56fe 14-12 \u6240\u793a\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a
\\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\]\u56fe 14-12 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
Note
\u6839\u636e\u5b9a\u4e49\u597d\u7684 \\(dp\\) \u8868\uff0c\u601d\u8003\u539f\u95ee\u9898\u548c\u5b50\u95ee\u9898\u7684\u5173\u7cfb\uff0c\u627e\u51fa\u901a\u8fc7\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6765\u6784\u9020\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u7684\u65b9\u6cd5\uff0c\u5373\u6700\u4f18\u5b50\u7ed3\u6784\u3002
\u4e00\u65e6\u6211\u4eec\u627e\u5230\u4e86\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002
\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
\u5728\u672c\u9898\u4e2d\uff0c\u5904\u5728\u9996\u884c\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u5de6\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u5904\u5728\u9996\u5217\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u4e0a\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u56e0\u6b64\u9996\u884c \\(i = 0\\) \u548c\u9996\u5217 \\(j = 0\\) \u662f\u8fb9\u754c\u6761\u4ef6\u3002
\u5982\u56fe 14-13 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u662f\u7531\u5176\u5de6\u65b9\u683c\u5b50\u548c\u4e0a\u65b9\u683c\u5b50\u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u5faa\u73af\u6765\u904d\u5386\u77e9\u9635\uff0c\u5916\u5faa\u73af\u904d\u5386\u5404\u884c\uff0c\u5185\u5faa\u73af\u904d\u5386\u5404\u5217\u3002
\u56fe 14-13 \u00a0 \u8fb9\u754c\u6761\u4ef6\u4e0e\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
Note
\u8fb9\u754c\u6761\u4ef6\u5728\u52a8\u6001\u89c4\u5212\u4e2d\u7528\u4e8e\u521d\u59cb\u5316 \\(dp\\) \u8868\uff0c\u5728\u641c\u7d22\u4e2d\u7528\u4e8e\u526a\u679d\u3002
\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f\u7684\u6838\u5fc3\u662f\u8981\u4fdd\u8bc1\u5728\u8ba1\u7b97\u5f53\u524d\u95ee\u9898\u7684\u89e3\u65f6\uff0c\u6240\u6709\u5b83\u4f9d\u8d56\u7684\u66f4\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u90fd\u5df2\u7ecf\u88ab\u6b63\u786e\u5730\u8ba1\u7b97\u51fa\u6765\u3002
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u5df2\u7ecf\u53ef\u4ee5\u76f4\u63a5\u5199\u51fa\u52a8\u6001\u89c4\u5212\u4ee3\u7801\u3002\u7136\u800c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u601d\u60f3\uff0c\u56e0\u6b64\u6309\u7167\u201c\u66b4\u529b\u641c\u7d22 \\(\\rightarrow\\) \u8bb0\u5fc6\u5316\u641c\u7d22 \\(\\rightarrow\\) \u52a8\u6001\u89c4\u5212\u201d\u7684\u987a\u5e8f\u5b9e\u73b0\u66f4\u52a0\u7b26\u5408\u601d\u7ef4\u4e60\u60ef\u3002
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"\u4ece\u72b6\u6001 \\([i, j]\\) \u5f00\u59cb\u641c\u7d22\uff0c\u4e0d\u65ad\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u72b6\u6001 \\([i-1, j]\\) \u548c \\([i, j-1]\\) \uff0c\u9012\u5f52\u51fd\u6570\u5305\u62ec\u4ee5\u4e0b\u8981\u7d20\u3002
- \u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, j]\\) \u3002
- \u8fd4\u56de\u503c\uff1a\u4ece \\([0, 0]\\) \u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c \\(dp[i, j]\\) \u3002
- \u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53 \\(i = 0\\) \u4e14 \\(j = 0\\) \u65f6\uff0c\u8fd4\u56de\u4ee3\u4ef7 \\(grid[0, 0]\\) \u3002
- \u526a\u679d\uff1a\u5f53 \\(i < 0\\) \u65f6\u6216 \\(j < 0\\) \u65f6\u7d22\u5f15\u8d8a\u754c\uff0c\u6b64\u65f6\u8fd4\u56de\u4ee3\u4ef7 \\(+\\infty\\) \uff0c\u4ee3\u8868\u4e0d\u53ef\u884c\u3002
\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py
min_path_sum.cppdef min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -> int:\n \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 and j == 0:\n return grid[0][0]\n # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 or j < 0:\n return inf\n # \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n up = min_path_sum_dfs(grid, i - 1, j)\n left = min_path_sum_dfs(grid, i, j - 1)\n # \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return min(left, up) + grid[i][j]\n
min_path_sum.java/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(vector<vector<int>> &grid, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return INT_MAX;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFS(grid, i - 1, j);\n int left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n
min_path_sum.cs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int[][] grid, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Integer.MAX_VALUE;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFS(grid, i - 1, j);\n int left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return Math.min(left, up) + grid[i][j];\n}\n
min_path_sum.go/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return int.MaxValue;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = MinPathSumDFS(grid, i - 1, j);\n int left = MinPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return Math.Min(left, up) + grid[i][j];\n}\n
min_path_sum.swift/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 && j == 0 {\n return grid[0][0]\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return math.MaxInt\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n up := minPathSumDFS(grid, i-1, j)\n left := minPathSumDFS(grid, i, j-1)\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n
min_path_sum.js/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -> Int {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0, j == 0 {\n return grid[0][0]\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return .max\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return min(left, up) + grid[i][j]\n}\n
min_path_sum.ts/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(grid, i, j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i === 0 && j === 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Infinity;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n const up = minPathSumDFS(grid, i - 1, j);\n const left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return Math.min(left, up) + grid[i][j];\n}\n
min_path_sum.dart/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(\n grid: Array<Array<number>>,\n i: number,\n j: number\n): number {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i === 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Infinity;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n const up = minPathSumDFS(grid, i - 1, j);\n const left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return Math.min(left, up) + grid[i][j];\n}\n
min_path_sum.rs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(List<List<int>> grid, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n return BigInt.from(2).pow(31).toInt();\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFS(grid, i - 1, j);\n int left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return min(left, up) + grid[i][j];\n}\n
min_path_sum.c/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfn min_path_sum_dfs(grid: &Vec<Vec<i32>>, i: i32, j: i32) -> i32 {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 && j == 0 {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return i32::MAX;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n let up = min_path_sum_dfs(grid, i - 1, j);\n let left = min_path_sum_dfs(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n
min_path_sum.zig/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return INT_MAX;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFS(grid, i - 1, j);\n int left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\nfn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 and j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 or j < 0) {\n return std.math.maxInt(i32);\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n var up = minPathSumDFS(grid, i - 1, j);\n var left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n\u56fe 14-14 \u7ed9\u51fa\u4e86\u4ee5 \\(dp[2, 1]\\) \u4e3a\u6839\u8282\u70b9\u7684\u9012\u5f52\u6811\uff0c\u5176\u4e2d\u5305\u542b\u4e00\u4e9b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5176\u6570\u91cf\u4f1a\u968f\u7740\u7f51\u683c
grid\u7684\u5c3a\u5bf8\u53d8\u5927\u800c\u6025\u5267\u589e\u591a\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9020\u6210\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u539f\u56e0\u4e3a\uff1a\u5b58\u5728\u591a\u6761\u8def\u5f84\u53ef\u4ee5\u4ece\u5de6\u4e0a\u89d2\u5230\u8fbe\u67d0\u4e00\u5355\u5143\u683c\u3002
\u56fe 14-14 \u00a0 \u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811
\u6bcf\u4e2a\u72b6\u6001\u90fd\u6709\u5411\u4e0b\u548c\u5411\u53f3\u4e24\u79cd\u9009\u62e9\uff0c\u4ece\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\u603b\u5171\u9700\u8981 \\(m + n - 2\\) \u6b65\uff0c\u6240\u4ee5\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^{m + n})\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u8fd9\u79cd\u8ba1\u7b97\u65b9\u5f0f\u672a\u8003\u8651\u4e34\u8fd1\u7f51\u683c\u8fb9\u754c\u7684\u60c5\u51b5\uff0c\u5f53\u5230\u8fbe\u7f51\u7edc\u8fb9\u754c\u65f6\u53ea\u5269\u4e0b\u4e00\u79cd\u9009\u62e9\uff0c\u56e0\u6b64\u5b9e\u9645\u7684\u8def\u5f84\u6570\u91cf\u4f1a\u5c11\u4e00\u4e9b\u3002
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"\u6211\u4eec\u5f15\u5165\u4e00\u4e2a\u548c\u7f51\u683c
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.pygrid\u76f8\u540c\u5c3a\u5bf8\u7684\u8bb0\u5fc6\u5217\u8868mem\uff0c\u7528\u4e8e\u8bb0\u5f55\u5404\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u8fdb\u884c\u526a\u679d\uff1a
min_path_sum.cppdef min_path_sum_dfs_mem(\n grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -> int:\n \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 and j == 0:\n return grid[0][0]\n # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 or j < 0:\n return inf\n # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][j] != -1:\n return mem[i][j]\n # \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n # \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = min(left, up) + grid[i][j]\n return mem[i][j]\n
min_path_sum.java/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(vector<vector<int>> &grid, vector<vector<int>> &mem, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return INT_MAX;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFSMem(grid, mem, i - 1, j);\n int left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n return mem[i][j];\n}\n
min_path_sum.cs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Integer.MAX_VALUE;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFSMem(grid, mem, i - 1, j);\n int left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = Math.min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.go/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return int.MaxValue;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = Math.Min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.swift/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 && j == 0 {\n return grid[0][0]\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return math.MaxInt\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][j] != -1 {\n return mem[i][j]\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n up := minPathSumDFSMem(grid, mem, i-1, j)\n left := minPathSumDFSMem(grid, mem, i, j-1)\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n return mem[i][j]\n}\n
min_path_sum.js/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -> Int {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0, j == 0 {\n return grid[0][0]\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return .max\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][j] != -1 {\n return mem[i][j]\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n let up = minPathSumDFSMem(grid: grid, mem: &mem, i: i - 1, j: j)\n let left = minPathSumDFSMem(grid: grid, mem: &mem, i: i, j: j - 1)\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = min(left, up) + grid[i][j]\n return mem[i][j]\n}\n
min_path_sum.ts/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i === 0 && j === 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Infinity;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] !== -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n const up = minPathSumDFSMem(grid, mem, i - 1, j);\n const left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = Math.min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.dart/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(\n grid: Array<Array<number>>,\n mem: Array<Array<number>>,\n i: number,\n j: number\n): number {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i === 0 && j === 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Infinity;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n const up = minPathSumDFSMem(grid, mem, i - 1, j);\n const left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = Math.min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.rs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(List<List<int>> grid, List<List<int>> mem, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n return BigInt.from(2).pow(31).toInt();\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFSMem(grid, mem, i - 1, j);\n int left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.c/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn min_path_sum_dfs_mem(grid: &Vec<Vec<i32>>, mem: &mut Vec<Vec<i32>>, i: i32, j: i32) -> i32 {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 && j == 0 {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return i32::MAX;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i as usize][j as usize] != -1 {\n return mem[i as usize][j as usize];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n mem[i as usize][j as usize]\n}\n
min_path_sum.zig/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return INT_MAX;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFSMem(grid, mem, i - 1, j);\n int left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n return mem[i][j];\n}\n// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 and j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 or j < 0) {\n return std.math.maxInt(i32);\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {\n return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n var up = minPathSumDFSMem(grid, mem, i - 1, j);\n var left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n\u5982\u56fe 14-15 \u6240\u793a\uff0c\u5728\u5f15\u5165\u8bb0\u5fc6\u5316\u540e\uff0c\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u72b6\u6001\u603b\u6570\uff0c\u5373\u7f51\u683c\u5c3a\u5bf8 \\(O(nm)\\) \u3002
\u56fe 14-15 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py
min_path_sum.cppdef min_path_sum_dp(grid: list[list[int]]) -> int:\n \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n, m = len(grid), len(grid[0])\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * m for _ in range(n)]\n dp[0][0] = grid[0][0]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in range(1, m):\n dp[0][j] = dp[0][j - 1] + grid[0][j]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for i in range(1, n):\n dp[i][0] = dp[i - 1][0] + grid[i][0]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in range(1, n):\n for j in range(1, m):\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n return dp[n - 1][m - 1]\n
min_path_sum.java/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(vector<vector<int>> &grid) {\n int n = grid.size(), m = grid[0].size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n, vector<int>(m));\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.cs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int[][] grid) {\n int n = grid.length, m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n][m];\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.go/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDP(int[][] grid) {\n int n = grid.Length, m = grid[0].Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n, m];\n dp[0, 0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0, j] = dp[0, j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n }\n }\n return dp[n - 1, m - 1];\n}\n
min_path_sum.swift/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid [][]int) int {\n n, m := len(grid), len(grid[0])\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n)\n for i := 0; i < n; i++ {\n dp[i] = make([]int, m)\n }\n dp[0][0] = grid[0][0]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j := 1; j < m; j++ {\n dp[0][j] = dp[0][j-1] + grid[0][j]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for i := 1; i < n; i++ {\n dp[i][0] = dp[i-1][0] + grid[i][0]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i < n; i++ {\n for j := 1; j < m; j++ {\n dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n }\n }\n return dp[n-1][m-1]\n}\n
min_path_sum.js/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid: [[Int]]) -> Int {\n let n = grid.count\n let m = grid[0].count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n dp[0][0] = grid[0][0]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in stride(from: 1, to: m, by: 1) {\n dp[0][j] = dp[0][j - 1] + grid[0][j]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for i in stride(from: 1, to: n, by: 1) {\n dp[i][0] = dp[i - 1][0] + grid[i][0]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in stride(from: 1, to: n, by: 1) {\n for j in stride(from: 1, to: m, by: 1) {\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n }\n }\n return dp[n - 1][m - 1]\n}\n
min_path_sum.ts/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid) {\n const n = grid.length,\n m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n }, () =>\n Array.from({ length: m }, () => 0)\n );\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (let j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (let i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i < n; i++) {\n for (let j = 1; j < m; j++) {\n dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.dart/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid: Array<Array<number>>): number {\n const n = grid.length,\n m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n }, () =>\n Array.from({ length: m }, () => 0)\n );\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (let j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (let i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i < n; i++) {\n for (let j: number = 1; j < m; j++) {\n dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.rs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(List<List<int>> grid) {\n int n = grid.length, m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n, (i) => List.filled(m, 0));\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.c/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp(grid: &Vec<Vec<i32>>) -> i32 {\n let (n, m) = (grid.len(), grid[0].len());\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; m]; n];\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in 1..m {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for i in 1..n {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in 1..n {\n for j in 1..m {\n dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n dp[n - 1][m - 1]\n}\n
min_path_sum.zig/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc(n * sizeof(int *));\n for (int i = 0; i < n; i++) {\n dp[i] = calloc(m, sizeof(int));\n }\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n int res = dp[n - 1][m - 1];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i < n; i++) {\n free(dp[i]);\n }\n return res;\n}\n// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\nfn minPathSumDP(comptime grid: anytype) i32 {\n comptime var n = grid.len;\n comptime var m = grid[0].len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][m]i32{[_]i32{0} ** m} ** n;\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (1..m) |j| {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (1..n) |i| {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (1..n) |i| {\n for (1..m) |j| {\n dp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n\u56fe 14-16 \u5c55\u793a\u4e86\u6700\u5c0f\u8def\u5f84\u548c\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\uff0c\u5176\u904d\u5386\u4e86\u6574\u4e2a\u7f51\u683c\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002
\u6570\u7ec4
<1><2><3><4><5><6><7><8><9><10><11><12>dp\u5927\u5c0f\u4e3a \\(n \\times m\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002\u56fe 14-16 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u53ea\u4e0e\u5176\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u683c\u5b50\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u53ea\u7528\u4e00\u4e2a\u5355\u884c\u6570\u7ec4\u6765\u5b9e\u73b0 \\(dp\\) \u8868\u3002
\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.pydp\u53ea\u80fd\u8868\u793a\u4e00\u884c\u7684\u72b6\u6001\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u6cd5\u63d0\u524d\u521d\u59cb\u5316\u9996\u5217\u72b6\u6001\uff0c\u800c\u662f\u5728\u904d\u5386\u6bcf\u884c\u65f6\u66f4\u65b0\u5b83\uff1a
min_path_sum.cppdef min_path_sum_dp_comp(grid: list[list[int]]) -> int:\n \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n, m = len(grid), len(grid[0])\n # \u521d\u59cb\u5316 dp \u8868\n dp = [0] * m\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0]\n for j in range(1, m):\n dp[j] = dp[j - 1] + grid[0][j]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in range(1, n):\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in range(1, m):\n dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n return dp[m - 1]\n
min_path_sum.java/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(vector<vector<int>> &grid) {\n int n = grid.size(), m = grid[0].size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(m);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.cs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int[][] grid) {\n int n = grid.length, m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[m];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.go/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDPComp(int[][] grid) {\n int n = grid.Length, m = grid[0].Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[m];\n dp[0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.swift/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid [][]int) int {\n n, m := len(grid), len(grid[0])\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, m)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0]\n for j := 1; j < m; j++ {\n dp[j] = dp[j-1] + grid[0][j]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i < n; i++ {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j := 1; j < m; j++ {\n dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n }\n }\n return dp[m-1]\n}\n
min_path_sum.js/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid: [[Int]]) -> Int {\n let n = grid.count\n let m = grid[0].count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: 0, count: m)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0]\n for j in stride(from: 1, to: m, by: 1) {\n dp[j] = dp[j - 1] + grid[0][j]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in stride(from: 1, to: n, by: 1) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in stride(from: 1, to: m, by: 1) {\n dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n }\n }\n return dp[m - 1]\n}\n
min_path_sum.ts/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid) {\n const n = grid.length,\n m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = new Array(m);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (let j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (let i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (let j = 1; j < m; j++) {\n dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.dart/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid: Array<Array<number>>): number {\n const n = grid.length,\n m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = new Array(m);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (let j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (let i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (let j = 1; j < m; j++) {\n dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.rs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(List<List<int>> grid) {\n int n = grid.length, m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(m, 0);\n dp[0] = grid[0][0];\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.c/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp_comp(grid: &Vec<Vec<i32>>) -> i32 {\n let (n, m) = (grid.len(), grid[0].len());\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; m];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for j in 1..m {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in 1..n {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in 1..m {\n dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n dp[m - 1]\n}\n
min_path_sum.zig/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(m, sizeof(int));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n int res = dp[m - 1];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/edit_distance_problem/","title":"14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898","text":"// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minPathSumDPComp(comptime grid: anytype) i32 {\n comptime var n = grid.len;\n comptime var m = grid[0].len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** m;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (1..m) |j| {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (1..n) |i| {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n for (1..m) |j| {\n dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n\u7f16\u8f91\u8ddd\u79bb\uff0c\u4e5f\u79f0 Levenshtein \u8ddd\u79bb\uff0c\u6307\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u4e92\u76f8\u8f6c\u6362\u7684\u6700\u5c11\u4fee\u6539\u6b21\u6570\uff0c\u901a\u5e38\u7528\u4e8e\u5728\u4fe1\u606f\u68c0\u7d22\u548c\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\u5ea6\u91cf\u4e24\u4e2a\u5e8f\u5217\u7684\u76f8\u4f3c\u5ea6\u3002
Question
\u8f93\u5165\u4e24\u4e2a\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \uff0c\u8fd4\u56de\u5c06 \\(s\\) \u8f6c\u6362\u4e3a \\(t\\) \u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002
\u4f60\u53ef\u4ee5\u5728\u4e00\u4e2a\u5b57\u7b26\u4e32\u4e2d\u8fdb\u884c\u4e09\u79cd\u7f16\u8f91\u64cd\u4f5c\uff1a\u63d2\u5165\u4e00\u4e2a\u5b57\u7b26\u3001\u5220\u9664\u4e00\u4e2a\u5b57\u7b26\u3001\u5c06\u5b57\u7b26\u66ff\u6362\u4e3a\u4efb\u610f\u4e00\u4e2a\u5b57\u7b26\u3002
\u5982\u56fe 14-27 \u6240\u793a\uff0c\u5c06
kitten\u8f6c\u6362\u4e3asitting\u9700\u8981\u7f16\u8f91 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u4e0e 1 \u6b21\u6dfb\u52a0\u64cd\u4f5c\uff1b\u5c06hello\u8f6c\u6362\u4e3aalgo\u9700\u8981 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u548c 1 \u6b21\u5220\u9664\u64cd\u4f5c\u3002\u56fe 14-27 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u793a\u4f8b\u6570\u636e
\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u53ef\u4ee5\u5f88\u81ea\u7136\u5730\u7528\u51b3\u7b56\u6811\u6a21\u578b\u6765\u89e3\u91ca\u3002\u5b57\u7b26\u4e32\u5bf9\u5e94\u6811\u8282\u70b9\uff0c\u4e00\u8f6e\u51b3\u7b56\uff08\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\uff09\u5bf9\u5e94\u6811\u7684\u4e00\u6761\u8fb9\u3002
\u5982\u56fe 14-28 \u6240\u793a\uff0c\u5728\u4e0d\u9650\u5236\u64cd\u4f5c\u7684\u60c5\u51b5\u4e0b\uff0c\u6bcf\u4e2a\u8282\u70b9\u90fd\u53ef\u4ee5\u6d3e\u751f\u51fa\u8bb8\u591a\u6761\u8fb9\uff0c\u6bcf\u6761\u8fb9\u5bf9\u5e94\u4e00\u79cd\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u4ece
hello\u8f6c\u6362\u5230algo\u6709\u8bb8\u591a\u79cd\u53ef\u80fd\u7684\u8def\u5f84\u3002\u4ece\u51b3\u7b56\u6811\u7684\u89d2\u5ea6\u770b\uff0c\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u8282\u70b9
hello\u548c\u8282\u70b9algo\u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u3002\u56fe 14-28 \u00a0 \u57fa\u4e8e\u51b3\u7b56\u6811\u6a21\u578b\u8868\u793a\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898
"},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868
\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u662f\u5bf9\u5b57\u7b26\u4e32 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\u3002
\u6211\u4eec\u5e0c\u671b\u5728\u7f16\u8f91\u64cd\u4f5c\u7684\u8fc7\u7a0b\u4e2d\uff0c\u95ee\u9898\u7684\u89c4\u6a21\u9010\u6e10\u7f29\u5c0f\uff0c\u8fd9\u6837\u624d\u80fd\u6784\u5efa\u5b50\u95ee\u9898\u3002\u8bbe\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \u7684\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \uff0c\u6211\u4eec\u5148\u8003\u8651\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u3002
- \u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u76f8\u540c\uff0c\u6211\u4eec\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u76f4\u63a5\u8003\u8651 \\(s[n-2]\\) \u548c \\(t[m-2]\\) \u3002
- \u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u4e0d\u540c\uff0c\u6211\u4eec\u9700\u8981\u5bf9 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\uff08\u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\uff09\uff0c\u4f7f\u5f97\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26\u76f8\u540c\uff0c\u4ece\u800c\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u8003\u8651\u89c4\u6a21\u66f4\u5c0f\u7684\u95ee\u9898\u3002
\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u5728\u5b57\u7b26\u4e32 \\(s\\) \u4e2d\u8fdb\u884c\u7684\u6bcf\u4e00\u8f6e\u51b3\u7b56\uff08\u7f16\u8f91\u64cd\u4f5c\uff09\uff0c\u90fd\u4f1a\u4f7f\u5f97 \\(s\\) \u548c \\(t\\) \u4e2d\u5269\u4f59\u7684\u5f85\u5339\u914d\u5b57\u7b26\u53d1\u751f\u53d8\u5316\u3002\u56e0\u6b64\uff0c\u72b6\u6001\u4e3a\u5f53\u524d\u5728 \\(s\\) \u548c \\(t\\) \u4e2d\u8003\u8651\u7684\u7b2c \\(i\\) \u548c\u7b2c \\(j\\) \u4e2a\u5b57\u7b26\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002
\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\uff1a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002
\u81f3\u6b64\uff0c\u5f97\u5230\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((i+1) \\times (j+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002
\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
\u8003\u8651\u5b50\u95ee\u9898 \\(dp[i, j]\\) \uff0c\u5176\u5bf9\u5e94\u7684\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5c3e\u90e8\u5b57\u7b26\u4e3a \\(s[i-1]\\) \u548c \\(t[j-1]\\) \uff0c\u53ef\u6839\u636e\u4e0d\u540c\u7f16\u8f91\u64cd\u4f5c\u5206\u4e3a\u56fe 14-29 \u6240\u793a\u7684\u4e09\u79cd\u60c5\u51b5\u3002
- \u5728 \\(s[i-1]\\) \u4e4b\u540e\u6dfb\u52a0 \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i, j-1]\\) \u3002
- \u5220\u9664 \\(s[i-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j]\\) \u3002
- \u5c06 \\(s[i-1]\\) \u66ff\u6362\u4e3a \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j-1]\\) \u3002
\u56fe 14-29 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u72b6\u6001\u8f6c\u79fb
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u5f97\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\\(dp[i, j]\\) \u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(dp[i, j-1]\\)\u3001\\(dp[i-1, j]\\)\u3001\\(dp[i-1, j-1]\\) \u4e09\u8005\u4e2d\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u518d\u52a0\u4e0a\u672c\u6b21\u7684\u7f16\u8f91\u6b65\u6570 \\(1\\) \u3002\u5bf9\u5e94\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a
\\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\]\u8bf7\u6ce8\u610f\uff0c\u5f53 \\(s[i-1]\\) \u548c \\(t[j-1]\\) \u76f8\u540c\u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a
\\[ dp[i, j] = dp[i-1, j-1] \\]\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
\u5f53\u4e24\u5b57\u7b26\u4e32\u90fd\u4e3a\u7a7a\u65f6\uff0c\u7f16\u8f91\u6b65\u6570\u4e3a \\(0\\) \uff0c\u5373 \\(dp[0, 0] = 0\\) \u3002\u5f53 \\(s\\) \u4e3a\u7a7a\u4f46 \\(t\\) \u4e0d\u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(t\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u884c \\(dp[0, j] = j\\) \u3002\u5f53 \\(s\\) \u4e0d\u4e3a\u7a7a\u4f46 \\(t\\) \u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(s\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u5217 \\(dp[i, 0] = i\\) \u3002
\u89c2\u5bdf\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u89e3 \\(dp[i, j]\\) \u4f9d\u8d56\u5de6\u65b9\u3001\u4e0a\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u89e3\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002
"},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig edit_distance.py
edit_distance.cppdef edit_distance_dp(s: str, t: str) -> int:\n \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n, m = len(s), len(t)\n dp = [[0] * (m + 1) for _ in range(n + 1)]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for i in range(1, n + 1):\n dp[i][0] = i\n for j in range(1, m + 1):\n dp[0][j] = j\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in range(1, n + 1):\n for j in range(1, m + 1):\n if s[i - 1] == t[j - 1]:\n # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1]\n else:\n # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n return dp[n][m]\n
edit_distance.java/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(string s, string t) {\n int n = s.length(), m = t.length();\n vector<vector<int>> dp(n + 1, vector<int>(m + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.cs/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n int n = s.length(), m = t.length();\n int[][] dp = new int[n + 1][m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s.charAt(i - 1) == t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.go/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDP(string s, string t) {\n int n = s.Length, m = t.Length;\n int[,] dp = new int[n + 1, m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i, 0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0, j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i, j] = dp[i - 1, j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n }\n }\n }\n return dp[n, m];\n}\n
edit_distance.swift/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s string, t string) int {\n n := len(s)\n m := len(t)\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, m+1)\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for i := 1; i <= n; i++ {\n dp[i][0] = i\n }\n for j := 1; j <= m; j++ {\n dp[0][j] = j\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i <= n; i++ {\n for j := 1; j <= m; j++ {\n if s[i-1] == t[j-1] {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i-1][j-1]\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n }\n }\n }\n return dp[n][m]\n}\n
edit_distance.js/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s: String, t: String) -> Int {\n let n = s.utf8CString.count\n let m = t.utf8CString.count\n var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for i in stride(from: 1, through: n, by: 1) {\n dp[i][0] = i\n }\n for j in stride(from: 1, through: m, by: 1) {\n dp[0][j] = j\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in stride(from: 1, through: n, by: 1) {\n for j in stride(from: 1, through: m, by: 1) {\n if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1]\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n }\n }\n }\n return dp[n][m]\n}\n
edit_distance.ts/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s, t) {\n const n = s.length,\n m = t.length;\n const dp = Array.from({ length: n + 1 }, () => new Array(m + 1).fill(0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (let i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (let j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i <= n; i++) {\n for (let j = 1; j <= m; j++) {\n if (s.charAt(i - 1) === t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] =\n Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.dart/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s: string, t: string): number {\n const n = s.length,\n m = t.length;\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: m + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (let i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (let j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i <= n; i++) {\n for (let j = 1; j <= m; j++) {\n if (s.charAt(i - 1) === t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] =\n Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.rs/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n int n = s.length, m = t.length;\n List<List<int>> dp = List.generate(n + 1, (_) => List.filled(m + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.c/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp(s: &str, t: &str) -> i32 {\n let (n, m) = (s.len(), t.len());\n let mut dp = vec![vec![0; m + 1]; n + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for i in 1..= n {\n dp[i][0] = i as i32;\n }\n for j in 1..m {\n dp[0][j] = j as i32;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in 1..=n {\n for j in 1..=m {\n if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n dp[n][m]\n}\n
edit_distance.zig/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(char *s, char *t, int n, int m) {\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(m + 1, sizeof(int));\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n int res = dp[n][m];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n return res;\n}\n// \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\nfn editDistanceDP(comptime s: []const u8, comptime t: []const u8) i32 {\n comptime var n = s.len;\n comptime var m = t.len;\n var dp = [_][m + 1]i32{[_]i32{0} ** (m + 1)} ** (n + 1);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (1..n + 1) |i| {\n dp[i][0] = @intCast(i);\n }\n for (1..m + 1) |j| {\n dp[0][j] = @intCast(j);\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (1..n + 1) |i| {\n for (1..m + 1) |j| {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = @min(@min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n\u5982\u56fe 14-30 \u6240\u793a\uff0c\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\u4e0e\u80cc\u5305\u95ee\u9898\u975e\u5e38\u7c7b\u4f3c\uff0c\u90fd\u53ef\u4ee5\u770b\u4f5c\u586b\u5199\u4e00\u4e2a\u4e8c\u7ef4\u7f51\u683c\u7684\u8fc7\u7a0b\u3002
<1><2><3><4><5><6><7><8><9><10><11><12><13><14><15>\u56fe 14-30 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
"},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"\u7531\u4e8e \\(dp[i,j]\\) \u662f\u7531\u4e0a\u65b9 \\(dp[i-1, j]\\)\u3001\u5de6\u65b9 \\(dp[i, j-1]\\)\u3001\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \u8f6c\u79fb\u800c\u6765\u7684\uff0c\u800c\u6b63\u5e8f\u904d\u5386\u4f1a\u4e22\u5931\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \uff0c\u5012\u5e8f\u904d\u5386\u65e0\u6cd5\u63d0\u524d\u6784\u5efa \\(dp[i, j-1]\\) \uff0c\u56e0\u6b64\u4e24\u79cd\u904d\u5386\u987a\u5e8f\u90fd\u4e0d\u53ef\u53d6\u3002
\u4e3a\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf
PythonC++JavaC#GoSwiftJSTSDartRustCZig edit_distance.pyleftup\u6765\u6682\u5b58\u5de6\u4e0a\u65b9\u7684\u89e3 \\(dp[i-1, j-1]\\) \uff0c\u4ece\u800c\u53ea\u9700\u8003\u8651\u5de6\u65b9\u548c\u4e0a\u65b9\u7684\u89e3\u3002\u6b64\u65f6\u7684\u60c5\u51b5\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u53ef\u4f7f\u7528\u6b63\u5e8f\u904d\u5386\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
edit_distance.cppdef edit_distance_dp_comp(s: str, t: str) -> int:\n \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n, m = len(s), len(t)\n dp = [0] * (m + 1)\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in range(1, m + 1):\n dp[j] = j\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in range(1, n + 1):\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n leftup = dp[0] # \u6682\u5b58 dp[i-1, j-1]\n dp[0] += 1\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in range(1, m + 1):\n temp = dp[j]\n if s[i - 1] == t[j - 1]:\n # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup\n else:\n # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n leftup = temp # \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n return dp[m]\n
edit_distance.java/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(string s, string t) {\n int n = s.length(), m = t.length();\n vector<int> dp(m + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.cs/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n int n = s.length(), m = t.length();\n int[] dp = new int[m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s.charAt(i - 1) == t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.go/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDPComp(string s, string t) {\n int n = s.Length, m = t.Length;\n int[] dp = new int[m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.swift/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s string, t string) int {\n n := len(s)\n m := len(t)\n dp := make([]int, m+1)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j := 1; j <= m; j++ {\n dp[j] = j\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i := 1; i <= n; i++ {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n leftUp := dp[0] // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j := 1; j <= m; j++ {\n temp := dp[j]\n if s[i-1] == t[j-1] {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftUp\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n }\n leftUp = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m]\n}\n
edit_distance.js/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s: String, t: String) -> Int {\n let n = s.utf8CString.count\n let m = t.utf8CString.count\n var dp = Array(repeating: 0, count: m + 1)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in stride(from: 1, through: m, by: 1) {\n dp[j] = j\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in stride(from: 1, through: n, by: 1) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n var leftup = dp[0] // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in stride(from: 1, through: m, by: 1) {\n let temp = dp[j]\n if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n }\n leftup = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m]\n}\n
edit_distance.ts/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s, t) {\n const n = s.length,\n m = t.length;\n const dp = new Array(m + 1).fill(0);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (let j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (let i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (let j = 1; j <= m; j++) {\n const temp = dp[j];\n if (s.charAt(i - 1) === t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.dart/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s: string, t: string): number {\n const n = s.length,\n m = t.length;\n const dp = new Array(m + 1).fill(0);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (let j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (let i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (let j = 1; j <= m; j++) {\n const temp = dp[j];\n if (s.charAt(i - 1) === t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.rs/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n int n = s.length, m = t.length;\n List<int> dp = List.filled(m + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.c/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp_comp(s: &str, t: &str) -> i32 {\n let (n, m) = (s.len(), t.len());\n let mut dp = vec![0; m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in 1..m {\n dp[j] = j as i32;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in 1..=n {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n let mut leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i as i32;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in 1..=m {\n let temp = dp[j];\n if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n dp[m]\n}\n
edit_distance.zig/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n int *dp = calloc(m + 1, sizeof(int));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n int res = dp[m];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212","text":"// \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn editDistanceDPComp(comptime s: []const u8, comptime t: []const u8) i32 {\n comptime var n = s.len;\n comptime var m = t.len;\n var dp = [_]i32{0} ** (m + 1);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (1..m + 1) |j| {\n dp[j] = @intCast(j);\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (1..n + 1) |i| {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n var leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = @intCast(i);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (1..m + 1) |j| {\n var temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = @min(@min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n\u300c\u52a8\u6001\u89c4\u5212 dynamic programming\u300d\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u7b97\u6cd5\u8303\u5f0f\uff0c\u5b83\u5c06\u4e00\u4e2a\u95ee\u9898\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\uff0c\u4ece\u800c\u5927\u5e45\u63d0\u5347\u65f6\u95f4\u6548\u7387\u3002
\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u4ece\u4e00\u4e2a\u7ecf\u5178\u4f8b\u9898\u5165\u624b\uff0c\u5148\u7ed9\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c2\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u518d\u9010\u6b65\u5bfc\u51fa\u66f4\u9ad8\u6548\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002
\u722c\u697c\u68af
\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f
\u5982\u56fe 14-1 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e00\u4e2a \\(3\\) \u9636\u697c\u68af\uff0c\u5171\u6709 \\(3\\) \u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002
\u56fe 14-1 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf
\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u56de\u6eaf\u6765\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c06\u722c\u697c\u68af\u60f3\u8c61\u4e3a\u4e00\u4e2a\u591a\u8f6e\u9009\u62e9\u7684\u8fc7\u7a0b\uff1a\u4ece\u5730\u9762\u51fa\u53d1\uff0c\u6bcf\u8f6e\u9009\u62e9\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u6bcf\u5f53\u5230\u8fbe\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u65b9\u6848\u6570\u91cf\u52a0 \\(1\\) \uff0c\u5f53\u8d8a\u8fc7\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u5176\u526a\u679d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_backtrack.py
climbing_stairs_backtrack.cppdef backtrack(choices: list[int], state: int, n: int, res: list[int]) -> int:\n \"\"\"\u56de\u6eaf\"\"\"\n # \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if state == n:\n res[0] += 1\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices:\n # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if state + choice > n:\n continue\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res)\n # \u56de\u9000\n\ndef climbing_stairs_backtrack(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u56de\u6eaf\"\"\"\n choices = [1, 2] # \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n state = 0 # \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n res = [0] # \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res)\n return res[0]\n
climbing_stairs_backtrack.java/* \u56de\u6eaf */\nvoid backtrack(vector<int> &choices, int state, int n, vector<int> &res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n)\n res[0]++;\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (auto &choice : choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n)\n continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n vector<int> choices = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n vector<int> res = {0}; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res[0];\n}\n
climbing_stairs_backtrack.cs/* \u56de\u6eaf */\nvoid backtrack(List<Integer> choices, int state, int n, List<Integer> res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n)\n res.set(0, res.get(0) + 1);\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (Integer choice : choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n)\n continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n List<Integer> choices = Arrays.asList(1, 2); // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n List<Integer> res = new ArrayList<>();\n res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res.get(0);\n}\n
climbing_stairs_backtrack.go/* \u56de\u6eaf */\nvoid Backtrack(List<int> choices, int state, int n, List<int> res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n)\n res[0]++;\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n foreach (int choice in choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n)\n continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n Backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint ClimbingStairsBacktrack(int n) {\n List<int> choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n List<int> res = [0]; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n Backtrack(choices, state, n, res);\n return res[0];\n}\n
climbing_stairs_backtrack.swift/* \u56de\u6eaf */\nfunc backtrack(choices []int, state, n int, res []int) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if state == n {\n res[0] = res[0] + 1\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for _, choice := range choices {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if state+choice > n {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state+choice, n, res)\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n int) int {\n // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n choices := []int{1, 2}\n // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n state := 0\n res := make([]int, 1)\n // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n res[0] = 0\n backtrack(choices, state, n, res)\n return res[0]\n}\n
climbing_stairs_backtrack.js/* \u56de\u6eaf */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if state == n {\n res[0] += 1\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if state + choice > n {\n continue\n }\n backtrack(choices: choices, state: state + choice, n: n, res: &res)\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n: Int) -> Int {\n let choices = [1, 2] // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n let state = 0 // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n var res: [Int] = []\n res.append(0) // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices: choices, state: state, n: n, res: &res)\n return res[0]\n}\n
climbing_stairs_backtrack.ts/* \u56de\u6eaf */\nfunction backtrack(choices, state, n, res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state === n) res.set(0, res.get(0) + 1);\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (const choice of choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n) continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n) {\n const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n const res = new Map();\n res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res.get(0);\n}\n
climbing_stairs_backtrack.dart/* \u56de\u6eaf */\nfunction backtrack(\n choices: number[],\n state: number,\n n: number,\n res: Map<0, any>\n): void {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state === n) res.set(0, res.get(0) + 1);\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (const choice of choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n) continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n: number): number {\n const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n const res = new Map();\n res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res.get(0);\n}\n
climbing_stairs_backtrack.rs/* \u56de\u6eaf */\nvoid backtrack(List<int> choices, int state, int n, List<int> res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n) {\n res[0]++;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int choice in choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n) continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n List<int> choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n List<int> res = [];\n res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res[0];\n}\n
climbing_stairs_backtrack.c/* \u56de\u6eaf */\nfn backtrack(choices: &[i32], state: i32, n: i32, res: &mut [i32]) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if state == n { res[0] = res[0] + 1; }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for &choice in choices {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if state + choice > n { continue; }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfn climbing_stairs_backtrack(n: usize) -> i32 {\n let choices = vec![ 1, 2 ]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n let state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n let mut res = Vec::new();\n res.push(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(&choices, state, n as i32, &mut res);\n res[0]\n}\n
climbing_stairs_backtrack.zig/* \u56de\u6eaf */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n)\n res[0]++;\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < len; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n)\n continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res, len);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n int choices[2] = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n int *res = (int *)malloc(sizeof(int));\n *res = 0; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n int len = sizeof(choices) / sizeof(int);\n backtrack(choices, state, n, res, len);\n int result = *res;\n free(res);\n return result;\n}\n
"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411","title":"14.1.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"// \u56de\u6eaf\nfn backtrack(choices: []i32, state: i32, n: i32, res: std.ArrayList(i32)) void {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n) {\n res.items[0] = res.items[0] + 1;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (choices) |choice| {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n// \u722c\u697c\u68af\uff1a\u56de\u6eaf\nfn climbingStairsBacktrack(n: usize) !i32 {\n var choices = [_]i32{ 1, 2 }; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n var state: i32 = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n var res = std.ArrayList(i32).init(std.heap.page_allocator);\n defer res.deinit();\n try res.append(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(&choices, state, @intCast(n), res);\n return res.items[0];\n}\n\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u5e76\u4e0d\u663e\u5f0f\u5730\u5bf9\u95ee\u9898\u8fdb\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c06\u6c42\u89e3\u95ee\u9898\u770b\u4f5c\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\uff0c\u901a\u8fc7\u8bd5\u63a2\u548c\u526a\u679d\uff0c\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002
\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u4ece\u95ee\u9898\u5206\u89e3\u7684\u89d2\u5ea6\u5206\u6790\u8fd9\u9053\u9898\u3002\u8bbe\u722c\u5230\u7b2c \\(i\\) \u9636\u5171\u6709 \\(dp[i]\\) \u79cd\u65b9\u6848\uff0c\u90a3\u4e48 \\(dp[i]\\) \u5c31\u662f\u539f\u95ee\u9898\uff0c\u5176\u5b50\u95ee\u9898\u5305\u62ec\uff1a
\\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\]\u7531\u4e8e\u6bcf\u8f6e\u53ea\u80fd\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u56e0\u6b64\u5f53\u6211\u4eec\u7ad9\u5728\u7b2c \\(i\\) \u9636\u697c\u68af\u4e0a\u65f6\uff0c\u4e0a\u4e00\u8f6e\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u4e0a\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ea\u80fd\u4ece\u7b2c \\(i -1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u8fc8\u5411\u7b2c \\(i\\) \u9636\u3002
\u7531\u6b64\u4fbf\u53ef\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u63a8\u8bba\uff1a\u722c\u5230\u7b2c \\(i - 1\\) \u9636\u7684\u65b9\u6848\u6570\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u9636\u7684\u65b9\u6848\u6570\u5c31\u7b49\u4e8e\u722c\u5230\u7b2c \\(i\\) \u9636\u7684\u65b9\u6848\u6570\u3002\u516c\u5f0f\u5982\u4e0b\uff1a
\\[ dp[i] = dp[i-1] + dp[i-2] \\]\u8fd9\u610f\u5473\u7740\u5728\u722c\u697c\u68af\u95ee\u9898\u4e2d\uff0c\u5404\u4e2a\u5b50\u95ee\u9898\u4e4b\u95f4\u5b58\u5728\u9012\u63a8\u5173\u7cfb\uff0c\u539f\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u7684\u89e3\u6784\u5efa\u5f97\u6765\u3002\u56fe 14-2 \u5c55\u793a\u4e86\u8be5\u9012\u63a8\u5173\u7cfb\u3002
\u56fe 14-2 \u00a0 \u65b9\u6848\u6570\u91cf\u9012\u63a8\u5173\u7cfb
\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9012\u63a8\u516c\u5f0f\u5f97\u5230\u66b4\u529b\u641c\u7d22\u89e3\u6cd5\u3002\u4ee5 \\(dp[n]\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u9012\u5f52\u5730\u5c06\u4e00\u4e2a\u8f83\u5927\u95ee\u9898\u62c6\u89e3\u4e3a\u4e24\u4e2a\u8f83\u5c0f\u95ee\u9898\u7684\u548c\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898 \\(dp[1]\\) \u548c \\(dp[2]\\) \u65f6\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373 \\(dp[1] = 1\\)\u3001\\(dp[2] = 2\\) \uff0c\u8868\u793a\u722c\u5230\u7b2c \\(1\\)\u3001\\(2\\) \u9636\u5206\u522b\u6709 \\(1\\)\u3001\\(2\\) \u79cd\u65b9\u6848\u3002
\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5b83\u548c\u6807\u51c6\u56de\u6eaf\u4ee3\u7801\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f46\u66f4\u52a0\u7b80\u6d01\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dfs.py
climbing_stairs_dfs.cppdef dfs(i: int) -> int:\n \"\"\"\u641c\u7d22\"\"\"\n # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 or i == 2:\n return i\n # dp[i] = dp[i-1] + dp[i-2]\n count = dfs(i - 1) + dfs(i - 2)\n return count\n\ndef climbing_stairs_dfs(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u641c\u7d22\"\"\"\n return dfs(n)\n
climbing_stairs_dfs.java/* \u641c\u7d22 */\nint dfs(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n return dfs(n);\n}\n
climbing_stairs_dfs.cs/* \u641c\u7d22 */\nint dfs(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n return dfs(n);\n}\n
climbing_stairs_dfs.go/* \u641c\u7d22 */\nint DFS(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = DFS(i - 1) + DFS(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint ClimbingStairsDFS(int n) {\n return DFS(n);\n}\n
climbing_stairs_dfs.swift/* \u641c\u7d22 */\nfunc dfs(i int) int {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 {\n return i\n }\n // dp[i] = dp[i-1] + dp[i-2]\n count := dfs(i-1) + dfs(i-2)\n return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n int) int {\n return dfs(n)\n}\n
climbing_stairs_dfs.js/* \u641c\u7d22 */\nfunc dfs(i: Int) -> Int {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 {\n return i\n }\n // dp[i] = dp[i-1] + dp[i-2]\n let count = dfs(i: i - 1) + dfs(i: i - 2)\n return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n: Int) -> Int {\n dfs(i: n)\n}\n
climbing_stairs_dfs.ts/* \u641c\u7d22 */\nfunction dfs(i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i === 1 || i === 2) return i;\n // dp[i] = dp[i-1] + dp[i-2]\n const count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n) {\n return dfs(n);\n}\n
climbing_stairs_dfs.dart/* \u641c\u7d22 */\nfunction dfs(i: number): number {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i === 1 || i === 2) return i;\n // dp[i] = dp[i-1] + dp[i-2]\n const count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n: number): number {\n return dfs(n);\n}\n
climbing_stairs_dfs.rs/* \u641c\u7d22 */\nint dfs(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2) return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n return dfs(n);\n}\n
climbing_stairs_dfs.c/* \u641c\u7d22 */\nfn dfs(i: usize) -> i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 { return i as i32; }\n // dp[i] = dp[i-1] + dp[i-2]\n let count = dfs(i - 1) + dfs(i - 2);\n count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfn climbing_stairs_dfs(n: usize) -> i32 {\n dfs(n) \n}\n
climbing_stairs_dfs.zig/* \u641c\u7d22 */\nint dfs(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n return dfs(n);\n}\n// \u641c\u7d22\nfn dfs(i: usize) i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 or i == 2) {\n return @intCast(i);\n }\n // dp[i] = dp[i-1] + dp[i-2]\n var count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u641c\u7d22\nfn climbingStairsDFS(comptime n: usize) i32 {\n return dfs(n);\n}\n\u56fe 14-3 \u5c55\u793a\u4e86\u66b4\u529b\u641c\u7d22\u5f62\u6210\u7684\u9012\u5f52\u6811\u3002\u5bf9\u4e8e\u95ee\u9898 \\(dp[n]\\) \uff0c\u5176\u9012\u5f52\u6811\u7684\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002\u6307\u6570\u9636\u5c5e\u4e8e\u7206\u70b8\u5f0f\u589e\u957f\uff0c\u5982\u679c\u6211\u4eec\u8f93\u5165\u4e00\u4e2a\u6bd4\u8f83\u5927\u7684 \\(n\\) \uff0c\u5219\u4f1a\u9677\u5165\u6f2b\u957f\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002
\u56fe 14-3 \u00a0 \u722c\u697c\u68af\u5bf9\u5e94\u9012\u5f52\u6811
\u89c2\u5bdf\u56fe 14-3 \uff0c\u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u201c\u91cd\u53e0\u5b50\u95ee\u9898\u201d\u5bfc\u81f4\u7684\u3002\u4f8b\u5982 \\(dp[9]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\\(dp[8]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u4e24\u8005\u90fd\u5305\u542b\u5b50\u95ee\u9898 \\(dp[7]\\) \u3002
\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5b50\u95ee\u9898\u4e2d\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5b50\u5b50\u5b59\u5b59\u65e0\u7a77\u5c3d\u4e5f\u3002\u7edd\u5927\u90e8\u5206\u8ba1\u7b97\u8d44\u6e90\u90fd\u6d6a\u8d39\u5728\u8fd9\u4e9b\u91cd\u53e0\u7684\u95ee\u9898\u4e0a\u3002
"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412","title":"14.1.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"\u4e3a\u4e86\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u6240\u6709\u7684\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u6570\u7ec4
mem\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u526a\u679d\u3002- \u5f53\u9996\u6b21\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u5c06\u5176\u8bb0\u5f55\u81f3
mem[i]\uff0c\u4ee5\u4fbf\u4e4b\u540e\u4f7f\u7528\u3002 - \u5f53\u518d\u6b21\u9700\u8981\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u4fbf\u53ef\u76f4\u63a5\u4ece
mem[i]\u4e2d\u83b7\u53d6\u7ed3\u679c\uff0c\u4ece\u800c\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u8be5\u5b50\u95ee\u9898\u3002
\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dfs_mem.py
climbing_stairs_dfs_mem.cppdef dfs(i: int, mem: list[int]) -> int:\n \"\"\"\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 or i == 2:\n return i\n # \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if mem[i] != -1:\n return mem[i]\n # dp[i] = dp[i-1] + dp[i-2]\n count = dfs(i - 1, mem) + dfs(i - 2, mem)\n # \u8bb0\u5f55 dp[i]\n mem[i] = count\n return count\n\ndef climbing_stairs_dfs_mem(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n # mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n mem = [-1] * (n + 1)\n return dfs(n, mem)\n
climbing_stairs_dfs_mem.java/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, vector<int> &mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1)\n return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n vector<int> mem(n + 1, -1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.cs/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1)\n return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n int[] mem = new int[n + 1];\n Arrays.fill(mem, -1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.go/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint DFS(int i, int[] mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1)\n return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint ClimbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n int[] mem = new int[n + 1];\n Array.Fill(mem, -1);\n return DFS(n, mem);\n}\n
climbing_stairs_dfs_mem.swift/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfsMem(i int, mem []int) int {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 {\n return i\n }\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if mem[i] != -1 {\n return mem[i]\n }\n // dp[i] = dp[i-1] + dp[i-2]\n count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n // \u8bb0\u5f55 dp[i]\n mem[i] = count\n return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n int) int {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n mem := make([]int, n+1)\n for i := range mem {\n mem[i] = -1\n }\n return dfsMem(n, mem)\n}\n
climbing_stairs_dfs_mem.js/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfs(i: Int, mem: inout [Int]) -> Int {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 {\n return i\n }\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if mem[i] != -1 {\n return mem[i]\n }\n // dp[i] = dp[i-1] + dp[i-2]\n let count = dfs(i: i - 1, mem: &mem) + dfs(i: i - 2, mem: &mem)\n // \u8bb0\u5f55 dp[i]\n mem[i] = count\n return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n: Int) -> Int {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n var mem = Array(repeating: -1, count: n + 1)\n return dfs(i: n, mem: &mem)\n}\n
climbing_stairs_dfs_mem.ts/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i, mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i === 1 || i === 2) return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1) return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n const mem = new Array(n + 1).fill(-1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.dart/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i: number, mem: number[]): number {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i === 1 || i === 2) return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1) return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n: number): number {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n const mem = new Array(n + 1).fill(-1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.rs/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, List<int> mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2) return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1) return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n List<int> mem = List.filled(n + 1, -1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.c/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn dfs(i: usize, mem: &mut [i32]) -> i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 { return i as i32; }\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if mem[i] != -1 { return mem[i]; }\n // dp[i] = dp[i-1] + dp[i-2]\n let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn climbing_stairs_dfs_mem(n: usize) -> i32 {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n let mut mem = vec![-1; n + 1];\n dfs(n, &mut mem)\n}\n
climbing_stairs_dfs_mem.zig/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int *mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1)\n return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n int *mem = (int *)malloc((n + 1) * sizeof(int));\n for (int i = 0; i <= n; i++) {\n mem[i] = -1;\n }\n int result = dfs(n, mem);\n free(mem);\n return result;\n}\n// \u8bb0\u5fc6\u5316\u641c\u7d22\nfn dfs(i: usize, mem: []i32) i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 or i == 2) {\n return @intCast(i);\n }\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1) {\n return mem[i];\n }\n // dp[i] = dp[i-1] + dp[i-2]\n var count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn climbingStairsDFSMem(comptime n: usize) i32 {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n var mem = [_]i32{ -1 } ** (n + 1);\n return dfs(n, &mem);\n}\n\u89c2\u5bdf\u56fe 14-4 \uff0c\u7ecf\u8fc7\u8bb0\u5fc6\u5316\u5904\u7406\u540e\uff0c\u6240\u6709\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u8fd9\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u98de\u8dc3\u3002
\u56fe 14-4 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u5bf9\u5e94\u9012\u5f52\u6811
"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413","title":"14.1.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"\u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u201c\u4ece\u9876\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u4eec\u4ece\u539f\u95ee\u9898\uff08\u6839\u8282\u70b9\uff09\u5f00\u59cb\uff0c\u9012\u5f52\u5730\u5c06\u8f83\u5927\u5b50\u95ee\u9898\u5206\u89e3\u4e3a\u8f83\u5c0f\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u53f6\u8282\u70b9\uff09\u3002\u4e4b\u540e\uff0c\u901a\u8fc7\u56de\u6eaf\u9010\u5c42\u6536\u96c6\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002
\u4e0e\u4e4b\u76f8\u53cd\uff0c\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u201c\u4ece\u5e95\u81f3\u9876\u201d\u7684\u65b9\u6cd5\uff1a\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u5f00\u59cb\uff0c\u8fed\u4ee3\u5730\u6784\u5efa\u66f4\u5927\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002
\u7531\u4e8e\u52a8\u6001\u89c4\u5212\u4e0d\u5305\u542b\u56de\u6eaf\u8fc7\u7a0b\uff0c\u56e0\u6b64\u53ea\u9700\u4f7f\u7528\u5faa\u73af\u8fed\u4ee3\u5b9e\u73b0\uff0c\u65e0\u987b\u4f7f\u7528\u9012\u5f52\u3002\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u4e00\u4e2a\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dp.pydp\u6765\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5b83\u8d77\u5230\u4e86\u4e0e\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u6570\u7ec4mem\u76f8\u540c\u7684\u8bb0\u5f55\u4f5c\u7528\uff1a
climbing_stairs_dp.cppdef climbing_stairs_dp(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n if n == 1 or n == 2:\n return n\n # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp = [0] * (n + 1)\n # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1], dp[2] = 1, 2\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in range(3, n + 1):\n dp[i] = dp[i - 1] + dp[i - 2]\n return dp[n]\n
climbing_stairs_dp.java/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n if (n == 1 || n == 2)\n return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n vector<int> dp(n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.cs/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n if (n == 1 || n == 2)\n return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[] dp = new int[n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.go/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDP(int n) {\n if (n == 1 || n == 2)\n return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[] dp = new int[n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.swift/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n int) int {\n if n == 1 || n == 2 {\n return n\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp := make([]int, n+1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1\n dp[2] = 2\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n dp[i] = dp[i-1] + dp[i-2]\n }\n return dp[n]\n}\n
climbing_stairs_dp.js/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n: Int) -> Int {\n if n == 1 || n == 2 {\n return n\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = Array(repeating: 0, count: n + 1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1\n dp[2] = 2\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in stride(from: 3, through: n, by: 1) {\n dp[i] = dp[i - 1] + dp[i - 2]\n }\n return dp[n]\n}\n
climbing_stairs_dp.ts/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n) {\n if (n === 1 || n === 2) return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = new Array(n + 1).fill(-1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.dart/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n: number): number {\n if (n === 1 || n === 2) return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = new Array(n + 1).fill(-1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.rs/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n if (n == 1 || n == 2) return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n List<int> dp = List.filled(n + 1, 0);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.c/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp(n: usize) -> i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if n == 1 || n == 2 { return n as i32; }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n let mut dp = vec![-1; n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in 3..=n {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n dp[n]\n}\n
climbing_stairs_dp.zig/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n if (n == 1 || n == 2)\n return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int *dp = (int *)malloc((n + 1) * sizeof(int));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n int result = dp[n];\n free(dp);\n return result;\n}\n// \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsDP(comptime n: usize) i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (n == 1 or n == 2) {\n return @intCast(n);\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = [_]i32{-1} ** (n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (3..n + 1) |i| {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n\u56fe 14-5 \u6a21\u62df\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u6267\u884c\u8fc7\u7a0b\u3002
\u56fe 14-5 \u00a0 \u722c\u697c\u68af\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
\u4e0e\u56de\u6eaf\u7b97\u6cd5\u4e00\u6837\uff0c\u52a8\u6001\u89c4\u5212\u4e5f\u4f7f\u7528\u201c\u72b6\u6001\u201d\u6982\u5ff5\u6765\u8868\u793a\u95ee\u9898\u6c42\u89e3\u7684\u7279\u5b9a\u9636\u6bb5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\u4ee5\u53ca\u76f8\u5e94\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u4f8b\u5982\uff0c\u722c\u697c\u68af\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570 \\(i\\) \u3002
\u6839\u636e\u4ee5\u4e0a\u5185\u5bb9\uff0c\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u52a8\u6001\u89c4\u5212\u7684\u5e38\u7528\u672f\u8bed\u3002
- \u5c06\u6570\u7ec4
dp\u79f0\u4e3a\u300c\\(dp\\) \u8868\u300d\uff0c\\(dp[i]\\) \u8868\u793a\u72b6\u6001 \\(i\\) \u5bf9\u5e94\u5b50\u95ee\u9898\u7684\u89e3\u3002 - \u5c06\u6700\u5c0f\u5b50\u95ee\u9898\u5bf9\u5e94\u7684\u72b6\u6001\uff08\u7b2c \\(1\\) \u9636\u548c\u7b2c \\(2\\) \u9636\u697c\u68af\uff09\u79f0\u4e3a\u300c\u521d\u59cb\u72b6\u6001\u300d\u3002
- \u5c06\u9012\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u79f0\u4e3a\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\u3002
\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u7531\u4e8e \\(dp[i]\\) \u53ea\u4e0e \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u987b\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dp.pydp\u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u800c\u53ea\u9700\u4e24\u4e2a\u53d8\u91cf\u6eda\u52a8\u524d\u8fdb\u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
climbing_stairs_dp.cppdef climbing_stairs_dp_comp(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n if n == 1 or n == 2:\n return n\n a, b = 1, 2\n for _ in range(3, n + 1):\n a, b = b, a + b\n return b\n
climbing_stairs_dp.java/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n if (n == 1 || n == 2)\n return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.cs/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n if (n == 1 || n == 2)\n return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.go/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDPComp(int n) {\n if (n == 1 || n == 2)\n return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.swift/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n int) int {\n if n == 1 || n == 2 {\n return n\n }\n a, b := 1, 2\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n a, b = b, a+b\n }\n return b\n}\n
climbing_stairs_dp.js/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n: Int) -> Int {\n if n == 1 || n == 2 {\n return n\n }\n var a = 1\n var b = 2\n for _ in stride(from: 3, through: n, by: 1) {\n (a, b) = (b, a + b)\n }\n return b\n}\n
climbing_stairs_dp.ts/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n) {\n if (n === 1 || n === 2) return n;\n let a = 1,\n b = 2;\n for (let i = 3; i <= n; i++) {\n const tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.dart/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n: number): number {\n if (n === 1 || n === 2) return n;\n let a = 1,\n b = 2;\n for (let i = 3; i <= n; i++) {\n const tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.rs/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n if (n == 1 || n == 2) return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.c/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp_comp(n: usize) -> i32 {\n if n == 1 || n == 2 { return n as i32; }\n let (mut a, mut b) = (1, 2);\n for _ in 3..=n {\n let tmp = b;\n b = a + b;\n a = tmp;\n }\n b\n}\n
climbing_stairs_dp.zig/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n if (n == 1 || n == 2)\n return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n// \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn climbingStairsDPComp(comptime n: usize) i32 {\n if (n == 1 or n == 2) {\n return @intCast(n);\n }\n var a: i32 = 1;\n var b: i32 = 2;\n for (3..n + 1) |_| {\n var tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u7531\u4e8e\u7701\u53bb\u4e86\u6570\u7ec4
dp\u5360\u7528\u7684\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002\u5728\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e2d\uff0c\u5f53\u524d\u72b6\u6001\u5f80\u5f80\u4ec5\u4e0e\u524d\u9762\u6709\u9650\u4e2a\u72b6\u6001\u6709\u5173\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72b6\u6001\uff0c\u901a\u8fc7\u201c\u964d\u7ef4\u201d\u6765\u8282\u7701\u5185\u5b58\u7a7a\u95f4\u3002\u8fd9\u79cd\u7a7a\u95f4\u4f18\u5316\u6280\u5de7\u88ab\u79f0\u4e3a\u201c\u6eda\u52a8\u53d8\u91cf\u201d\u6216\u201c\u6eda\u52a8\u6570\u7ec4\u201d\u3002
"},{"location":"chapter_dynamic_programming/knapsack_problem/","title":"14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898","text":"\u80cc\u5305\u95ee\u9898\u662f\u4e00\u4e2a\u975e\u5e38\u597d\u7684\u52a8\u6001\u89c4\u5212\u5165\u95e8\u9898\u76ee\uff0c\u662f\u52a8\u6001\u89c4\u5212\u4e2d\u6700\u5e38\u89c1\u7684\u95ee\u9898\u5f62\u5f0f\u3002\u5176\u5177\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982 0-1 \u80cc\u5305\u95ee\u9898\u3001\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u3001\u591a\u91cd\u80cc\u5305\u95ee\u9898\u7b49\u3002
\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6765\u6c42\u89e3\u6700\u5e38\u89c1\u7684 0-1 \u80cc\u5305\u95ee\u9898\u3002
Question
\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002
\u89c2\u5bdf\u56fe 14-17 \uff0c\u7531\u4e8e\u7269\u54c1\u7f16\u53f7 \\(i\\) \u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u6570\u7ec4\u7d22\u5f15\u4ece \\(0\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u56e0\u6b64\u7269\u54c1 \\(i\\) \u5bf9\u5e94\u91cd\u91cf \\(wgt[i-1]\\) \u548c\u4ef7\u503c \\(val[i-1]\\) \u3002
\u56fe 14-17 \u00a0 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u6570\u636e
\u6211\u4eec\u53ef\u4ee5\u5c06 0-1 \u80cc\u5305\u95ee\u9898\u770b\u4f5c\u4e00\u4e2a\u7531 \\(n\\) \u8f6e\u51b3\u7b56\u7ec4\u6210\u7684\u8fc7\u7a0b\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u4f53\u90fd\u6709\u4e0d\u653e\u5165\u548c\u653e\u5165\u4e24\u79cd\u51b3\u7b56\uff0c\u56e0\u6b64\u8be5\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\u3002
\u8be5\u95ee\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u201c\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u201d\uff0c\u56e0\u6b64\u8f83\u5927\u6982\u7387\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\u3002
\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868
\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u54c1\u6765\u8bf4\uff0c\u4e0d\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff1b\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u51cf\u5c0f\u3002\u7531\u6b64\u53ef\u5f97\u72b6\u6001\u5b9a\u4e49\uff1a\u5f53\u524d\u7269\u54c1\u7f16\u53f7 \\(i\\) \u548c\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u8bb0\u4e3a \\([i, c]\\) \u3002
\u72b6\u6001 \\([i, c]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\uff0c\u8bb0\u4e3a \\(dp[i, c]\\) \u3002
\u5f85\u6c42\u89e3\u7684\u662f \\(dp[n, cap]\\) \uff0c\u56e0\u6b64\u9700\u8981\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((n+1) \\times (cap+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002
\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
\u5f53\u6211\u4eec\u505a\u51fa\u7269\u54c1 \\(i\\) \u7684\u51b3\u7b56\u540e\uff0c\u5269\u4f59\u7684\u662f\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u7684\u51b3\u7b56\uff0c\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002
- \u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c]\\) \u3002
- \u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u51cf\u5c11 \\(wgt[i-1]\\) \uff0c\u4ef7\u503c\u589e\u52a0 \\(val[i-1]\\) \uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c-wgt[i-1]]\\) \u3002
\u4e0a\u8ff0\u5206\u6790\u5411\u6211\u4eec\u63ed\u793a\u4e86\u672c\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6700\u5927\u4ef7\u503c \\(dp[i, c]\\) \u7b49\u4e8e\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \u548c\u653e\u5165\u7269\u54c1 \\(i\\) \u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\u3002\u7531\u6b64\u53ef\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a
\\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\]\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf \\(wgt[i - 1]\\) \u8d85\u51fa\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002
\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
\u5f53\u65e0\u7269\u54c1\u6216\u65e0\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u65f6\u6700\u5927\u4ef7\u503c\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217 \\(dp[i, 0]\\) \u548c\u9996\u884c \\(dp[0, c]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002
\u5f53\u524d\u72b6\u6001 \\([i, c]\\) \u4ece\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c]\\) \u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c-wgt[i-1]]\\) \u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u6309\u987a\u5e8f\u5b9e\u73b0\u66b4\u529b\u641c\u7d22\u3001\u8bb0\u5fc6\u5316\u641c\u7d22\u3001\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002
"},{"location":"chapter_dynamic_programming/knapsack_problem/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"\u641c\u7d22\u4ee3\u7801\u5305\u542b\u4ee5\u4e0b\u8981\u7d20\u3002
- \u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, c]\\) \u3002
- \u8fd4\u56de\u503c\uff1a\u5b50\u95ee\u9898\u7684\u89e3 \\(dp[i, c]\\) \u3002
- \u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53\u7269\u54c1\u7f16\u53f7\u8d8a\u754c \\(i = 0\\) \u6216\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\u4e3a \\(0\\) \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5e76\u8fd4\u56de\u4ef7\u503c \\(0\\) \u3002
- \u526a\u679d\uff1a\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf\u8d85\u51fa\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002
knapsack.cppdef knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -> int:\n \"\"\"0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 or c == 0:\n return 0\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c:\n return knapsack_dfs(wgt, val, i - 1, c)\n # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n no = knapsack_dfs(wgt, val, i - 1, c)\n yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n # \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return max(no, yes)\n
knapsack.java/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(vector<int> &wgt, vector<int> &val, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFS(wgt, val, i - 1, c);\n int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return max(no, yes);\n}\n
knapsack.cs/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFS(wgt, val, i - 1, c);\n int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return Math.max(no, yes);\n}\n
knapsack.go/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (weight[i - 1] > c) {\n return KnapsackDFS(weight, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = KnapsackDFS(weight, val, i - 1, c);\n int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return Math.Max(no, yes);\n}\n
knapsack.swift/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i-1] > c {\n return knapsackDFS(wgt, val, i-1, c)\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n no := knapsackDFS(wgt, val, i-1, c)\n yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return int(math.Max(float64(no), float64(yes)))\n}\n
knapsack.js/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -> Int {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c {\n return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return max(no, yes)\n}\n
knapsack.ts/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(wgt, val, i, c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i === 0 || c === 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n const no = knapsackDFS(wgt, val, i - 1, c);\n const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return Math.max(no, yes);\n}\n
knapsack.dart/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(\n wgt: Array<number>,\n val: Array<number>,\n i: number,\n c: number\n): number {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i === 0 || c === 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n const no = knapsackDFS(wgt, val, i - 1, c);\n const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return Math.max(no, yes);\n}\n
knapsack.rs/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(List<int> wgt, List<int> val, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFS(wgt, val, i - 1, c);\n int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return max(no, yes);\n}\n
knapsack.c/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfn knapsack_dfs(wgt: &[i32], val: &[i32], i: usize, c: usize) -> i32 {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c as i32 {\n return knapsack_dfs(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n let no = knapsack_dfs(wgt, val, i - 1, c);\n let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n std::cmp::max(no, yes)\n}\n
knapsack.zig/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFS(wgt, val, i - 1, c);\n int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return myMax(no, yes);\n}\n// 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\nfn knapsackDFS(wgt: []i32, val: []i32, i: usize, c: usize) i32 {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 or c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n var no = knapsackDFS(wgt, val, i - 1, c);\n var yes = knapsackDFS(wgt, val, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return @max(no, yes);\n}\n\u5982\u56fe 14-18 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u7269\u54c1\u90fd\u4f1a\u4ea7\u751f\u4e0d\u9009\u548c\u9009\u4e24\u6761\u641c\u7d22\u5206\u652f\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002
\u89c2\u5bdf\u9012\u5f52\u6811\uff0c\u5bb9\u6613\u53d1\u73b0\u5176\u4e2d\u5b58\u5728\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u4f8b\u5982 \\(dp[1, 10]\\) \u7b49\u3002\u800c\u5f53\u7269\u54c1\u8f83\u591a\u3001\u80cc\u5305\u5bb9\u91cf\u8f83\u5927\uff0c\u5c24\u5176\u662f\u76f8\u540c\u91cd\u91cf\u7684\u7269\u54c1\u8f83\u591a\u65f6\uff0c\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u6570\u91cf\u5c06\u4f1a\u5927\u5e45\u589e\u591a\u3002
\u56fe 14-18 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811
"},{"location":"chapter_dynamic_programming/knapsack_problem/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"\u4e3a\u4e86\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\uff0c\u6211\u4eec\u501f\u52a9\u8bb0\u5fc6\u5217\u8868
mem\u6765\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5176\u4e2dmem[i][c]\u5bf9\u5e94 \\(dp[i, c]\\) \u3002\u5f15\u5165\u8bb0\u5fc6\u5316\u4e4b\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5b50\u95ee\u9898\u6570\u91cf\uff0c\u4e5f\u5c31\u662f \\(O(n \\times cap)\\) \u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py
knapsack.cppdef knapsack_dfs_mem(\n wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -> int:\n \"\"\"0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 or c == 0:\n return 0\n # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][c] != -1:\n return mem[i][c]\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c:\n return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n # \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = max(no, yes)\n return mem[i][c]\n
knapsack.java/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(vector<int> &wgt, vector<int> &val, vector<vector<int>> &mem, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = max(no, yes);\n return mem[i][c];\n}\n
knapsack.cs/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = Math.max(no, yes);\n return mem[i][c];\n}\n
knapsack.go/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (weight[i - 1] > c) {\n return KnapsackDFSMem(weight, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = Math.Max(no, yes);\n return mem[i][c];\n}\n
knapsack.swift/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][c] != -1 {\n return mem[i][c]\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i-1] > c {\n return knapsackDFSMem(wgt, val, mem, i-1, c)\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n no := knapsackDFSMem(wgt, val, mem, i-1, c)\n yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = int(math.Max(float64(no), float64(yes)))\n return mem[i][c]\n}\n
knapsack.js/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -> Int {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][c] != -1 {\n return mem[i][c]\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c {\n return knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: i - 1, c: c)\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n let no = knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: i - 1, c: c)\n let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = max(no, yes)\n return mem[i][c]\n}\n
knapsack.ts/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i === 0 || c === 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] !== -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n const yes =\n knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = Math.max(no, yes);\n return mem[i][c];\n}\n
knapsack.dart/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(\n wgt: Array<number>,\n val: Array<number>,\n mem: Array<Array<number>>,\n i: number,\n c: number\n): number {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i === 0 || c === 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] !== -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n const yes =\n knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = Math.max(no, yes);\n return mem[i][c];\n}\n
knapsack.rs/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(\n List<int> wgt,\n List<int> val,\n List<List<int>> mem,\n int i,\n int c,\n) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = max(no, yes);\n return mem[i][c];\n}\n
knapsack.c/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn knapsack_dfs_mem(wgt: &[i32], val: &[i32], mem: &mut Vec<Vec<i32>>, i: usize, c: usize) -> i32 {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][c] != -1 {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c as i32 {\n return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = std::cmp::max(no, yes);\n mem[i][c]\n}\n
knapsack.zig/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = myMax(no, yes);\n return mem[i][c];\n}\n// 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn knapsackDFSMem(wgt: []i32, val: []i32, mem: anytype, i: usize, c: usize) i32 {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 or c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n var no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n var yes = knapsackDFSMem(wgt, val, mem, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = @max(no, yes);\n return mem[i][c];\n}\n\u56fe 14-19 \u5c55\u793a\u4e86\u5728\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u88ab\u526a\u6389\u7684\u641c\u7d22\u5206\u652f\u3002
\u56fe 14-19 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811
"},{"location":"chapter_dynamic_programming/knapsack_problem/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"\u52a8\u6001\u89c4\u5212\u5b9e\u8d28\u4e0a\u5c31\u662f\u5728\u72b6\u6001\u8f6c\u79fb\u4e2d\u586b\u5145 \\(dp\\) \u8868\u7684\u8fc7\u7a0b\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py
knapsack.cppdef knapsack_dp(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(wgt)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * (cap + 1) for _ in range(n + 1)]\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n for c in range(1, cap + 1):\n if wgt[i - 1] > c:\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c]\n else:\n # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n return dp[n][cap]\n
knapsack.java/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(vector<int> &wgt, vector<int> &val, int cap) {\n int n = wgt.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n + 1, vector<int>(cap + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.cs/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n + 1][cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.go/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n int n = weight.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n + 1, cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (weight[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i, c] = dp[i - 1, c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n }\n }\n }\n return dp[n, cap];\n}\n
knapsack.swift/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt, val []int, cap int) int {\n n := len(wgt)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, cap+1)\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n for c := 1; c <= cap; c++ {\n if wgt[i-1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i-1][c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n }\n }\n }\n return dp[n][cap]\n}\n
knapsack.js/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -> Int {\n let n = wgt.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for c in stride(from: 1, through: cap, by: 1) {\n if wgt[i - 1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n }\n }\n }\n return dp[n][cap]\n}\n
knapsack.ts/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(wgt, val, cap) {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array(n + 1)\n .fill(0)\n .map(() => Array(cap + 1).fill(0));\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(\n dp[i - 1][c],\n dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n );\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.dart/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(\n wgt: Array<number>,\n val: Array<number>,\n cap: number\n): number {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: cap + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(\n dp[i - 1][c],\n dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n );\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.rs/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(List<int> wgt, List<int> val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(cap + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.c/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {\n let n = wgt.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; cap + 1]; n + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for c in 1..=cap {\n if wgt[i - 1] > c as i32 {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1]);\n }\n }\n }\n dp[n][cap]\n}\n
knapsack.zig/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n int n = wgtSize;\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(cap + 1, sizeof(int));\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n int res = dp[n][cap];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n return res;\n}\n// 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn knapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n comptime var n = wgt.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..cap + 1) |c| {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = @max(dp[i - 1][c], dp[i - 1][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n\u5982\u56fe 14-20 \u6240\u793a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u7531\u6570\u7ec4
<1><2><3><4><5><6><7><8><9><10><11><12><13><14>dp\u5927\u5c0f\u51b3\u5b9a\uff0c\u5373 \\(O(n \\times cap)\\) \u3002\u56fe 14-20 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
"},{"location":"chapter_dynamic_programming/knapsack_problem/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u90fd\u53ea\u4e0e\u5176\u4e0a\u4e00\u884c\u7684\u72b6\u6001\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6570\u7ec4\u6eda\u52a8\u524d\u8fdb\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \u3002
\u8fdb\u4e00\u6b65\u601d\u8003\uff0c\u6211\u4eec\u80fd\u5426\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u5b9e\u73b0\u7a7a\u95f4\u4f18\u5316\u5462\uff1f\u89c2\u5bdf\u53ef\u77e5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u662f\u7531\u6b63\u4e0a\u65b9\u6216\u5de6\u4e0a\u65b9\u7684\u683c\u5b50\u8f6c\u79fb\u8fc7\u6765\u7684\u3002\u5047\u8bbe\u53ea\u6709\u4e00\u4e2a\u6570\u7ec4\uff0c\u5f53\u5f00\u59cb\u904d\u5386\u7b2c \\(i\\) \u884c\u65f6\uff0c\u8be5\u6570\u7ec4\u5b58\u50a8\u7684\u4ecd\u7136\u662f\u7b2c \\(i-1\\) \u884c\u7684\u72b6\u6001\u3002
- \u5982\u679c\u91c7\u53d6\u6b63\u5e8f\u904d\u5386\uff0c\u90a3\u4e48\u904d\u5386\u5230 \\(dp[i, j]\\) \u65f6\uff0c\u5de6\u4e0a\u65b9 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) \u503c\u53ef\u80fd\u5df2\u7ecf\u88ab\u8986\u76d6\uff0c\u6b64\u65f6\u5c31\u65e0\u6cd5\u5f97\u5230\u6b63\u786e\u7684\u72b6\u6001\u8f6c\u79fb\u7ed3\u679c\u3002
- \u5982\u679c\u91c7\u53d6\u5012\u5e8f\u904d\u5386\uff0c\u5219\u4e0d\u4f1a\u53d1\u751f\u8986\u76d6\u95ee\u9898\uff0c\u72b6\u6001\u8f6c\u79fb\u53ef\u4ee5\u6b63\u786e\u8fdb\u884c\u3002
\u56fe 14-21 \u5c55\u793a\u4e86\u5728\u5355\u4e2a\u6570\u7ec4\u4e0b\u4ece\u7b2c \\(i = 1\\) \u884c\u8f6c\u6362\u81f3\u7b2c \\(i = 2\\) \u884c\u7684\u8fc7\u7a0b\u3002\u8bf7\u601d\u8003\u6b63\u5e8f\u904d\u5386\u548c\u5012\u5e8f\u904d\u5386\u7684\u533a\u522b\u3002
<1><2><3><4><5><6>\u56fe 14-21 \u00a0 0-1 \u80cc\u5305\u7684\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4ec5\u9700\u5c06\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.pydp\u7684\u7b2c\u4e00\u7ef4 \\(i\\) \u76f4\u63a5\u5220\u9664\uff0c\u5e76\u4e14\u628a\u5185\u5faa\u73af\u66f4\u6539\u4e3a\u5012\u5e8f\u904d\u5386\u5373\u53ef\uff1a
knapsack.cppdef knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(wgt)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [0] * (cap + 1)\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n # \u5012\u5e8f\u904d\u5386\n for c in range(cap, 0, -1):\n if wgt[i - 1] > c:\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c]\n else:\n # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n return dp[cap]\n
knapsack.java/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(vector<int> &wgt, vector<int> &val, int cap) {\n int n = wgt.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(cap + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.cs/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.go/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n int n = weight.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c > 0; c--) {\n if (weight[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.swift/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n n := len(wgt)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, cap+1)\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n // \u5012\u5e8f\u904d\u5386\n for c := cap; c >= 1; c-- {\n if wgt[i-1] <= c {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n }\n }\n }\n return dp[cap]\n}\n
knapsack.js/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -> Int {\n let n = wgt.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: 0, count: cap + 1)\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n // \u5012\u5e8f\u904d\u5386\n for c in stride(from: cap, through: 1, by: -1) {\n if wgt[i - 1] <= c {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n }\n }\n }\n return dp[cap]\n}\n
knapsack.ts/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(wgt, val, cap) {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array(cap + 1).fill(0);\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (let c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.dart/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(\n wgt: Array<number>,\n val: Array<number>,\n cap: number\n): number {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array(cap + 1).fill(0);\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (let c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.rs/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(List<int> wgt, List<int> val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(cap + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.c/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp_comp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {\n let n = wgt.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n // \u5012\u5e8f\u904d\u5386\n for c in (1..=cap).rev() {\n if wgt[i - 1] <= c as i32 {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n }\n }\n }\n dp[cap]\n}\n
knapsack.zig/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n int n = wgtSize;\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(cap + 1, sizeof(int));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n int res = dp[cap];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/summary/","title":"14.7 \u00a0 \u5c0f\u7ed3","text":"// 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn knapsackDPComp(wgt: []i32, val: []i32, comptime cap: usize) i32 {\n var n = wgt.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** (cap + 1);\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n // \u5012\u5e8f\u904d\u5386\n var c = cap;\n while (c > 0) : (c -= 1) {\n if (wgt[i - 1] < c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n- \u52a8\u6001\u89c4\u5212\u5bf9\u95ee\u9898\u8fdb\u884c\u5206\u89e3\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u89c4\u907f\u91cd\u590d\u8ba1\u7b97\uff0c\u63d0\u9ad8 \u8ba1\u7b97\u6548\u7387\u3002
- \u4e0d\u8003\u8651\u65f6\u95f4\u7684\u524d\u63d0\u4e0b\uff0c\u6240\u6709\u52a8\u6001\u89c4\u5212\u95ee\u9898\u90fd\u53ef\u4ee5\u7528\u56de\u6eaf\uff08\u66b4\u529b\u641c\u7d22\uff09\u8fdb\u884c\u6c42\u89e3\uff0c\u4f46\u9012\u5f52\u6811\u4e2d\u5b58\u5728\u5927\u91cf\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u6548\u7387\u6781\u4f4e\u3002\u901a\u8fc7\u5f15\u5165\u8bb0\u5fc6\u5316\u5217\u8868\uff0c\u53ef\u4ee5\u5b58\u50a8\u6240\u6709\u8ba1\u7b97\u8fc7\u7684\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u4ece\u800c\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002
- \u8bb0\u5fc6\u5316\u9012\u5f52\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u9012\u5f52\u5f0f\u89e3\u6cd5\uff0c\u800c\u4e0e\u4e4b\u5bf9\u5e94\u7684\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u4ece\u5e95\u81f3\u9876\u7684\u9012\u63a8\u5f0f\u89e3\u6cd5\uff0c\u5176\u5982\u540c\u201c\u586b\u5199\u8868\u683c\u201d\u4e00\u6837\u3002\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u4ec5\u4f9d\u8d56\u67d0\u4e9b\u5c40\u90e8\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u6d88\u9664 \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\uff0c\u4ece\u800c\u964d\u4f4e\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
- \u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u5177\u6709\u4e0d\u540c\u7684\u6027\u8d28\u3002
- \u52a8\u6001\u89c4\u5212\u95ee\u9898\u6709\u4e09\u5927\u7279\u6027\uff1a\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002
- \u5982\u679c\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\uff0c\u5219\u5b83\u5c31\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u3002
- \u65e0\u540e\u6548\u6027\u6307\u5bf9\u4e8e\u4e00\u4e2a\u72b6\u6001\uff0c\u5176\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u8be5\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u4e0d\u5177\u6709\u65e0\u540e\u6548\u6027\uff0c\u65e0\u6cd5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u5feb\u901f\u6c42\u89e3\u3002
\u80cc\u5305\u95ee\u9898
- \u80cc\u5305\u95ee\u9898\u662f\u6700\u5178\u578b\u7684\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e4b\u4e00\uff0c\u5177\u6709 0-1 \u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305\u3001\u591a\u91cd\u80cc\u5305\u7b49\u53d8\u79cd\u3002
- 0-1 \u80cc\u5305\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\u3002\u6839\u636e\u4e0d\u653e\u5165\u80cc\u5305\u548c\u653e\u5165\u80cc\u5305\u4e24\u79cd\u51b3\u7b56\uff0c\u53ef\u5f97\u5230\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\uff0c\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u9700\u8981\u5012\u5e8f\u904d\u5386\u5217\u8868\uff0c\u907f\u514d\u5de6\u4e0a\u65b9\u72b6\u6001\u88ab\u8986\u76d6\u3002
- \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u6bcf\u79cd\u7269\u54c1\u7684\u9009\u53d6\u6570\u91cf\u65e0\u9650\u5236\uff0c\u56e0\u6b64\u9009\u62e9\u653e\u5165\u7269\u54c1\u7684\u72b6\u6001\u8f6c\u79fb\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\u3002\u7531\u4e8e\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u6b63\u5de6\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\u5e94\u5f53\u6b63\u5e8f\u904d\u5386\u3002
- \u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u4e2a\u53d8\u79cd\u3002\u5b83\u4ece\u6c42\u201c\u6700\u5927\u201d\u4ef7\u503c\u53d8\u4e3a\u6c42\u201c\u6700\u5c0f\u201d\u786c\u5e01\u6570\u91cf\uff0c\u56e0\u6b64\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\max()\\) \u5e94\u6539\u4e3a \\(\\min()\\) \u3002\u4ece\u8ffd\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u5230\u8ffd\u6c42\u201c\u6070\u597d\u201d\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u4f7f\u7528 \\(amt + 1\\) \u6765\u8868\u793a\u201c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u201d\u7684\u65e0\u6548\u89e3\u3002
- \u96f6\u94b1\u5151\u6362 II \u95ee\u9898\u4ece\u6c42\u201c\u6700\u5c11\u786c\u5e01\u6570\u91cf\u201d\u6539\u4e3a\u6c42\u201c\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u201d\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u76f8\u5e94\u5730\u4ece \\(\\min()\\) \u6539\u4e3a\u6c42\u548c\u8fd0\u7b97\u7b26\u3002
\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898
- \u7f16\u8f91\u8ddd\u79bb\uff08Levenshtein \u8ddd\u79bb\uff09\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\uff0c\u5176\u5b9a\u4e49\u4e3a\u4ece\u4e00\u4e2a\u5b57\u7b26\u4e32\u5230\u53e6\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u7f16\u8f91\u64cd\u4f5c\u5305\u62ec\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\u3002
- \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002\u5f53 \\(s[i] \\ne t[j]\\) \u65f6\uff0c\u5177\u6709\u4e09\u79cd\u51b3\u7b56\uff1a\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\uff0c\u5b83\u4eec\u90fd\u6709\u76f8\u5e94\u7684\u5269\u4f59\u5b50\u95ee\u9898\u3002\u636e\u6b64\u4fbf\u53ef\u4ee5\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u6784\u5efa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u800c\u5f53 \\(s[i] = t[j]\\) \u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\u3002
- \u5728\u7f16\u8f91\u8ddd\u79bb\u4e2d\uff0c\u72b6\u6001\u4f9d\u8d56\u5176\u6b63\u4e0a\u65b9\u3001\u6b63\u5de6\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u6b63\u5e8f\u6216\u5012\u5e8f\u904d\u5386\u90fd\u65e0\u6cd5\u6b63\u786e\u5730\u8fdb\u884c\u72b6\u6001\u8f6c\u79fb\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u53d8\u91cf\u6682\u5b58\u5de6\u4e0a\u65b9\u72b6\u6001\uff0c\u4ece\u800c\u8f6c\u5316\u5230\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7b49\u4ef7\u7684\u60c5\u51b5\uff0c\u53ef\u4ee5\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002
\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6c42\u89e3\u53e6\u4e00\u4e2a\u5e38\u89c1\u7684\u80cc\u5305\u95ee\u9898\uff1a\u5b8c\u5168\u80cc\u5305\uff0c\u518d\u4e86\u89e3\u5b83\u7684\u4e00\u79cd\u7279\u4f8b\uff1a\u96f6\u94b1\u5151\u6362\u3002
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","title":"14.5.1 \u00a0 \u5b8c\u5168\u80cc\u5305","text":"Question
\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 14-22 \u6240\u793a\u3002
\u56fe 14-22 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\uff0c\u533a\u522b\u4ec5\u5728\u4e8e\u4e0d\u9650\u5236\u7269\u54c1\u7684\u9009\u62e9\u6b21\u6570\u3002
- \u5728 0-1 \u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u53ea\u6709\u4e00\u4e2a\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u53ea\u80fd\u4ece\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002
- \u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u7684\u6570\u91cf\u662f\u65e0\u9650\u7684\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u4ecd\u53ef\u4ee5\u4ece\u524d \\(i\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002
\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u89c4\u5b9a\u4e0b\uff0c\u72b6\u6001 \\([i, c]\\) \u7684\u53d8\u5316\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5\u3002
- \u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i-1, c]\\) \u3002
- \u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i, c-wgt[i-1]]\\) \u3002
\u4ece\u800c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u53d8\u4e3a\uff1a
\\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5bf9\u6bd4\u4e24\u9053\u9898\u76ee\u7684\u4ee3\u7801\uff0c\u72b6\u6001\u8f6c\u79fb\u4e2d\u6709\u4e00\u5904\u4ece \\(i-1\\) \u53d8\u4e3a \\(i\\) \uff0c\u5176\u4f59\u5b8c\u5168\u4e00\u81f4\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig unbounded_knapsack.py
unbounded_knapsack.cppdef unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(wgt)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * (cap + 1) for _ in range(n + 1)]\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n for c in range(1, cap + 1):\n if wgt[i - 1] > c:\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c]\n else:\n # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n return dp[n][cap]\n
unbounded_knapsack.java/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(vector<int> &wgt, vector<int> &val, int cap) {\n int n = wgt.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n + 1, vector<int>(cap + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.cs/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n + 1][cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.go/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n int n = wgt.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n + 1, cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i, c] = dp[i - 1, c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n, cap];\n}\n
unbounded_knapsack.swift/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n n := len(wgt)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, cap+1)\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n for c := 1; c <= cap; c++ {\n if wgt[i-1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i-1][c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n }\n }\n }\n return dp[n][cap]\n}\n
unbounded_knapsack.js/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -> Int {\n let n = wgt.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for c in stride(from: 1, through: cap, by: 1) {\n if wgt[i - 1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n }\n }\n }\n return dp[n][cap]\n}\n
unbounded_knapsack.ts/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: cap + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(\n dp[i - 1][c],\n dp[i][c - wgt[i - 1]] + val[i - 1]\n );\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.dart/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(\n wgt: Array<number>,\n val: Array<number>,\n cap: number\n): number {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: cap + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(\n dp[i - 1][c],\n dp[i][c - wgt[i - 1]] + val[i - 1]\n );\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.rs/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(List<int> wgt, List<int> val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(cap + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.c/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {\n let n = wgt.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; cap + 1]; n + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for c in 1..=cap {\n if wgt[i - 1] > c as i32 {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.zig/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n int n = wgtSize;\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(cap + 1, sizeof(int));\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n int res = dp[n][cap];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"// \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n comptime var n = wgt.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..cap + 1) |c| {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = @max(dp[i - 1][c], dp[i][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u662f\u4ece\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u72b6\u6001\u8f6c\u79fb\u800c\u6765\u7684\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u5e94\u8be5\u5bf9 \\(dp\\) \u8868\u4e2d\u7684\u6bcf\u4e00\u884c\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002
\u8fd9\u4e2a\u904d\u5386\u987a\u5e8f\u4e0e 0-1 \u80cc\u5305\u6b63\u597d\u76f8\u53cd\u3002\u8bf7\u501f\u52a9\u56fe 14-23 \u6765\u7406\u89e3\u4e24\u8005\u7684\u533a\u522b\u3002
<1><2><3><4><5><6>\u56fe 14-23 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
\u4ee3\u7801\u5b9e\u73b0\u6bd4\u8f83\u7b80\u5355\uff0c\u4ec5\u9700\u5c06\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig unbounded_knapsack.pydp\u7684\u7b2c\u4e00\u7ef4\u5220\u9664\uff1a
unbounded_knapsack.cppdef unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(wgt)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [0] * (cap + 1)\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n # \u6b63\u5e8f\u904d\u5386\n for c in range(1, cap + 1):\n if wgt[i - 1] > c:\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c]\n else:\n # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n return dp[cap]\n
unbounded_knapsack.java/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(vector<int> &wgt, vector<int> &val, int cap) {\n int n = wgt.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(cap + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.cs/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.go/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n int n = wgt.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.swift/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n n := len(wgt)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, cap+1)\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n for c := 1; c <= cap; c++ {\n if wgt[i-1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n }\n }\n }\n return dp[cap]\n}\n
unbounded_knapsack.js/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -> Int {\n let n = wgt.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: 0, count: cap + 1)\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for c in stride(from: 1, through: cap, by: 1) {\n if wgt[i - 1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n }\n }\n }\n return dp[cap]\n}\n
unbounded_knapsack.ts/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: cap + 1 }, () => 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.dart/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(\n wgt: Array<number>,\n val: Array<number>,\n cap: number\n): number {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: cap + 1 }, () => 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.rs/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(List<int> wgt, List<int> val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(cap + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.c/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp_comp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {\n let n = wgt.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for c in 1..=cap {\n if wgt[i - 1] > c as i32 {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n }\n }\n }\n dp[cap]\n}\n
unbounded_knapsack.zig/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n int n = wgtSize;\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(cap + 1, sizeof(int));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n int res = dp[cap];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","title":"14.5.2 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898","text":"// \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDPComp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n comptime var n = wgt.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** (cap + 1);\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..cap + 1) |c| {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n\u80cc\u5305\u95ee\u9898\u662f\u4e00\u5927\u7c7b\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4ee3\u8868\uff0c\u5176\u62e5\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002
Question
\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 14-24 \u6240\u793a\u3002
\u56fe 14-24 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"\u96f6\u94b1\u5151\u6362\u53ef\u4ee5\u770b\u4f5c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u79cd\u7279\u6b8a\u60c5\u51b5\uff0c\u4e24\u8005\u5177\u6709\u4ee5\u4e0b\u8054\u7cfb\u4e0e\u4e0d\u540c\u70b9\u3002
- \u4e24\u9053\u9898\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\uff0c\u201c\u7269\u54c1\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u201d\u3001\u201c\u7269\u54c1\u91cd\u91cf\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u9762\u503c\u201d\u3001\u201c\u80cc\u5305\u5bb9\u91cf\u201d\u5bf9\u5e94\u201c\u76ee\u6807\u91d1\u989d\u201d\u3002
- \u4f18\u5316\u76ee\u6807\u76f8\u53cd\uff0c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u8981\u6700\u5927\u5316\u7269\u54c1\u4ef7\u503c\uff0c\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u8981\u6700\u5c0f\u5316\u786c\u5e01\u6570\u91cf\u3002
- \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u89e3\uff0c\u96f6\u94b1\u5151\u6362\u662f\u6c42\u201c\u6070\u597d\u201d\u51d1\u5230\u76ee\u6807\u91d1\u989d\u7684\u89e3\u3002
\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868
\u72b6\u6001 \\([i, a]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\uff0c\u8bb0\u4e3a \\(dp[i, a]\\) \u3002
\u4e8c\u7ef4 \\(dp\\) \u8868\u7684\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt+1)\\) \u3002
\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
\u672c\u9898\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5b58\u5728\u4ee5\u4e0b\u4e24\u70b9\u5dee\u5f02\u3002
- \u672c\u9898\u8981\u6c42\u6700\u5c0f\u503c\uff0c\u56e0\u6b64\u9700\u5c06\u8fd0\u7b97\u7b26 \\(\\max()\\) \u66f4\u6539\u4e3a \\(\\min()\\) \u3002
- \u4f18\u5316\u4e3b\u4f53\u662f\u786c\u5e01\u6570\u91cf\u800c\u975e\u5546\u54c1\u4ef7\u503c\uff0c\u56e0\u6b64\u5728\u9009\u4e2d\u786c\u5e01\u65f6\u6267\u884c \\(+1\\) \u5373\u53ef\u3002
\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u51d1\u51fa\u5b83\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002
\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u610f \\(> 0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u5373\u662f\u65e0\u6548\u89e3\u3002\u4e3a\u4f7f\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\min()\\) \u51fd\u6570\u80fd\u591f\u8bc6\u522b\u5e76\u8fc7\u6ee4\u65e0\u6548\u89e3\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528 \\(+ \\infty\\) \u6765\u8868\u793a\u5b83\u4eec\uff0c\u5373\u4ee4\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(+ \\infty\\) \u3002
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u5e76\u672a\u63d0\u4f9b \\(+ \\infty\\) \u53d8\u91cf\uff0c\u53ea\u80fd\u4f7f\u7528\u6574\u578b
int\u7684\u6700\u5927\u503c\u6765\u4ee3\u66ff\u3002\u800c\u8fd9\u53c8\u4f1a\u5bfc\u81f4\u5927\u6570\u8d8a\u754c\uff1a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(+ 1\\) \u64cd\u4f5c\u53ef\u80fd\u53d1\u751f\u6ea2\u51fa\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u6570\u5b57 \\(amt + 1\\) \u6765\u8868\u793a\u65e0\u6548\u89e3\uff0c\u56e0\u4e3a\u51d1\u51fa \\(amt\\) \u7684\u786c\u5e01\u6570\u91cf\u6700\u591a\u4e3a \\(amt\\) \u3002\u6700\u540e\u8fd4\u56de\u524d\uff0c\u5224\u65ad \\(dp[n, amt]\\) \u662f\u5426\u7b49\u4e8e \\(amt + 1\\) \uff0c\u82e5\u662f\u5219\u8fd4\u56de \\(-1\\) \uff0c\u4ee3\u8868\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change.py
coin_change.cppdef coin_change_dp(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(coins)\n MAX = amt + 1\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * (amt + 1) for _ in range(n + 1)]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for a in range(1, amt + 1):\n dp[0][a] = MAX\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in range(1, n + 1):\n for a in range(1, amt + 1):\n if coins[i - 1] > a:\n # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a]\n else:\n # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n return dp[n][amt] if dp[n][amt] != MAX else -1\n
coin_change.java/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(vector<int> &coins, int amt) {\n int n = coins.size();\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n + 1, vector<int>(amt + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n
coin_change.cs/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int[] coins, int amt) {\n int n = coins.length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n + 1][amt + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n
coin_change.go/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDP(int[] coins, int amt) {\n int n = coins.Length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n + 1, amt + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0, a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i, a] = dp[i - 1, a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n
coin_change.swift/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins []int, amt int) int {\n n := len(coins)\n max := amt + 1\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, amt+1)\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for a := 1; a <= amt; a++ {\n dp[0][a] = max\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i <= n; i++ {\n for a := 1; a <= amt; a++ {\n if coins[i-1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i-1][a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n }\n }\n }\n if dp[n][amt] != max {\n return dp[n][amt]\n }\n return -1\n}\n
coin_change.js/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins: [Int], amt: Int) -> Int {\n let n = coins.count\n let MAX = amt + 1\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for a in stride(from: 1, through: amt, by: 1) {\n dp[0][a] = MAX\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in stride(from: 1, through: n, by: 1) {\n for a in stride(from: 1, through: amt, by: 1) {\n if coins[i - 1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n }\n }\n }\n return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n
coin_change.ts/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins, amt) {\n const n = coins.length;\n const MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: amt + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (let a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n
coin_change.dart/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins: Array<number>, amt: number): number {\n const n = coins.length;\n const MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: amt + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (let a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n
coin_change.rs/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(List<int> coins, int amt) {\n int n = coins.length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(amt + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n
coin_change.c/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp(coins: &[i32], amt: usize) -> i32 {\n let n = coins.len();\n let max = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; amt + 1]; n + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for a in 1..= amt {\n dp[0][a] = max;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in 1..=n {\n for a in 1..=amt {\n if coins[i - 1] > a as i32 {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n }\n }\n }\n if dp[n][amt] != max { return dp[n][amt] as i32; } else { -1 }\n}\n
coin_change.zig/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n int n = coinsSize;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(amt + 1, sizeof(int));\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n free(dp);\n return res;\n}\n// \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeDP(comptime coins: []i32, comptime amt: usize) i32 {\n comptime var n = coins.len;\n comptime var max = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (1..amt + 1) |a| {\n dp[0][a] = max;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (1..n + 1) |i| {\n for (1..amt + 1) |a| {\n if (coins[i - 1] > @as(i32, @intCast(a))) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = @min(dp[i - 1][a], dp[i][a - @as(usize, @intCast(coins[i - 1]))] + 1);\n }\n }\n }\n if (dp[n][amt] != max) {\n return @intCast(dp[n][amt]);\n } else {\n return -1;\n }\n}\n\u56fe 14-25 \u5c55\u793a\u4e86\u96f6\u94b1\u5151\u6362\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\uff0c\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\u3002
<1><2><3><4><5><6><7><8><9><10><11><12><13><14><15>\u56fe 14-25 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"\u96f6\u94b1\u5151\u6362\u7684\u7a7a\u95f4\u4f18\u5316\u7684\u5904\u7406\u65b9\u5f0f\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e00\u81f4\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change.py
coin_change.cppdef coin_change_dp_comp(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(coins)\n MAX = amt + 1\n # \u521d\u59cb\u5316 dp \u8868\n dp = [MAX] * (amt + 1)\n dp[0] = 0\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n # \u6b63\u5e8f\u904d\u5386\n for a in range(1, amt + 1):\n if coins[i - 1] > a:\n # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n else:\n # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n return dp[amt] if dp[amt] != MAX else -1\n
coin_change.java/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(vector<int> &coins, int amt) {\n int n = coins.size();\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(amt + 1, MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1;\n}\n
coin_change.cs/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int[] coins, int amt) {\n int n = coins.length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[amt + 1];\n Arrays.fill(dp, MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1;\n}\n
coin_change.go/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDPComp(int[] coins, int amt) {\n int n = coins.Length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[amt + 1];\n Array.Fill(dp, MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1;\n}\n
coin_change.swift/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins []int, amt int) int {\n n := len(coins)\n max := amt + 1\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, amt+1)\n for i := 1; i <= amt; i++ {\n dp[i] = max\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n // \u5012\u5e8f\u904d\u5386\n for a := 1; a <= amt; a++ {\n if coins[i-1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n }\n }\n }\n if dp[amt] != max {\n return dp[amt]\n }\n return -1\n}\n
coin_change.js/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -> Int {\n let n = coins.count\n let MAX = amt + 1\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: MAX, count: amt + 1)\n dp[0] = 0\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for a in stride(from: 1, through: amt, by: 1) {\n if coins[i - 1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1\n}\n
coin_change.ts/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins, amt) {\n const n = coins.length;\n const MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: amt + 1 }, () => MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] !== MAX ? dp[amt] : -1;\n}\n
coin_change.dart/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins: Array<number>, amt: number): number {\n const n = coins.length;\n const MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: amt + 1 }, () => MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] !== MAX ? dp[amt] : -1;\n}\n
coin_change.rs/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(List<int> coins, int amt) {\n int n = coins.length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(amt + 1, MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1;\n}\n
coin_change.c/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp_comp(coins: &[i32], amt: usize) -> i32 {\n let n = coins.len();\n let max = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; amt + 1];\n dp.fill(max);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for a in 1..=amt {\n if coins[i - 1] > a as i32 {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n }\n }\n }\n if dp[amt] != max { return dp[amt] as i32; } else { -1 }\n}\n
coin_change.zig/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n int n = coinsSize;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(amt + 1, sizeof(int));\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n int res = dp[amt] != MAX ? dp[amt] : -1;\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","title":"14.5.3 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II","text":"// \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n comptime var n = coins.len;\n comptime var max = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** (amt + 1);\n @memset(&dp, max);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..amt + 1) |a| {\n if (coins[i - 1] > @as(i32, @intCast(a))) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = @min(dp[a], dp[a - @as(usize, @intCast(coins[i - 1]))] + 1);\n }\n }\n }\n if (dp[amt] != max) {\n return @intCast(dp[amt]);\n } else {\n return -1;\n }\n}\nQuestion
\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u3002\u793a\u4f8b\u5982\u56fe 14-26 \u6240\u793a\u3002
\u56fe 14-26 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II \u7684\u793a\u4f8b\u6570\u636e
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"\u76f8\u6bd4\u4e8e\u4e0a\u4e00\u9898\uff0c\u672c\u9898\u76ee\u6807\u662f\u6c42\u7ec4\u5408\u6570\u91cf\uff0c\u56e0\u6b64\u5b50\u95ee\u9898\u53d8\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u7ec4\u5408\u6570\u91cf\u3002\u800c \\(dp\\) \u8868\u4ecd\u7136\u662f\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt + 1)\\) \u7684\u4e8c\u7ef4\u77e9\u9635\u3002
\u5f53\u524d\u72b6\u6001\u7684\u7ec4\u5408\u6570\u91cf\u7b49\u4e8e\u4e0d\u9009\u5f53\u524d\u786c\u5e01\u4e0e\u9009\u5f53\u524d\u786c\u5e01\u8fd9\u4e24\u79cd\u51b3\u7b56\u7684\u7ec4\u5408\u6570\u91cf\u4e4b\u548c\u3002\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a
\\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\]\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u65e0\u987b\u9009\u62e9\u4efb\u4f55\u786c\u5e01\u5373\u53ef\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u5e94\u5c06\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u521d\u59cb\u5316\u4e3a \\(1\\) \u3002\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u4f55 \\(>0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_ii.py
coin_change_ii.cppdef coin_change_ii_dp(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(coins)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * (amt + 1) for _ in range(n + 1)]\n # \u521d\u59cb\u5316\u9996\u5217\n for i in range(n + 1):\n dp[i][0] = 1\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n for a in range(1, amt + 1):\n if coins[i - 1] > a:\n # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a]\n else:\n # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n return dp[n][amt]\n
coin_change_ii.java/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(vector<int> &coins, int amt) {\n int n = coins.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n + 1, vector<int>(amt + 1, 0));\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.cs/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int[] coins, int amt) {\n int n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n + 1][amt + 1];\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.go/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDP(int[] coins, int amt) {\n int n = coins.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n + 1, amt + 1];\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i, 0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i, a] = dp[i - 1, a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n }\n }\n }\n return dp[n, amt];\n}\n
coin_change_ii.swift/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins []int, amt int) int {\n n := len(coins)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, amt+1)\n }\n // \u521d\u59cb\u5316\u9996\u5217\n for i := 0; i <= n; i++ {\n dp[i][0] = 1\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i <= n; i++ {\n for a := 1; a <= amt; a++ {\n if coins[i-1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i-1][a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n }\n }\n }\n return dp[n][amt]\n}\n
coin_change_ii.js/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -> Int {\n let n = coins.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n // \u521d\u59cb\u5316\u9996\u5217\n for i in stride(from: 0, through: n, by: 1) {\n dp[i][0] = 1\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for a in stride(from: 1, through: amt, by: 1) {\n if coins[i - 1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n }\n }\n }\n return dp[n][amt]\n}\n
coin_change_ii.ts/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins, amt) {\n const n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: amt + 1 }, () => 0)\n );\n // \u521d\u59cb\u5316\u9996\u5217\n for (let i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.dart/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins: Array<number>, amt: number): number {\n const n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: amt + 1 }, () => 0)\n );\n // \u521d\u59cb\u5316\u9996\u5217\n for (let i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.rs/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(List<int> coins, int amt) {\n int n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(amt + 1, 0));\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.c/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp(coins: &[i32], amt: usize) -> i32 {\n let n = coins.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; amt + 1]; n + 1];\n // \u521d\u59cb\u5316\u9996\u5217\n for i in 0..= n {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for a in 1..=amt {\n if coins[i - 1] > a as i32 {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n }\n }\n }\n dp[n][amt]\n}\n
coin_change_ii.zig/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n int n = coinsSize;\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(amt + 1, sizeof(int));\n }\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n int res = dp[n][amt];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"// \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDP(comptime coins: []i32, comptime amt: usize) i32 {\n comptime var n = coins.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n // \u521d\u59cb\u5316\u9996\u5217\n for (0..n + 1) |i| {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..amt + 1) |a| {\n if (coins[i - 1] > @as(i32, @intCast(a))) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = dp[i - 1][a] + dp[i][a - @as(usize, @intCast(coins[i - 1]))];\n }\n }\n }\n return dp[n][amt];\n}\n\u7a7a\u95f4\u4f18\u5316\u5904\u7406\u65b9\u5f0f\u76f8\u540c\uff0c\u5220\u9664\u786c\u5e01\u7ef4\u5ea6\u5373\u53ef\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_ii.py
coin_change_ii.cppdef coin_change_ii_dp_comp(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(coins)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [0] * (amt + 1)\n dp[0] = 1\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n # \u6b63\u5e8f\u904d\u5386\n for a in range(1, amt + 1):\n if coins[i - 1] > a:\n # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n else:\n # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]]\n return dp[amt]\n
coin_change_ii.java/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(vector<int> &coins, int amt) {\n int n = coins.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(amt + 1, 0);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.cs/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int[] coins, int amt) {\n int n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[amt + 1];\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.go/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n int n = coins.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[amt + 1];\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.swift/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n n := len(coins)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, amt+1)\n dp[0] = 1\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n // \u5012\u5e8f\u904d\u5386\n for a := 1; a <= amt; a++ {\n if coins[i-1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a-coins[i-1]]\n }\n }\n }\n return dp[amt]\n}\n
coin_change_ii.js/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -> Int {\n let n = coins.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: 0, count: amt + 1)\n dp[0] = 1\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for a in stride(from: 1, through: amt, by: 1) {\n if coins[i - 1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]]\n }\n }\n }\n return dp[amt]\n}\n
coin_change_ii.ts/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins, amt) {\n const n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: amt + 1 }, () => 0);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.dart/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins: Array<number>, amt: number): number {\n const n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: amt + 1 }, () => 0);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.rs/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(List<int> coins, int amt) {\n int n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(amt + 1, 0);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.c/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp_comp(coins: &[i32], amt: usize) -> i32 {\n let n = coins.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; amt + 1];\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for a in 1..=amt {\n if coins[i - 1] > a as i32 {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n }\n }\n }\n dp[amt]\n}\n
coin_change_ii.zig/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n int n = coinsSize;\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(amt + 1, sizeof(int));\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n int res = dp[amt];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_graph/","title":"\u7b2c 9 \u7ae0 \u00a0 \u56fe","text":"// \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n comptime var n = coins.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** (amt + 1);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..amt + 1) |a| {\n if (coins[i - 1] > @as(i32, @intCast(a))) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = dp[a] + dp[a - @as(usize, @intCast(coins[i - 1]))];\n }\n }\n }\n return dp[amt];\n}\nAbstract
\u5728\u751f\u547d\u65c5\u9014\u4e2d\uff0c\u6211\u4eec\u5c31\u50cf\u662f\u4e00\u4e2a\u4e2a\u8282\u70b9\uff0c\u88ab\u65e0\u6570\u770b\u4e0d\u89c1\u7684\u8fb9\u76f8\u8fde\u3002
\u6bcf\u4e00\u6b21\u7684\u76f8\u8bc6\u4e0e\u76f8\u79bb\uff0c\u90fd\u5728\u8fd9\u5f20\u5de8\u5927\u7684\u7f51\u7edc\u56fe\u4e2d\u7559\u4e0b\u72ec\u7279\u7684\u5370\u8bb0\u3002
"},{"location":"chapter_graph/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 9.1 \u00a0 \u56fe
- 9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c
- 9.3 \u00a0 \u56fe\u7684\u904d\u5386
- 9.4 \u00a0 \u5c0f\u7ed3
\u300c\u56fe graph\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u300c\u9876\u70b9 vertex\u300d\u548c\u300c\u8fb9 edge\u300d\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe\u3002
\\[ \\begin{aligned} V & = \\{ 1, 2, 3, 4, 5 \\} \\newline E & = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G & = \\{ V, E \\} \\newline \\end{aligned} \\]\u5982\u679c\u5c06\u9876\u70b9\u770b\u4f5c\u8282\u70b9\uff0c\u5c06\u8fb9\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u56fe\u770b\u4f5c\u4e00\u79cd\u4ece\u94fe\u8868\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u5982\u56fe 9-1 \u6240\u793a\uff0c\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002
\u56fe 9-1 \u00a0 \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb
"},{"location":"chapter_graph/graph/#911","title":"9.1.1 \u00a0 \u56fe\u5e38\u89c1\u7c7b\u578b\u4e0e\u672f\u8bed","text":"\u6839\u636e\u8fb9\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u4e3a\u300c\u65e0\u5411\u56fe undirected graph\u300d\u548c\u300c\u6709\u5411\u56fe directed graph\u300d\uff0c\u5982\u56fe 9-2 \u6240\u793a\u3002
- \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u7684\u201c\u53cc\u5411\u201d\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\u3002
- \u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\u3002
\u56fe 9-2 \u00a0 \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe
\u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u53ef\u5206\u4e3a\u300c\u8fde\u901a\u56fe connected graph\u300d\u548c\u300c\u975e\u8fde\u901a\u56fe disconnected graph\u300d\uff0c\u5982\u56fe 9-3 \u6240\u793a\u3002
- \u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\u3002
- \u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002
\u56fe 9-3 \u00a0 \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe
\u6211\u4eec\u8fd8\u53ef\u4ee5\u4e3a\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u4ece\u800c\u5f97\u5230\u5982\u56fe 9-4 \u6240\u793a\u7684\u300c\u6709\u6743\u56fe weighted graph\u300d\u3002\u4f8b\u5982\u5728\u201c\u738b\u8005\u8363\u8000\u201d\u7b49\u624b\u6e38\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002
\u56fe 9-4 \u00a0 \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe
\u56fe\u6570\u636e\u7ed3\u6784\u5305\u542b\u4ee5\u4e0b\u5e38\u7528\u672f\u8bed\u3002
- \u300c\u90bb\u63a5 adjacency\u300d\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u5b58\u5728\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u8fd9\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2\u30013\u30015\u3002
- \u300c\u8def\u5f84 path\u300d\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u7ecf\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u6761\u8def\u5f84\u3002
- \u300c\u5ea6 degree\u300d\uff1a\u4e00\u4e2a\u9876\u70b9\u62e5\u6709\u7684\u8fb9\u6570\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u300c\u5165\u5ea6 in-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u300c\u51fa\u5ea6 out-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002
\u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u5f0f\u5305\u62ec\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u3002\u4ee5\u4e0b\u4f7f\u7528\u65e0\u5411\u56fe\u8fdb\u884c\u4e3e\u4f8b\u3002
"},{"location":"chapter_graph/graph/#1","title":"1. \u00a0 \u90bb\u63a5\u77e9\u9635","text":"\u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u300c\u90bb\u63a5\u77e9\u9635 adjacency matrix\u300d\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u662f\u5426\u5b58\u5728\u8fb9\u3002
\u5982\u56fe 9-5 \u6240\u793a\uff0c\u8bbe\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\)\u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u90a3\u4e48\u77e9\u9635\u5143\u7d20 \\(M[i, j] = 1\\) \u8868\u793a\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u5b58\u5728\u8fb9\uff0c\u53cd\u4e4b \\(M[i, j] = 0\\) \u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002
\u56fe 9-5 \u00a0 \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a
\u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002
- \u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u6b64\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002
- \u5bf9\u4e8e\u65e0\u5411\u56fe\uff0c\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002
- \u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) \u548c \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u53ef\u8868\u793a\u6709\u6743\u56fe\u3002
\u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u4ee5\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u591a\u3002
"},{"location":"chapter_graph/graph/#2","title":"2. \u00a0 \u90bb\u63a5\u8868","text":"\u300c\u90bb\u63a5\u8868 adjacency list\u300d\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u8282\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002\u56fe 9-6 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f7f\u7528\u90bb\u63a5\u8868\u5b58\u50a8\u7684\u56fe\u7684\u793a\u4f8b\u3002
\u56fe 9-6 \u00a0 \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a
\u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b9e\u9645\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u901a\u5e38\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u7136\u800c\uff0c\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002
\u89c2\u5bdf\u56fe 9-6 \uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u4e2d\u7684\u201c\u94fe\u5f0f\u5730\u5740\u201d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u91c7\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u6bd4\u5982\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff1b\u8fd8\u53ef\u4ee5\u628a\u94fe\u8868\u8f6c\u6362\u4e3a\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u81f3 \\(O(1)\\) \u3002
"},{"location":"chapter_graph/graph/#913","title":"9.1.3 \u00a0 \u56fe\u5e38\u89c1\u5e94\u7528","text":"\u5982\u8868 9-1 \u6240\u793a\uff0c\u8bb8\u591a\u73b0\u5b9e\u7cfb\u7edf\u53ef\u4ee5\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u76f8\u5e94\u7684\u95ee\u9898\u4e5f\u53ef\u4ee5\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002
\u8868 9-1 \u00a0 \u73b0\u5b9e\u751f\u6d3b\u4e2d\u5e38\u89c1\u7684\u56fe
\u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c","text":"\u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u53ef\u5206\u4e3a\u5bf9\u201c\u8fb9\u201d\u7684\u64cd\u4f5c\u548c\u5bf9\u201c\u9876\u70b9\u201d\u7684\u64cd\u4f5c\u3002\u5728\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u4e24\u79cd\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5b9e\u73b0\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002
"},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"\u7ed9\u5b9a\u4e00\u4e2a\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \u7684\u65e0\u5411\u56fe\uff0c\u5219\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u5f0f\u5982\u56fe 9-7 \u6240\u793a\u3002
- \u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
- \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
- \u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
- \u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868
vertices\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635adjMat\uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
\u56fe 9-7 \u00a0 \u90bb\u63a5\u77e9\u9635\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9
\u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_adjacency_matrix.py
graph_adjacency_matrix.cppclass GraphAdjMat:\n \"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n def __init__(self, vertices: list[int], edges: list[list[int]]):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n # \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n self.vertices: list[int] = []\n # \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n self.adj_mat: list[list[int]] = []\n # \u6dfb\u52a0\u9876\u70b9\n for val in vertices:\n self.add_vertex(val)\n # \u6dfb\u52a0\u8fb9\n # \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for e in edges:\n self.add_edge(e[0], e[1])\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n return len(self.vertices)\n\n def add_vertex(self, val: int):\n \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n n = self.size()\n # \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n self.vertices.append(val)\n # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n new_row = [0] * n\n self.adj_mat.append(new_row)\n # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for row in self.adj_mat:\n row.append(0)\n\n def remove_vertex(self, index: int):\n \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n if index >= self.size():\n raise IndexError()\n # \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n self.vertices.pop(index)\n # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n self.adj_mat.pop(index)\n # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for row in self.adj_mat:\n row.pop(index)\n\n def add_edge(self, i: int, j: int):\n \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\n raise IndexError()\n # \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n self.adj_mat[i][j] = 1\n self.adj_mat[j][i] = 1\n\n def remove_edge(self, i: int, j: int):\n \"\"\"\u5220\u9664\u8fb9\"\"\"\n # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\n raise IndexError()\n self.adj_mat[i][j] = 0\n self.adj_mat[j][i] = 0\n\n def print(self):\n \"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\n print(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\n print(\"\u90bb\u63a5\u77e9\u9635 =\")\n print_matrix(self.adj_mat)\n
graph_adjacency_matrix.java/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n vector<int> vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n vector<vector<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n GraphAdjMat(const vector<int> &vertices, const vector<vector<int>> &edges) {\n // \u6dfb\u52a0\u9876\u70b9\n for (int val : vertices) {\n addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (const vector<int> &edge : edges) {\n addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int size() const {\n return vertices.size();\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n void addVertex(int val) {\n int n = size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.push_back(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n adjMat.emplace_back(vector<int>(n, 0));\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (vector<int> &row : adjMat) {\n row.push_back(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n void removeVertex(int index) {\n if (index >= size()) {\n throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.erase(vertices.begin() + index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.erase(adjMat.begin() + index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (vector<int> &row : adjMat) {\n row.erase(row.begin() + index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n void addEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\n throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat[i][j] = 1;\n adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n void removeEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\n throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n }\n adjMat[i][j] = 0;\n adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n void print() {\n cout << \"\u9876\u70b9\u5217\u8868 = \";\n printVector(vertices);\n cout << \"\u90bb\u63a5\u77e9\u9635 =\" << endl;\n printVectorMatrix(adjMat);\n }\n};\n
graph_adjacency_matrix.cs/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n List<Integer> vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n List<List<Integer>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public GraphAdjMat(int[] vertices, int[][] edges) {\n this.vertices = new ArrayList<>();\n this.adjMat = new ArrayList<>();\n // \u6dfb\u52a0\u9876\u70b9\n for (int val : vertices) {\n addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (int[] e : edges) {\n addEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n public int size() {\n return vertices.size();\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public void addVertex(int val) {\n int n = size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.add(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n List<Integer> newRow = new ArrayList<>(n);\n for (int j = 0; j < n; j++) {\n newRow.add(0);\n }\n adjMat.add(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (List<Integer> row : adjMat) {\n row.add(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public void removeVertex(int index) {\n if (index >= size())\n throw new IndexOutOfBoundsException();\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.remove(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.remove(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (List<Integer> row : adjMat) {\n row.remove(index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n public void addEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\n throw new IndexOutOfBoundsException();\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat.get(i).set(j, 1);\n adjMat.get(j).set(i, 1);\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n public void removeEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\n throw new IndexOutOfBoundsException();\n adjMat.get(i).set(j, 0);\n adjMat.get(j).set(i, 0);\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n public void print() {\n System.out.print(\"\u9876\u70b9\u5217\u8868 = \");\n System.out.println(vertices);\n System.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\n PrintUtil.printMatrix(adjMat);\n }\n}\n
graph_adjacency_matrix.go/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n List<int> vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n List<List<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u51fd\u6570 */\n public GraphAdjMat(int[] vertices, int[][] edges) {\n this.vertices = [];\n this.adjMat = [];\n // \u6dfb\u52a0\u9876\u70b9\n foreach (int val in vertices) {\n AddVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n foreach (int[] e in edges) {\n AddEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int Size() {\n return vertices.Count;\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public void AddVertex(int val) {\n int n = Size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.Add(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n List<int> newRow = new(n);\n for (int j = 0; j < n; j++) {\n newRow.Add(0);\n }\n adjMat.Add(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n foreach (List<int> row in adjMat) {\n row.Add(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public void RemoveVertex(int index) {\n if (index >= Size())\n throw new IndexOutOfRangeException();\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.RemoveAt(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.RemoveAt(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n foreach (List<int> row in adjMat) {\n row.RemoveAt(index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n public void AddEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= Size() || j >= Size() || i == j)\n throw new IndexOutOfRangeException();\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat[i][j] = 1;\n adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n public void RemoveEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= Size() || j >= Size() || i == j)\n throw new IndexOutOfRangeException();\n adjMat[i][j] = 0;\n adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n public void Print() {\n Console.Write(\"\u9876\u70b9\u5217\u8868 = \");\n PrintUtil.PrintList(vertices);\n Console.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\n PrintUtil.PrintMatrix(adjMat);\n }\n}\n
graph_adjacency_matrix.swift/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n vertices []int\n // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n adjMat [][]int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n // \u6dfb\u52a0\u9876\u70b9\n n := len(vertices)\n adjMat := make([][]int, n)\n for i := range adjMat {\n adjMat[i] = make([]int, n)\n }\n // \u521d\u59cb\u5316\u56fe\n g := &graphAdjMat{\n vertices: vertices,\n adjMat: adjMat,\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for i := range edges {\n g.addEdge(edges[i][0], edges[i][1])\n }\n return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\n return len(g.vertices)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\n n := g.size()\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n g.vertices = append(g.vertices, val)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n newRow := make([]int, n)\n g.adjMat = append(g.adjMat, newRow)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for i := range g.adjMat {\n g.adjMat[i] = append(g.adjMat[i], 0)\n }\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\n if index >= g.size() {\n return\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for i := range g.adjMat {\n g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n }\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\n fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n g.adjMat[i][j] = 1\n g.adjMat[j][i] = 1\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\n fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n }\n g.adjMat[i][j] = 0\n g.adjMat[j][i] = 0\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\n fmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\n fmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\n for i := range g.adjMat {\n fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n }\n}\n
graph_adjacency_matrix.js/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n private var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n private var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init(vertices: [Int], edges: [[Int]]) {\n self.vertices = []\n adjMat = []\n // \u6dfb\u52a0\u9876\u70b9\n for val in vertices {\n addVertex(val: val)\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for e in edges {\n addEdge(i: e[0], j: e[1])\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n func size() -> Int {\n vertices.count\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n func addVertex(val: Int) {\n let n = size()\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.append(val)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n let newRow = Array(repeating: 0, count: n)\n adjMat.append(newRow)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for i in adjMat.indices {\n adjMat[i].append(0)\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n func removeVertex(index: Int) {\n if index >= size() {\n fatalError(\"\u8d8a\u754c\")\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.remove(at: index)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.remove(at: index)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for i in adjMat.indices {\n adjMat[i].remove(at: index)\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n func addEdge(i: Int, j: Int) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 || j < 0 || i >= size() || j >= size() || i == j {\n fatalError(\"\u8d8a\u754c\")\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat[i][j] = 1\n adjMat[j][i] = 1\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n func removeEdge(i: Int, j: Int) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 || j < 0 || i >= size() || j >= size() || i == j {\n fatalError(\"\u8d8a\u754c\")\n }\n adjMat[i][j] = 0\n adjMat[j][i] = 0\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n func print() {\n Swift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\n Swift.print(vertices)\n Swift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\n PrintUtil.printMatrix(matrix: adjMat)\n }\n}\n
graph_adjacency_matrix.ts/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u51fd\u6570 */\n constructor(vertices, edges) {\n this.vertices = [];\n this.adjMat = [];\n // \u6dfb\u52a0\u9876\u70b9\n for (const val of vertices) {\n this.addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (const e of edges) {\n this.addEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n size() {\n return this.vertices.length;\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n addVertex(val) {\n const n = this.size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n this.vertices.push(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n const newRow = [];\n for (let j = 0; j < n; j++) {\n newRow.push(0);\n }\n this.adjMat.push(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (const row of this.adjMat) {\n row.push(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n removeVertex(index) {\n if (index >= this.size()) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n this.vertices.splice(index, 1);\n\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n this.adjMat.splice(index, 1);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (const row of this.adjMat) {\n row.splice(index, 1);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n addEdge(i, j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n this.adjMat[i][j] = 1;\n this.adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n removeEdge(i, j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n this.adjMat[i][j] = 0;\n this.adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n print() {\n console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n }\n}\n
graph_adjacency_matrix.dart/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n vertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n adjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u51fd\u6570 */\n constructor(vertices: number[], edges: number[][]) {\n this.vertices = [];\n this.adjMat = [];\n // \u6dfb\u52a0\u9876\u70b9\n for (const val of vertices) {\n this.addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (const e of edges) {\n this.addEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n size(): number {\n return this.vertices.length;\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n addVertex(val: number): void {\n const n: number = this.size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n this.vertices.push(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n const newRow: number[] = [];\n for (let j: number = 0; j < n; j++) {\n newRow.push(0);\n }\n this.adjMat.push(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (const row of this.adjMat) {\n row.push(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n removeVertex(index: number): void {\n if (index >= this.size()) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n this.vertices.splice(index, 1);\n\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n this.adjMat.splice(index, 1);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (const row of this.adjMat) {\n row.splice(index, 1);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n addEdge(i: number, j: number): void {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n this.adjMat[i][j] = 1;\n this.adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n removeEdge(i: number, j: number): void {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n this.adjMat[i][j] = 0;\n this.adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n print(): void {\n console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n }\n}\n
graph_adjacency_matrix.rs/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n List<int> vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n List<List<int>> adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u65b9\u6cd5 */\n GraphAdjMat(List<int> vertices, List<List<int>> edges) {\n this.vertices = [];\n this.adjMat = [];\n // \u6dfb\u52a0\u9876\u70b9\n for (int val in vertices) {\n addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (List<int> e in edges) {\n addEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int size() {\n return vertices.length;\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n void addVertex(int val) {\n int n = size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.add(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n List<int> newRow = List.filled(n, 0, growable: true);\n adjMat.add(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (List<int> row in adjMat) {\n row.add(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n void removeVertex(int index) {\n if (index >= size()) {\n throw IndexError;\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.removeAt(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.removeAt(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (List<int> row in adjMat) {\n row.removeAt(index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n void addEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\n throw IndexError;\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat[i][j] = 1;\n adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n void removeEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\n throw IndexError;\n }\n adjMat[i][j] = 0;\n adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n void printAdjMat() {\n print(\"\u9876\u70b9\u5217\u8868 = $vertices\");\n print(\"\u90bb\u63a5\u77e9\u9635 = \");\n printMatrix(adjMat);\n }\n}\n
graph_adjacency_matrix.c/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjMat {\n // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n pub vertices: Vec<i32>,\n // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n pub adj_mat: Vec<Vec<i32>>,\n}\n\nimpl GraphAdjMat {\n /* \u6784\u9020\u65b9\u6cd5 */\n pub fn new(vertices: Vec<i32>, edges: Vec<[usize; 2]>) -> Self {\n let mut graph = GraphAdjMat {\n vertices: vec![],\n adj_mat: vec![],\n };\n // \u6dfb\u52a0\u9876\u70b9\n for val in vertices {\n graph.add_vertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for edge in edges {\n graph.add_edge(edge[0], edge[1])\n }\n\n graph\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n pub fn size(&self) -> usize {\n self.vertices.len()\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n pub fn add_vertex(&mut self, val: i32) {\n let n = self.size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n self.vertices.push(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n self.adj_mat.push(vec![0; n]);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for row in &mut self.adj_mat {\n row.push(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n pub fn remove_vertex(&mut self, index: usize) {\n if index >= self.size() {\n panic!(\"index error\")\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n self.vertices.remove(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n self.adj_mat.remove(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for row in &mut self.adj_mat {\n row.remove(index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n pub fn add_edge(&mut self, i: usize, j: usize) {\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i >= self.size() || j >= self.size() || i == j {\n panic!(\"index error\")\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n self.adj_mat[i][j] = 1;\n self.adj_mat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n pub fn remove_edge(&mut self, i: usize, j: usize) {\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i >= self.size() || j >= self.size() || i == j {\n panic!(\"index error\")\n }\n self.adj_mat[i][j] = 0;\n self.adj_mat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n pub fn print(&self) {\n println!(\"\u9876\u70b9\u5217\u8868 = {:?}\", self.vertices);\n println!(\"\u90bb\u63a5\u77e9\u9635 =\");\n println!(\"[\");\n for row in &self.adj_mat {\n println!(\" {:?},\", row);\n }\n println!(\"]\")\n }\n}\n
graph_adjacency_matrix.zig/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7ed3\u6784\u4f53 */\ntypedef struct {\n int vertices[MAX_SIZE];\n int adjMat[MAX_SIZE][MAX_SIZE];\n int size;\n} GraphAdjMat;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjMat *newGraphAdjMat() {\n GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n graph->size = 0;\n for (int i = 0; i < MAX_SIZE; i++) {\n for (int j = 0; j < MAX_SIZE; j++) {\n graph->adjMat[i][j] = 0;\n }\n }\n return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n free(graph);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjMat *graph, int val) {\n if (graph->size == MAX_SIZE) {\n fprintf(stderr, \"\u56fe\u7684\u9876\u70b9\u6570\u91cf\u5df2\u8fbe\u6700\u5927\u503c\\n\");\n return;\n }\n // \u6dfb\u52a0\u7b2c n \u4e2a\u9876\u70b9\uff0c\u5e76\u5c06\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\n int n = graph->size;\n graph->vertices[n] = val;\n for (int i = 0; i <= n; i++) {\n graph->adjMat[n][i] = graph->adjMat[i][n] = 0;\n }\n graph->size++;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n if (index < 0 || index >= graph->size) {\n fprintf(stderr, \"\u9876\u70b9\u7d22\u5f15\u8d8a\u754c\\n\");\n return;\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n for (int i = index; i < graph->size - 1; i++) {\n graph->vertices[i] = graph->vertices[i + 1];\n }\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n for (int i = index; i < graph->size - 1; i++) {\n for (int j = 0; j < graph->size; j++) {\n graph->adjMat[i][j] = graph->adjMat[i + 1][j];\n }\n }\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (int i = 0; i < graph->size; i++) {\n for (int j = index; j < graph->size - 1; j++) {\n graph->adjMat[i][j] = graph->adjMat[i][j + 1];\n }\n }\n graph->size--;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n if (i < 0 || j < 0 || i >= graph->size || j >= graph->size || i == j) {\n fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n return;\n }\n graph->adjMat[i][j] = 1;\n graph->adjMat[j][i] = 1;\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n if (i < 0 || j < 0 || i >= graph->size || j >= graph->size || i == j) {\n fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n return;\n }\n graph->adjMat[i][j] = 0;\n graph->adjMat[j][i] = 0;\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n printf(\"\u9876\u70b9\u5217\u8868 = \");\n printArray(graph->vertices, graph->size);\n printf(\"\u90bb\u63a5\u77e9\u9635 =\\n\");\n for (int i = 0; i < graph->size; i++) {\n printArray(graph->adjMat[i], graph->size);\n }\n}\n
"},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"[class]{GraphAdjMat}-[func]{}\n\u8bbe\u65e0\u5411\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\)\u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u53ef\u6839\u636e\u56fe 9-8 \u6240\u793a\u7684\u65b9\u6cd5\u5b9e\u73b0\u5404\u79cd\u64cd\u4f5c\u3002
- \u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u672b\u5c3e\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
- \u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u627e\u5e76\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
- \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\uff0c\u5e76\u5c06\u65b0\u589e\u9876\u70b9\u4f5c\u4e3a\u94fe\u8868\u5934\u8282\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
- \u5220\u9664\u9876\u70b9\uff1a\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
- \u521d\u59cb\u5316\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u521b\u5efa \\(n\\) \u4e2a\u9876\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
\u56fe 9-8 \u00a0 \u90bb\u63a5\u8868\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9
\u4ee5\u4e0b\u662f\u90bb\u63a5\u8868\u7684\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u6bd4\u56fe 9-8 \uff0c\u5b9e\u9645\u4ee3\u7801\u6709\u4ee5\u4e0b\u4e0d\u540c\u3002
- \u4e3a\u4e86\u65b9\u4fbf\u6dfb\u52a0\u4e0e\u5220\u9664\u9876\u70b9\uff0c\u4ee5\u53ca\u7b80\u5316\u4ee3\u7801\uff0c\u6211\u4eec\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u6765\u4ee3\u66ff\u94fe\u8868\u3002
- \u4f7f\u7528\u54c8\u5e0c\u8868\u6765\u5b58\u50a8\u90bb\u63a5\u8868\uff0c
key\u4e3a\u9876\u70b9\u5b9e\u4f8b\uff0cvalue\u4e3a\u8be5\u9876\u70b9\u7684\u90bb\u63a5\u9876\u70b9\u5217\u8868\uff08\u94fe\u8868\uff09\u3002
\u53e6\u5916\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528
PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_adjacency_list.pyVertex\u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u662f\uff1a\u5982\u679c\u4e0e\u90bb\u63a5\u77e9\u9635\u4e00\u6837\uff0c\u7528\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u5047\u8bbe\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u6240\u6709\u5927\u4e8e \\(i\\) \u7684\u7d22\u5f15\u5168\u90e8\u51cf \\(1\\) \uff0c\u6548\u7387\u5f88\u4f4e\u3002\u800c\u5982\u679c\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684Vertex\u5b9e\u4f8b\uff0c\u5220\u9664\u67d0\u4e00\u9876\u70b9\u4e4b\u540e\u5c31\u65e0\u987b\u6539\u52a8\u5176\u4ed6\u9876\u70b9\u4e86\u3002
graph_adjacency_list.cppclass GraphAdjList:\n \"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n def __init__(self, edges: list[list[Vertex]]):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n # \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n self.adj_list = dict[Vertex, list[Vertex]]()\n # \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for edge in edges:\n self.add_vertex(edge[0])\n self.add_vertex(edge[1])\n self.add_edge(edge[0], edge[1])\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n return len(self.adj_list)\n\n def add_edge(self, vet1: Vertex, vet2: Vertex):\n \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n raise ValueError()\n # \u6dfb\u52a0\u8fb9 vet1 - vet2\n self.adj_list[vet1].append(vet2)\n self.adj_list[vet2].append(vet1)\n\n def remove_edge(self, vet1: Vertex, vet2: Vertex):\n \"\"\"\u5220\u9664\u8fb9\"\"\"\n if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n raise ValueError()\n # \u5220\u9664\u8fb9 vet1 - vet2\n self.adj_list[vet1].remove(vet2)\n self.adj_list[vet2].remove(vet1)\n\n def add_vertex(self, vet: Vertex):\n \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n if vet in self.adj_list:\n return\n # \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n self.adj_list[vet] = []\n\n def remove_vertex(self, vet: Vertex):\n \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n if vet not in self.adj_list:\n raise ValueError()\n # \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n self.adj_list.pop(vet)\n # \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for vertex in self.adj_list:\n if vet in self.adj_list[vertex]:\n self.adj_list[vertex].remove(vet)\n\n def print(self):\n \"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\n print(\"\u90bb\u63a5\u8868 =\")\n for vertex in self.adj_list:\n tmp = [v.val for v in self.adj_list[vertex]]\n print(f\"{vertex.val}: {tmp},\")\n
graph_adjacency_list.java/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n public:\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n unordered_map<Vertex *, vector<Vertex *>> adjList;\n\n /* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\n void remove(vector<Vertex *> &vec, Vertex *vet) {\n for (int i = 0; i < vec.size(); i++) {\n if (vec[i] == vet) {\n vec.erase(vec.begin() + i);\n break;\n }\n }\n }\n\n /* \u6784\u9020\u65b9\u6cd5 */\n GraphAdjList(const vector<vector<Vertex *>> &edges) {\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for (const vector<Vertex *> &edge : edges) {\n addVertex(edge[0]);\n addVertex(edge[1]);\n addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int size() {\n return adjList.size();\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n void addEdge(Vertex *vet1, Vertex *vet2) {\n if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList[vet1].push_back(vet2);\n adjList[vet2].push_back(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n void removeEdge(Vertex *vet1, Vertex *vet2) {\n if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n // \u5220\u9664\u8fb9 vet1 - vet2\n remove(adjList[vet1], vet2);\n remove(adjList[vet2], vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n void addVertex(Vertex *vet) {\n if (adjList.count(vet))\n return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList[vet] = vector<Vertex *>();\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n void removeVertex(Vertex *vet) {\n if (!adjList.count(vet))\n throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.erase(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (auto &adj : adjList) {\n remove(adj.second, vet);\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n void print() {\n cout << \"\u90bb\u63a5\u8868 =\" << endl;\n for (auto &adj : adjList) {\n const auto &key = adj.first;\n const auto &vec = adj.second;\n cout << key->val << \": \";\n printVector(vetsToVals(vec));\n }\n }\n};\n
graph_adjacency_list.cs/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n Map<Vertex, List<Vertex>> adjList;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public GraphAdjList(Vertex[][] edges) {\n this.adjList = new HashMap<>();\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for (Vertex[] edge : edges) {\n addVertex(edge[0]);\n addVertex(edge[1]);\n addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n public int size() {\n return adjList.size();\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n public void addEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n throw new IllegalArgumentException();\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList.get(vet1).add(vet2);\n adjList.get(vet2).add(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n public void removeEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n throw new IllegalArgumentException();\n // \u5220\u9664\u8fb9 vet1 - vet2\n adjList.get(vet1).remove(vet2);\n adjList.get(vet2).remove(vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public void addVertex(Vertex vet) {\n if (adjList.containsKey(vet))\n return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList.put(vet, new ArrayList<>());\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public void removeVertex(Vertex vet) {\n if (!adjList.containsKey(vet))\n throw new IllegalArgumentException();\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.remove(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (List<Vertex> list : adjList.values()) {\n list.remove(vet);\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n public void print() {\n System.out.println(\"\u90bb\u63a5\u8868 =\");\n for (Map.Entry<Vertex, List<Vertex>> pair : adjList.entrySet()) {\n List<Integer> tmp = new ArrayList<>();\n for (Vertex vertex : pair.getValue())\n tmp.add(vertex.val);\n System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n }\n }\n}\n
graph_adjacency_list.go/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n public Dictionary<Vertex, List<Vertex>> adjList;\n\n /* \u6784\u9020\u51fd\u6570 */\n public GraphAdjList(Vertex[][] edges) {\n adjList = [];\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n foreach (Vertex[] edge in edges) {\n AddVertex(edge[0]);\n AddVertex(edge[1]);\n AddEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int Size() {\n return adjList.Count;\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n public void AddEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n throw new InvalidOperationException();\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList[vet1].Add(vet2);\n adjList[vet2].Add(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n public void RemoveEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n throw new InvalidOperationException();\n // \u5220\u9664\u8fb9 vet1 - vet2\n adjList[vet1].Remove(vet2);\n adjList[vet2].Remove(vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public void AddVertex(Vertex vet) {\n if (adjList.ContainsKey(vet))\n return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList.Add(vet, []);\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public void RemoveVertex(Vertex vet) {\n if (!adjList.ContainsKey(vet))\n throw new InvalidOperationException();\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.Remove(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n foreach (List<Vertex> list in adjList.Values) {\n list.Remove(vet);\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n public void Print() {\n Console.WriteLine(\"\u90bb\u63a5\u8868 =\");\n foreach (KeyValuePair<Vertex, List<Vertex>> pair in adjList) {\n List<int> tmp = [];\n foreach (Vertex vertex in pair.Value)\n tmp.Add(vertex.val);\n Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n }\n }\n}\n
graph_adjacency_list.swift/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n adjList map[Vertex][]Vertex\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n g := &graphAdjList{\n adjList: make(map[Vertex][]Vertex),\n }\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for _, edge := range edges {\n g.addVertex(edge[0])\n g.addVertex(edge[1])\n g.addEdge(edge[0], edge[1])\n }\n return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\n return len(g.adjList)\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n _, ok1 := g.adjList[vet1]\n _, ok2 := g.adjList[vet2]\n if !ok1 || !ok2 || vet1 == vet2 {\n panic(\"error\")\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\n g.adjList[vet1] = append(g.adjList[vet1], vet2)\n g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n _, ok1 := g.adjList[vet1]\n _, ok2 := g.adjList[vet2]\n if !ok1 || !ok2 || vet1 == vet2 {\n panic(\"error\")\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n _, ok := g.adjList[vet]\n if ok {\n return\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n _, ok := g.adjList[vet]\n if !ok {\n panic(\"error\")\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n delete(g.adjList, vet)\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for v, list := range g.adjList {\n g.adjList[v] = DeleteSliceElms(list, vet)\n }\n}\n\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\n var builder strings.Builder\n fmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\n for k, v := range g.adjList {\n builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n for _, vet := range v {\n builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n }\n fmt.Println(builder.String())\n builder.Reset()\n }\n}\n
graph_adjacency_list.js/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n public private(set) var adjList: [Vertex: [Vertex]]\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public init(edges: [[Vertex]]) {\n adjList = [:]\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for edge in edges {\n addVertex(vet: edge[0])\n addVertex(vet: edge[1])\n addEdge(vet1: edge[0], vet2: edge[1])\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n public func size() -> Int {\n adjList.count\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n public func addEdge(vet1: Vertex, vet2: Vertex) {\n if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n fatalError(\"\u53c2\u6570\u9519\u8bef\")\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList[vet1]?.append(vet2)\n adjList[vet2]?.append(vet1)\n }\n\n /* \u5220\u9664\u8fb9 */\n public func removeEdge(vet1: Vertex, vet2: Vertex) {\n if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n fatalError(\"\u53c2\u6570\u9519\u8bef\")\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n adjList[vet1]?.removeAll(where: { $0 == vet2 })\n adjList[vet2]?.removeAll(where: { $0 == vet1 })\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public func addVertex(vet: Vertex) {\n if adjList[vet] != nil {\n return\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList[vet] = []\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public func removeVertex(vet: Vertex) {\n if adjList[vet] == nil {\n fatalError(\"\u53c2\u6570\u9519\u8bef\")\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.removeValue(forKey: vet)\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for key in adjList.keys {\n adjList[key]?.removeAll(where: { $0 == vet })\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n public func print() {\n Swift.print(\"\u90bb\u63a5\u8868 =\")\n for pair in adjList {\n var tmp: [Int] = []\n for vertex in pair.value {\n tmp.append(vertex.val)\n }\n Swift.print(\"\\(pair.key.val): \\(tmp),\")\n }\n }\n}\n
graph_adjacency_list.ts/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n adjList;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(edges) {\n this.adjList = new Map();\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for (const edge of edges) {\n this.addVertex(edge[0]);\n this.addVertex(edge[1]);\n this.addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n size() {\n return this.adjList.size;\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n addEdge(vet1, vet2) {\n if (\n !this.adjList.has(vet1) ||\n !this.adjList.has(vet2) ||\n vet1 === vet2\n ) {\n throw new Error('Illegal Argument Exception');\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n this.adjList.get(vet1).push(vet2);\n this.adjList.get(vet2).push(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n removeEdge(vet1, vet2) {\n if (\n !this.adjList.has(vet1) ||\n !this.adjList.has(vet2) ||\n vet1 === vet2\n ) {\n throw new Error('Illegal Argument Exception');\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n addVertex(vet) {\n if (this.adjList.has(vet)) return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n this.adjList.set(vet, []);\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n removeVertex(vet) {\n if (!this.adjList.has(vet)) {\n throw new Error('Illegal Argument Exception');\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n this.adjList.delete(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (const set of this.adjList.values()) {\n const index = set.indexOf(vet);\n if (index > -1) {\n set.splice(index, 1);\n }\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n print() {\n console.log('\u90bb\u63a5\u8868 =');\n for (const [key, value] of this.adjList) {\n const tmp = [];\n for (const vertex of value) {\n tmp.push(vertex.val);\n }\n console.log(key.val + ': ' + tmp.join());\n }\n }\n}\n
graph_adjacency_list.dart/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n adjList: Map<Vertex, Vertex[]>;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(edges: Vertex[][]) {\n this.adjList = new Map();\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for (const edge of edges) {\n this.addVertex(edge[0]);\n this.addVertex(edge[1]);\n this.addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n size(): number {\n return this.adjList.size;\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n addEdge(vet1: Vertex, vet2: Vertex): void {\n if (\n !this.adjList.has(vet1) ||\n !this.adjList.has(vet2) ||\n vet1 === vet2\n ) {\n throw new Error('Illegal Argument Exception');\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n this.adjList.get(vet1).push(vet2);\n this.adjList.get(vet2).push(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n removeEdge(vet1: Vertex, vet2: Vertex): void {\n if (\n !this.adjList.has(vet1) ||\n !this.adjList.has(vet2) ||\n vet1 === vet2\n ) {\n throw new Error('Illegal Argument Exception');\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n addVertex(vet: Vertex): void {\n if (this.adjList.has(vet)) return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n this.adjList.set(vet, []);\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n removeVertex(vet: Vertex): void {\n if (!this.adjList.has(vet)) {\n throw new Error('Illegal Argument Exception');\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n this.adjList.delete(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (const set of this.adjList.values()) {\n const index: number = set.indexOf(vet);\n if (index > -1) {\n set.splice(index, 1);\n }\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n print(): void {\n console.log('\u90bb\u63a5\u8868 =');\n for (const [key, value] of this.adjList.entries()) {\n const tmp = [];\n for (const vertex of value) {\n tmp.push(vertex.val);\n }\n console.log(key.val + ': ' + tmp.join());\n }\n }\n}\n
graph_adjacency_list.rs/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n Map<Vertex, List<Vertex>> adjList = {};\n\n /* \u6784\u9020\u65b9\u6cd5 */\n GraphAdjList(List<List<Vertex>> edges) {\n for (List<Vertex> edge in edges) {\n addVertex(edge[0]);\n addVertex(edge[1]);\n addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int size() {\n return adjList.length;\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n void addEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.containsKey(vet1) ||\n !adjList.containsKey(vet2) ||\n vet1 == vet2) {\n throw ArgumentError;\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList[vet1]!.add(vet2);\n adjList[vet2]!.add(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n void removeEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.containsKey(vet1) ||\n !adjList.containsKey(vet2) ||\n vet1 == vet2) {\n throw ArgumentError;\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n adjList[vet1]!.remove(vet2);\n adjList[vet2]!.remove(vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n void addVertex(Vertex vet) {\n if (adjList.containsKey(vet)) return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList[vet] = [];\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n void removeVertex(Vertex vet) {\n if (!adjList.containsKey(vet)) {\n throw ArgumentError;\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.remove(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n adjList.forEach((key, value) {\n value.remove(vet);\n });\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n void printAdjList() {\n print(\"\u90bb\u63a5\u8868 =\");\n adjList.forEach((key, value) {\n List<int> tmp = [];\n for (Vertex vertex in value) {\n tmp.add(vertex.val);\n }\n print(\"${key.val}: $tmp,\");\n });\n }\n}\n
graph_adjacency_list.c/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n pub adj_list: HashMap<Vertex, Vec<Vertex>>,\n}\n\nimpl GraphAdjList {\n /* \u6784\u9020\u65b9\u6cd5 */\n pub fn new(edges: Vec<[Vertex; 2]>) -> Self {\n let mut graph = GraphAdjList {\n adj_list: HashMap::new(),\n };\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for edge in edges {\n graph.add_vertex(edge[0]);\n graph.add_vertex(edge[1]);\n graph.add_edge(edge[0], edge[1]);\n }\n\n graph\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n #[allow(unused)]\n pub fn size(&self) -> usize {\n self.adj_list.len()\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n pub fn add_edge(&mut self, vet1: Vertex, vet2: Vertex) {\n if !self.adj_list.contains_key(&vet1) || !self.adj_list.contains_key(&vet2) || vet1 == vet2\n {\n panic!(\"value error\");\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n self.adj_list.get_mut(&vet1).unwrap().push(vet2);\n self.adj_list.get_mut(&vet2).unwrap().push(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n #[allow(unused)]\n pub fn remove_edge(&mut self, vet1: Vertex, vet2: Vertex) {\n if !self.adj_list.contains_key(&vet1) || !self.adj_list.contains_key(&vet2) || vet1 == vet2\n {\n panic!(\"value error\");\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n self.adj_list\n .get_mut(&vet1)\n .unwrap()\n .retain(|&vet| vet != vet2);\n self.adj_list\n .get_mut(&vet2)\n .unwrap()\n .retain(|&vet| vet != vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n pub fn add_vertex(&mut self, vet: Vertex) {\n if self.adj_list.contains_key(&vet) {\n return;\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n self.adj_list.insert(vet, vec![]);\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n #[allow(unused)]\n pub fn remove_vertex(&mut self, vet: Vertex) {\n if !self.adj_list.contains_key(&vet) {\n panic!(\"value error\");\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n self.adj_list.remove(&vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for list in self.adj_list.values_mut() {\n list.retain(|&v| v != vet);\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n pub fn print(&self) {\n println!(\"\u90bb\u63a5\u8868 =\");\n for (vertex, list) in &self.adj_list {\n let list = list.iter().map(|vertex| vertex.val).collect::<Vec<i32>>();\n println!(\"{}: {:?},\", vertex.val, list);\n }\n }\n}\n
graph_adjacency_list.zig/* \u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct AdjListNode {\n Vertex *vertex; // \u9876\u70b9\n struct AdjListNode *next; // \u540e\u7ee7\u8282\u70b9\n} AdjListNode;\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n for (int i = 0; i < graph->size; i++) {\n if (graph->heads[i]->vertex == vet) {\n return graph->heads[i];\n }\n }\n return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n node->vertex = vet;\n // \u5934\u63d2\u6cd5\n node->next = head->next;\n head->next = node;\n}\n\n/* \u5220\u9664\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n AdjListNode *pre = head;\n AdjListNode *cur = head->next;\n // \u5728\u94fe\u8868\u4e2d\u641c\u7d22 vet \u5bf9\u5e94\u8282\u70b9\n while (cur != NULL && cur->vertex != vet) {\n pre = cur;\n cur = cur->next;\n }\n if (cur == NULL)\n return;\n // \u5c06 vet \u5bf9\u5e94\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\n pre->next = cur->next;\n // \u91ca\u653e\u5185\u5b58\n free(cur);\n}\n\n/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntypedef struct {\n AdjListNode *heads[MAX_SIZE]; // \u8282\u70b9\u6570\u7ec4\n int size; // \u8282\u70b9\u6570\u91cf\n} GraphAdjList;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjList *newGraphAdjList() {\n GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n if (!graph) {\n return NULL;\n }\n graph->size = 0;\n for (int i = 0; i < MAX_SIZE; i++) {\n graph->heads[i] = NULL;\n }\n return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjList(GraphAdjList *graph) {\n for (int i = 0; i < graph->size; i++) {\n AdjListNode *cur = graph->heads[i];\n while (cur != NULL) {\n AdjListNode *next = cur->next;\n if (cur != graph->heads[i]) {\n free(cur);\n }\n cur = next;\n }\n free(graph->heads[i]->vertex);\n free(graph->heads[i]);\n }\n free(graph);\n}\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n for (int i = 0; i < graph->size; i++) {\n if (graph->heads[i]->vertex == vet) {\n return graph->heads[i];\n }\n }\n return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n AdjListNode *head1 = findNode(graph, vet1);\n AdjListNode *head2 = findNode(graph, vet2);\n assert(head1 != NULL && head2 != NULL && head1 != head2);\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n addEdgeHelper(head1, vet2);\n addEdgeHelper(head2, vet1);\n}\n\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n AdjListNode *head1 = findNode(graph, vet1);\n AdjListNode *head2 = findNode(graph, vet2);\n assert(head1 != NULL && head2 != NULL);\n // \u5220\u9664\u8fb9 vet1 - vet2\n removeEdgeHelper(head1, head2->vertex);\n removeEdgeHelper(head2, head1->vertex);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n assert(graph != NULL && graph->size < MAX_SIZE);\n AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n head->vertex = vet;\n head->next = NULL;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n graph->heads[graph->size++] = head;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n AdjListNode *node = findNode(graph, vet);\n assert(node != NULL);\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n AdjListNode *cur = node, *pre = NULL;\n while (cur) {\n pre = cur;\n cur = cur->next;\n free(pre);\n }\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (int i = 0; i < graph->size; i++) {\n cur = graph->heads[i];\n pre = NULL;\n while (cur) {\n pre = cur;\n cur = cur->next;\n if (cur && cur->vertex == vet) {\n pre->next = cur->next;\n free(cur);\n break;\n }\n }\n }\n // \u5c06\u8be5\u9876\u70b9\u4e4b\u540e\u7684\u9876\u70b9\u5411\u524d\u79fb\u52a8\uff0c\u4ee5\u586b\u8865\u7a7a\u7f3a\n int i;\n for (i = 0; i < graph->size; i++) {\n if (graph->heads[i] == node)\n break;\n }\n for (int j = i; j < graph->size - 1; j++) {\n graph->heads[j] = graph->heads[j + 1];\n }\n graph->size--;\n free(vet);\n}\n
"},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3 \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"[class]{GraphAdjList}-[func]{}\n\u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u8868 9-2 \u5bf9\u6bd4\u4e86\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u3002
\u8868 9-2 \u00a0 \u90bb\u63a5\u77e9\u9635\u4e0e\u90bb\u63a5\u8868\u5bf9\u6bd4
\u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\)\u89c2\u5bdf\u8868 9-2 \uff0c\u4f3c\u4e4e\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u6548\u7387\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7efc\u5408\u6765\u770b\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u539f\u5219\uff0c\u800c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u7684\u539f\u5219\u3002
"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 \u56fe\u7684\u904d\u5386","text":"\u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\u3002\u663e\u7136\uff0c\u6811\u7684\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u7684\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u79cd\u7279\u4f8b\u3002
\u56fe\u548c\u6811\u90fd\u9700\u8981\u5e94\u7528\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u53ef\u5206\u4e3a\u4e24\u79cd\uff1a\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal\u300d\u3002\u5b83\u4eec\u4e5f\u5e38\u88ab\u79f0\u4e3a\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 breadth-first search\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22 depth-first search\u300d\uff0c\u7b80\u79f0 BFS \u548c DFS \u3002
"},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1 \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u67d0\u4e2a\u8282\u70b9\u51fa\u53d1\uff0c\u59cb\u7ec8\u4f18\u5148\u8bbf\u95ee\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5982\u56fe 9-9 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u9996\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u7136\u540e\u904d\u5386\u4e0b\u4e00\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6bd5\u3002
\u56fe 9-9 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386
"},{"location":"chapter_graph/graph_traversal/#1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"BFS \u901a\u5e38\u501f\u52a9\u961f\u5217\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u7684\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u5f02\u66f2\u540c\u5de5\u3002
- \u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9
startVet\u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\u3002 - \u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u7136\u540e\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\u3002
- \u5faa\u73af\u6b65\u9aa4
2.\uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u88ab\u8bbf\u95ee\u5b8c\u6bd5\u540e\u7ed3\u675f\u3002
\u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_bfs.pyvisited\u6765\u8bb0\u5f55\u54ea\u4e9b\u8282\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002
graph_bfs.cppdef graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]:\n \"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS\"\"\"\n # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n res = []\n # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n visited = set[Vertex]([start_vet])\n # \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n que = deque[Vertex]([start_vet])\n # \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while len(que) > 0:\n vet = que.popleft() # \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.append(vet) # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for adj_vet in graph.adj_list[vet]:\n if adj_vet in visited:\n continue # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n que.append(adj_vet) # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.add(adj_vet) # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n # \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res\n
graph_bfs.java/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex *> graphBFS(GraphAdjList &graph, Vertex *startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n vector<Vertex *> res;\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n unordered_set<Vertex *> visited = {startVet};\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n queue<Vertex *> que;\n que.push(startVet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (!que.empty()) {\n Vertex *vet = que.front();\n que.pop(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (auto adjVet : graph.adjList[vet]) {\n if (visited.count(adjVet))\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.cs/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = new ArrayList<>();\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n Set<Vertex> visited = new HashSet<>();\n visited.add(startVet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n Queue<Vertex> que = new LinkedList<>();\n que.offer(startVet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (!que.isEmpty()) {\n Vertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex adjVet : graph.adjList.get(vet)) {\n if (visited.contains(adjVet))\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n que.offer(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.go/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> GraphBFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n HashSet<Vertex> visited = [startVet];\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n Queue<Vertex> que = new();\n que.Enqueue(startVet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (que.Count > 0) {\n Vertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.Add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n foreach (Vertex adjVet in graph.adjList[vet]) {\n if (visited.Contains(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.Enqueue(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.Add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.swift/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n res := make([]Vertex, 0)\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n visited := make(map[Vertex]struct{})\n visited[startVet] = struct{}{}\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\n queue := make([]Vertex, 0)\n queue = append(queue, startVet)\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n for len(queue) > 0 {\n // \u961f\u9996\u9876\u70b9\u51fa\u961f\n vet := queue[0]\n queue = queue[1:]\n // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n res = append(res, vet)\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for _, adjVet := range g.adjList[vet] {\n _, isExist := visited[adjVet]\n // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n if !isExist {\n queue = append(queue, adjVet)\n visited[adjVet] = struct{}{}\n }\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res\n}\n
graph_bfs.js/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n var res: [Vertex] = []\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n var visited: Set<Vertex> = [startVet]\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n var que: [Vertex] = [startVet]\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while !que.isEmpty {\n let vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for adjVet in graph.adjList[vet] ?? [] {\n if visited.contains(adjVet) {\n continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res\n}\n
graph_bfs.ts/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n const res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n const visited = new Set();\n visited.add(startVet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n const que = [startVet];\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (que.length) {\n const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (const adjVet of graph.adjList.get(vet) ?? []) {\n if (visited.has(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.dart/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n const res: Vertex[] = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n const visited: Set<Vertex> = new Set();\n visited.add(startVet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n const que = [startVet];\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (que.length) {\n const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (const adjVet of graph.adjList.get(vet) ?? []) {\n if (visited.has(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\n visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.rs/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n // \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n Set<Vertex> visited = {};\n visited.add(startVet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n Queue<Vertex> que = Queue();\n que.add(startVet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (que.isNotEmpty) {\n Vertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex adjVet in graph.adjList[vet]!) {\n if (visited.contains(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.c/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> Vec<Vertex> {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n let mut res = vec![];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n let mut visited = HashSet::new();\n visited.insert(start_vet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n let mut que = VecDeque::new();\n que.push_back(start_vet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while !que.is_empty() {\n let vet = que.pop_front().unwrap(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n if let Some(adj_vets) = graph.adj_list.get(&vet) {\n for &adj_vet in adj_vets {\n if visited.contains(&adj_vet) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.push_back(adj_vet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.insert(adj_vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n res\n}\n
graph_bfs.zig/* \u8282\u70b9\u961f\u5217\u7ed3\u6784\u4f53 */\ntypedef struct {\n Vertex *vertices[MAX_SIZE];\n int front, rear, size;\n} Queue;\n\n/* \u6784\u9020\u51fd\u6570 */\nQueue *newQueue() {\n Queue *q = (Queue *)malloc(sizeof(Queue));\n q->front = q->rear = q->size = 0;\n return q;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nint isEmpty(Queue *q) {\n return q->size == 0;\n}\n\n/* \u5165\u961f\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\n q->vertices[q->rear] = vet;\n q->rear = (q->rear + 1) % MAX_SIZE;\n q->size++;\n}\n\n/* \u51fa\u961f\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\n Vertex *vet = q->vertices[q->front];\n q->front = (q->front + 1) % MAX_SIZE;\n q->size--;\n return vet;\n}\n\n/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n for (int i = 0; i < size; i++) {\n if (visited[i] == vet)\n return 1;\n }\n return 0;\n}\n\n/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n Queue *queue = newQueue();\n enqueue(queue, startVet);\n visited[(*visitedSize)++] = startVet;\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (!isEmpty(queue)) {\n Vertex *vet = dequeue(queue); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res[(*resSize)++] = vet; // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n AdjListNode *node = findNode(graph, vet);\n while (node != NULL) {\n // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n if (!isVisited(visited, *visitedSize, node->vertex)) {\n enqueue(queue, node->vertex); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited[(*visitedSize)++] = node->vertex; // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n node = node->next;\n }\n }\n // \u91ca\u653e\u5185\u5b58\n free(queue);\n}\n[class]{}-[func]{graphBFS}\n\u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u56fe 9-10 \u6765\u52a0\u6df1\u7406\u89e3\u3002
<1><2><3><4><5><6><7><8><9><10><11>\u56fe 9-10 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4
\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f
\u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u987a\u5e8f\u904d\u5386\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u3002\u4ee5\u56fe 9-10 \u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\)\u3001\\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\uff0c\u9876\u70b9 \\(2\\)\u3001\\(4\\)\u3001\\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3002
"},{"location":"chapter_graph/graph_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u5e76\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002
\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868
"},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2 \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"res\uff0c\u54c8\u5e0c\u8868visited\uff0c\u961f\u5217que\u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5982\u56fe 9-11 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u5f53\u524d\u9876\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u8fd4\u56de\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5c3d\u5934\u5e76\u8fd4\u56de\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u3002
\u56fe 9-11 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386
"},{"location":"chapter_graph/graph_traversal/#1_1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"\u8fd9\u79cd\u201c\u8d70\u5230\u5c3d\u5934\u518d\u8fd4\u56de\u201d\u7684\u7b97\u6cd5\u8303\u5f0f\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u5728\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u4e2d\uff0c\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_dfs.pyvisited\u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002
graph_dfs.cppdef dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570\"\"\"\n res.append(vet) # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet) # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for adjVet in graph.adj_list[vet]:\n if adjVet in visited:\n continue # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n # \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]:\n \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS\"\"\"\n # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n res = []\n # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n visited = set[Vertex]()\n dfs(graph, visited, res, start_vet)\n return res\n
graph_dfs.java/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &graph, unordered_set<Vertex *> &visited, vector<Vertex *> &res, Vertex *vet) {\n res.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex *adjVet : graph.adjList[vet]) {\n if (visited.count(adjVet))\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex *> graphDFS(GraphAdjList &graph, Vertex *startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n vector<Vertex *> res;\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n unordered_set<Vertex *> visited;\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.cs/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set<Vertex> visited, List<Vertex> res, Vertex vet) {\n res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex adjVet : graph.adjList.get(vet)) {\n if (visited.contains(adjVet))\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = new ArrayList<>();\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n Set<Vertex> visited = new HashSet<>();\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.go/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid DFS(GraphAdjList graph, HashSet<Vertex> visited, List<Vertex> res, Vertex vet) {\n res.Add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n foreach (Vertex adjVet in graph.adjList[vet]) {\n if (visited.Contains(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9 \n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n DFS(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> GraphDFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n HashSet<Vertex> visited = [];\n DFS(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.swift/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n // append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n *res = append(*res, vet)\n visited[vet] = struct{}{}\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for _, adjVet := range g.adjList[vet] {\n _, isExist := visited[adjVet]\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n if !isExist {\n dfs(g, visited, res, adjVet)\n }\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n res := make([]Vertex, 0)\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n visited := make(map[Vertex]struct{})\n dfs(g, visited, &res, startVet)\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res\n}\n
graph_dfs.js/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set<Vertex>, res: inout [Vertex], vet: Vertex) {\n res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for adjVet in graph.adjList[vet] ?? [] {\n if visited.contains(adjVet) {\n continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph: graph, visited: &visited, res: &res, vet: adjVet)\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n var res: [Vertex] = []\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n var visited: Set<Vertex> = []\n dfs(graph: graph, visited: &visited, res: &res, vet: startVet)\n return res\n}\n
graph_dfs.ts/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (const adjVet of graph.adjList.get(vet)) {\n if (visited.has(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n const res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n const visited = new Set();\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.dart/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\n graph: GraphAdjList,\n visited: Set<Vertex>,\n res: Vertex[],\n vet: Vertex\n): void {\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (const adjVet of graph.adjList.get(vet)) {\n if (visited.has(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n const res: Vertex[] = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n const visited: Set<Vertex> = new Set();\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.rs/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\n GraphAdjList graph,\n Set<Vertex> visited,\n List<Vertex> res,\n Vertex vet,\n) {\n res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex adjVet in graph.adjList[vet]!) {\n if (visited.contains(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n Set<Vertex> visited = {};\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.c/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfn dfs(graph: &GraphAdjList, visited: &mut HashSet<Vertex>, res: &mut Vec<Vertex>, vet: Vertex) {\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.insert(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n if let Some(adj_vets) = graph.adj_list.get(&vet) {\n for &adj_vet in adj_vets {\n if visited.contains(&adj_vet) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adj_vet);\n }\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> Vec<Vertex> {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n let mut res = vec![];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n let mut visited = HashSet::new();\n dfs(&graph, &mut visited, &mut res, start_vet);\n\n res\n}\n
graph_dfs.zig/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n for (int i = 0; i < size; i++) {\n if (res[i] == vet) {\n return 1;\n }\n }\n return 0;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n res[(*resSize)++] = vet;\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n AdjListNode *node = findNode(graph, vet);\n while (node != NULL) {\n // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n if (!isVisited(res, *resSize, node->vertex)) {\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, res, resSize, node->vertex);\n }\n node = node->next;\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n dfs(graph, res, resSize, startVet);\n}\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{graphDFS}\n\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 9-12 \u6240\u793a\u3002
- \u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u8868\u793a\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\u3002
- \u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u65b9\u6cd5\u7684\u4f4d\u7f6e\u3002
\u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8bae\u5c06\u56fe 9-12 \u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\u6a21\u62df\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002
<1><2><3><4><5><6><7><8><9><10><11>\u56fe 9-12 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4
\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f
\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u90bb\u63a5\u9876\u70b9\u7684\u987a\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e71\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002
\u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u5b83\u4eec\u5c55\u793a\u4e86\u4e09\u79cd\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u7136\u800c\u8fd9\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002
"},{"location":"chapter_graph/graph_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002
\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868
"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_graph/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"res\uff0c\u54c8\u5e0c\u8868visited\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002- \u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002
- \u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002
- \u6709\u5411\u56fe\u7684\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u5747\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002
- \u90bb\u63a5\u77e9\u9635\u5229\u7528\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u5728\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u95f4\u5360\u7528\u8f83\u591a\u3002
- \u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u4e8e\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002
- \u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002
- \u4ece\u7b97\u6cd5\u601d\u60f3\u7684\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002
- \u56fe\u53ef\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002
- \u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002
- \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002
- \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u65f6\u518d\u56de\u6eaf\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002
\u8def\u5f84\u7684\u5b9a\u4e49\u662f\u9876\u70b9\u5e8f\u5217\u8fd8\u662f\u8fb9\u5e8f\u5217\uff1f
\u7ef4\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8bed\u8a00\u7248\u672c\u7684\u5b9a\u4e49\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. \u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f84\u88ab\u89c6\u4e3a\u4e00\u4e2a\u8fb9\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u3002\u8fd9\u662f\u56e0\u4e3a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u591a\u6761\u8fb9\u8fde\u63a5\uff0c\u6b64\u65f6\u6bcf\u6761\u8fb9\u90fd\u5bf9\u5e94\u4e00\u6761\u8def\u5f84\u3002
\u975e\u8fde\u901a\u56fe\u4e2d\u662f\u5426\u4f1a\u6709\u65e0\u6cd5\u904d\u5386\u5230\u7684\u70b9\uff1f
\u5728\u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002\u904d\u5386\u975e\u8fde\u901a\u56fe\u9700\u8981\u8bbe\u7f6e\u591a\u4e2a\u8d77\u70b9\uff0c\u4ee5\u904d\u5386\u5230\u56fe\u7684\u6240\u6709\u8fde\u901a\u5206\u91cf\u3002
\u5728\u90bb\u63a5\u8868\u4e2d\uff0c\u201c\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u6240\u6709\u9876\u70b9\u201d\u7684\u9876\u70b9\u987a\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f
\u53ef\u4ee5\u662f\u4efb\u610f\u987a\u5e8f\u3002\u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u9700\u8981\u6309\u7167\u6307\u5b9a\u89c4\u5219\u6765\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9876\u70b9\u6dfb\u52a0\u7684\u6b21\u5e8f\uff0c\u6216\u8005\u6309\u7167\u9876\u70b9\u503c\u5927\u5c0f\u7684\u987a\u5e8f\u7b49\uff0c\u8fd9\u6837\u6709\u52a9\u4e8e\u5feb\u901f\u67e5\u627e\u201c\u5e26\u6709\u67d0\u79cd\u6781\u503c\u201d\u7684\u9876\u70b9\u3002
"},{"location":"chapter_greedy/","title":"\u7b2c 15 \u7ae0 \u00a0 \u8d2a\u5fc3","text":"Abstract
\u5411\u65e5\u8475\u671d\u7740\u592a\u9633\u8f6c\u52a8\uff0c\u65f6\u523b\u8ffd\u6c42\u81ea\u8eab\u6210\u957f\u7684\u6700\u5927\u53ef\u80fd\u3002
\u8d2a\u5fc3\u7b56\u7565\u5728\u4e00\u8f6e\u8f6e\u7684\u7b80\u5355\u9009\u62e9\u4e2d\uff0c\u9010\u6b65\u5bfc\u5411\u6700\u4f73\u7b54\u6848\u3002
"},{"location":"chapter_greedy/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5
- 15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898
- 15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898
- 15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898
- 15.5 \u00a0 \u5c0f\u7ed3
Question
\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u4f46\u53ef\u4ee5\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u4ef7\u503c\u6839\u636e\u9009\u62e9\u7684\u91cd\u91cf\u6bd4\u4f8b\u8ba1\u7b97\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 15-3 \u6240\u793a\u3002
\u56fe 15-3 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e
\u5206\u6570\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u6574\u4f53\u4e0a\u975e\u5e38\u76f8\u4f3c\uff0c\u72b6\u6001\u5305\u542b\u5f53\u524d\u7269\u54c1 \\(i\\) \u548c\u5bb9\u91cf \\(c\\) \uff0c\u76ee\u6807\u662f\u6c42\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u6700\u5927\u4ef7\u503c\u3002
\u4e0d\u540c\u70b9\u5728\u4e8e\uff0c\u672c\u9898\u5141\u8bb8\u53ea\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\u3002\u5982\u56fe 15-4 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u7269\u54c1\u4efb\u610f\u5730\u8fdb\u884c\u5207\u5206\uff0c\u5e76\u6309\u7167\u91cd\u91cf\u6bd4\u4f8b\u6765\u8ba1\u7b97\u76f8\u5e94\u4ef7\u503c\u3002
- \u5bf9\u4e8e\u7269\u54c1 \\(i\\) \uff0c\u5b83\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c\u4e3a \\(val[i-1] / wgt[i-1]\\) \uff0c\u7b80\u79f0\u5355\u4f4d\u4ef7\u503c\u3002
- \u5047\u8bbe\u653e\u5165\u4e00\u90e8\u5206\u7269\u54c1 \\(i\\) \uff0c\u91cd\u91cf\u4e3a \\(w\\) \uff0c\u5219\u80cc\u5305\u589e\u52a0\u7684\u4ef7\u503c\u4e3a \\(w \\times val[i-1] / wgt[i-1]\\) \u3002
\u56fe 15-4 \u00a0 \u7269\u54c1\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c
"},{"location":"chapter_greedy/fractional_knapsack_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"\u6700\u5927\u5316\u80cc\u5305\u5185\u7269\u54c1\u603b\u4ef7\u503c\uff0c\u672c\u8d28\u4e0a\u662f\u6700\u5927\u5316\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u7269\u54c1\u4ef7\u503c\u3002\u7531\u6b64\u4fbf\u53ef\u63a8\u7406\u51fa\u56fe 15-5 \u6240\u793a\u7684\u8d2a\u5fc3\u7b56\u7565\u3002
- \u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\u3002
- \u904d\u5386\u6240\u6709\u7269\u54c1\uff0c\u6bcf\u8f6e\u8d2a\u5fc3\u5730\u9009\u62e9\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\u3002
- \u82e5\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u4f7f\u7528\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u586b\u6ee1\u80cc\u5305\u3002
\u56fe 15-5 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565
"},{"location":"chapter_greedy/fractional_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u6211\u4eec\u5efa\u7acb\u4e86\u4e00\u4e2a\u7269\u54c1\u7c7b
PythonC++JavaC#GoSwiftJSTSDartRustCZig fractional_knapsack.pyItem\uff0c\u4ee5\u4fbf\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u8fdb\u884c\u6392\u5e8f\u3002\u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u5f53\u80cc\u5305\u5df2\u6ee1\u65f6\u8df3\u51fa\u5e76\u8fd4\u56de\u89e3\uff1a
fractional_knapsack.cppclass Item:\n \"\"\"\u7269\u54c1\"\"\"\n\n def __init__(self, w: int, v: int):\n self.w = w # \u7269\u54c1\u91cd\u91cf\n self.v = v # \u7269\u54c1\u4ef7\u503c\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"\u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3\"\"\"\n # \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n items = [Item(w, v) for w, v in zip(wgt, val)]\n # \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort(key=lambda item: item.v / item.w, reverse=True)\n # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n res = 0\n for item in items:\n if item.w <= cap:\n # \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v\n cap -= item.w\n else:\n # \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (item.v / item.w) * cap\n # \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break\n return res\n
fractional_knapsack.java/* \u7269\u54c1 */\nclass Item {\n public:\n int w; // \u7269\u54c1\u91cd\u91cf\n int v; // \u7269\u54c1\u4ef7\u503c\n\n Item(int w, int v) : w(w), v(v) {\n }\n};\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(vector<int> &wgt, vector<int> &val, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n vector<Item> items;\n for (int i = 0; i < wgt.size(); i++) {\n items.push_back(Item(wgt[i], val[i]));\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n sort(items.begin(), items.end(), [](Item &a, Item &b) { return (double)a.v / a.w > (double)b.v / b.w; });\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n double res = 0;\n for (auto &item : items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (double)item.v / item.w * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.cs/* \u7269\u54c1 */\nclass Item {\n int w; // \u7269\u54c1\u91cd\u91cf\n int v; // \u7269\u54c1\u4ef7\u503c\n\n public Item(int w, int v) {\n this.w = w;\n this.v = v;\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n Item[] items = new Item[wgt.length];\n for (int i = 0; i < wgt.length; i++) {\n items[i] = new Item(wgt[i], val[i]);\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n Arrays.sort(items, Comparator.comparingDouble(item -> -((double) item.v / item.w)));\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n double res = 0;\n for (Item item : items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (double) item.v / item.w * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.go/* \u7269\u54c1 */\nclass Item(int w, int v) {\n public int w = w; // \u7269\u54c1\u91cd\u91cf\n public int v = v; // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n Item[] items = new Item[wgt.Length];\n for (int i = 0; i < wgt.Length; i++) {\n items[i] = new Item(wgt[i], val[i]);\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n Array.Sort(items, (x, y) => (y.v / y.w).CompareTo(x.v / x.w));\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n double res = 0;\n foreach (Item item in items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (double)item.v / item.w * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.swift/* \u7269\u54c1 */\ntype Item struct {\n w int // \u7269\u54c1\u91cd\u91cf\n v int // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n items := make([]Item, len(wgt))\n for i := 0; i < len(wgt); i++ {\n items[i] = Item{wgt[i], val[i]}\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n sort.Slice(items, func(i, j int) bool {\n return float64(items[i].v)/float64(items[i].w) > float64(items[j].v)/float64(items[j].w)\n })\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n res := 0.0\n for _, item := range items {\n if item.w <= cap {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += float64(item.v)\n cap -= item.w\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += float64(item.v) / float64(item.w) * float64(cap)\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break\n }\n }\n return res\n}\n
fractional_knapsack.js/* \u7269\u54c1 */\nclass Item {\n var w: Int // \u7269\u54c1\u91cd\u91cf\n var v: Int // \u7269\u54c1\u4ef7\u503c\n\n init(w: Int, v: Int) {\n self.w = w\n self.v = v\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -> Double {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort(by: { -(Double($0.v) / Double($0.w)) < -(Double($1.v) / Double($1.w)) })\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n var res = 0.0\n var cap = cap\n for item in items {\n if item.w <= cap {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += Double(item.v)\n cap -= item.w\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += Double(item.v) / Double(item.w) * Double(cap)\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break\n }\n }\n return res\n}\n
fractional_knapsack.ts/* \u7269\u54c1 */\nclass Item {\n constructor(w, v) {\n this.w = w; // \u7269\u54c1\u91cd\u91cf\n this.v = v; // \u7269\u54c1\u4ef7\u503c\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt, val, cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n const items = wgt.map((w, i) => new Item(w, val[i]));\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort((a, b) => b.v / b.w - a.v / a.w);\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n let res = 0;\n for (const item of items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (item.v / item.w) * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.dart/* \u7269\u54c1 */\nclass Item {\n w: number; // \u7269\u54c1\u91cd\u91cf\n v: number; // \u7269\u54c1\u4ef7\u503c\n\n constructor(w: number, v: number) {\n this.w = w;\n this.v = v;\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n const items: Item[] = wgt.map((w, i) => new Item(w, val[i]));\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort((a, b) => b.v / b.w - a.v / a.w);\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n let res = 0;\n for (const item of items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (item.v / item.w) * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.rs/* \u7269\u54c1 */\nclass Item {\n int w; // \u7269\u54c1\u91cd\u91cf\n int v; // \u7269\u54c1\u4ef7\u503c\n\n Item(this.w, this.v);\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(List<int> wgt, List<int> val, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n List<Item> items = List.generate(wgt.length, (i) => Item(wgt[i], val[i]));\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort((a, b) => (b.v / b.w).compareTo(a.v / a.w));\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n double res = 0;\n for (Item item in items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += item.v / item.w * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.c/* \u7269\u54c1 */\nstruct Item {\n w: i32, // \u7269\u54c1\u91cd\u91cf\n v: i32, // \u7269\u54c1\u4ef7\u503c\n}\n\nimpl Item {\n fn new(w: i32, v: i32) -> Self {\n Self { w, v }\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfn fractional_knapsack(wgt: &[i32], val: &[i32], mut cap: i32) -> f64 {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n let mut items = wgt\n .iter()\n .zip(val.iter())\n .map(|(&w, &v)| Item::new(w, v))\n .collect::<Vec<Item>>();\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort_by(|a, b| {\n (b.v as f64 / b.w as f64)\n .partial_cmp(&(a.v as f64 / a.w as f64))\n .unwrap()\n });\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n let mut res = 0.0;\n for item in &items {\n if item.w <= cap {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v as f64;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += item.v as f64 / item.w as f64 * cap as f64;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n res\n}\n
fractional_knapsack.zig/* \u7269\u54c1 */\ntypedef struct {\n int w; // \u7269\u54c1\u91cd\u91cf\n int v; // \u7269\u54c1\u4ef7\u503c\n} Item;\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n Item *items = malloc(sizeof(Item) * itemCount);\n for (int i = 0; i < itemCount; i++) {\n items[i] = (Item){.w = wgt[i], .v = val[i]};\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n float res = 0.0;\n for (int i = 0; i < itemCount; i++) {\n if (items[i].w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += items[i].v;\n cap -= items[i].w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (float)cap / items[i].w * items[i].v;\n cap = 0;\n break;\n }\n }\n free(items);\n return res;\n}\n[class]{Item}-[func]{}\n\n[class]{}-[func]{fractionalKnapsack}\n\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u9700\u8981\u904d\u5386\u6574\u4e2a\u7269\u54c1\u5217\u8868\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7269\u54c1\u6570\u91cf\u3002
\u7531\u4e8e\u521d\u59cb\u5316\u4e86\u4e00\u4e2a
"},{"location":"chapter_greedy/fractional_knapsack_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"Item\u5bf9\u8c61\u5217\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u91c7\u7528\u53cd\u8bc1\u6cd5\u3002\u5047\u8bbe\u7269\u54c1 \\(x\\) \u662f\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\uff0c\u4f7f\u7528\u67d0\u7b97\u6cd5\u6c42\u5f97\u6700\u5927\u4ef7\u503c\u4e3a
res\uff0c\u4f46\u8be5\u89e3\u4e2d\u4e0d\u5305\u542b\u7269\u54c1 \\(x\\) \u3002\u73b0\u5728\u4ece\u80cc\u5305\u4e2d\u62ff\u51fa\u5355\u4f4d\u91cd\u91cf\u7684\u4efb\u610f\u7269\u54c1\uff0c\u5e76\u66ff\u6362\u4e3a\u5355\u4f4d\u91cd\u91cf\u7684\u7269\u54c1 \\(x\\) \u3002\u7531\u4e8e\u7269\u54c1 \\(x\\) \u7684\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\uff0c\u56e0\u6b64\u66ff\u6362\u540e\u7684\u603b\u4ef7\u503c\u4e00\u5b9a\u5927\u4e8e
res\u3002\u8fd9\u4e0eres\u662f\u6700\u4f18\u89e3\u77db\u76fe\uff0c\u8bf4\u660e\u6700\u4f18\u89e3\u4e2d\u5fc5\u987b\u5305\u542b\u7269\u54c1 \\(x\\) \u3002\u5bf9\u4e8e\u8be5\u89e3\u4e2d\u7684\u5176\u4ed6\u7269\u54c1\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u6784\u5efa\u51fa\u4e0a\u8ff0\u77db\u76fe\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u5355\u4f4d\u4ef7\u503c\u66f4\u5927\u7684\u7269\u54c1\u603b\u662f\u66f4\u4f18\u9009\u62e9\uff0c\u8fd9\u8bf4\u660e\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002
\u5982\u56fe 15-6 \u6240\u793a\uff0c\u5982\u679c\u5c06\u7269\u54c1\u91cd\u91cf\u548c\u7269\u54c1\u5355\u4f4d\u4ef7\u503c\u5206\u522b\u770b\u4f5c\u4e00\u5f20\u4e8c\u7ef4\u56fe\u8868\u7684\u6a2a\u8f74\u548c\u7eb5\u8f74\uff0c\u5219\u5206\u6570\u80cc\u5305\u95ee\u9898\u53ef\u8f6c\u5316\u4e3a\u201c\u6c42\u5728\u6709\u9650\u6a2a\u8f74\u533a\u95f4\u4e0b\u7684\u6700\u5927\u56f4\u6210\u9762\u79ef\u201d\u3002\u8fd9\u4e2a\u7c7b\u6bd4\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u4ece\u51e0\u4f55\u89d2\u5ea6\u7406\u89e3\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002
\u56fe 15-6 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u51e0\u4f55\u8868\u793a
"},{"location":"chapter_greedy/greedy_algorithm/","title":"15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5","text":"\u300c\u8d2a\u5fc3\u7b97\u6cd5 greedy algorithm\u300d\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u7684\u7b97\u6cd5\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u95ee\u9898\u7684\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\uff0c\u90fd\u9009\u62e9\u5f53\u524d\u770b\u8d77\u6765\u6700\u4f18\u7684\u9009\u62e9\uff0c\u5373\u8d2a\u5fc3\u5730\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002\u8d2a\u5fc3\u7b97\u6cd5\u7b80\u6d01\u4e14\u9ad8\u6548\uff0c\u5728\u8bb8\u591a\u5b9e\u9645\u95ee\u9898\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002
\u8d2a\u5fc3\u7b97\u6cd5\u548c\u52a8\u6001\u89c4\u5212\u90fd\u5e38\u7528\u4e8e\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u3002\u5b83\u4eec\u4e4b\u95f4\u5b58\u5728\u4e00\u4e9b\u76f8\u4f3c\u4e4b\u5904\uff0c\u6bd4\u5982\u90fd\u4f9d\u8d56\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28\uff0c\u4f46\u5de5\u4f5c\u539f\u7406\u4e0d\u540c\u3002
- \u52a8\u6001\u89c4\u5212\u4f1a\u6839\u636e\u4e4b\u524d\u9636\u6bb5\u7684\u6240\u6709\u51b3\u7b56\u6765\u8003\u8651\u5f53\u524d\u51b3\u7b56\uff0c\u5e76\u4f7f\u7528\u8fc7\u53bb\u5b50\u95ee\u9898\u7684\u89e3\u6765\u6784\u5efa\u5f53\u524d\u5b50\u95ee\u9898\u7684\u89e3\u3002
- \u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4f1a\u8003\u8651\u8fc7\u53bb\u7684\u51b3\u7b56\uff0c\u800c\u662f\u4e00\u8def\u5411\u524d\u5730\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u4e0d\u65ad\u7f29\u5c0f\u95ee\u9898\u8303\u56f4\uff0c\u76f4\u81f3\u95ee\u9898\u88ab\u89e3\u51b3\u3002
\u6211\u4eec\u5148\u901a\u8fc7\u4f8b\u9898\u201c\u96f6\u94b1\u5151\u6362\u201d\u4e86\u89e3\u8d2a\u5fc3\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002\u8fd9\u9053\u9898\u5df2\u7ecf\u5728\u201c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u76f8\u4fe1\u4f60\u5bf9\u5b83\u5e76\u4e0d\u964c\u751f\u3002
Question
\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u5219\u8fd4\u56de \\(-1\\) \u3002
\u672c\u9898\u91c7\u53d6\u7684\u8d2a\u5fc3\u7b56\u7565\u5982\u56fe 15-1 \u6240\u793a\u3002\u7ed9\u5b9a\u76ee\u6807\u91d1\u989d\uff0c\u6211\u4eec\u8d2a\u5fc3\u5730\u9009\u62e9\u4e0d\u5927\u4e8e\u4e14\u6700\u63a5\u8fd1\u5b83\u7684\u786c\u5e01\uff0c\u4e0d\u65ad\u5faa\u73af\u8be5\u6b65\u9aa4\uff0c\u76f4\u81f3\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u4e3a\u6b62\u3002
\u56fe 15-1 \u00a0 \u96f6\u94b1\u5151\u6362\u7684\u8d2a\u5fc3\u7b56\u7565
\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u4f60\u53ef\u80fd\u4f1a\u4e0d\u7531\u5730\u53d1\u51fa\u611f\u53f9\uff1aSo clean \uff01\u8d2a\u5fc3\u7b97\u6cd5\u4ec5\u7528\u7ea6\u5341\u884c\u4ee3\u7801\u5c31\u89e3\u51b3\u4e86\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_greedy.py
coin_change_greedy.cppdef coin_change_greedy(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3\"\"\"\n # \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n i = len(coins) - 1\n count = 0\n # \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while amt > 0:\n # \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while i > 0 and coins[i] > amt:\n i -= 1\n # \u9009\u62e9 coins[i]\n amt -= coins[i]\n count += 1\n # \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return count if amt == 0 else -1\n
coin_change_greedy.java/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(vector<int> &coins, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = coins.size() - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
coin_change_greedy.cs/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int[] coins, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = coins.length - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
coin_change_greedy.go/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint CoinChangeGreedy(int[] coins, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = coins.Length - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
coin_change_greedy.swift/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins []int, amt int) int {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n i := len(coins) - 1\n count := 0\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n for amt > 0 {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n for i > 0 && coins[i] > amt {\n i--\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i]\n count++\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n if amt != 0 {\n return -1\n }\n return count\n}\n
coin_change_greedy.js/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -> Int {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n var i = coins.count - 1\n var count = 0\n var amt = amt\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while amt > 0 {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while i > 0 && coins[i] > amt {\n i -= 1\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i]\n count += 1\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1\n}\n
coin_change_greedy.ts/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins, amt) {\n // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n let i = coins.length - 1;\n let count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt === 0 ? count : -1;\n}\n
coin_change_greedy.dart/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n let i = coins.length - 1;\n let count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt === 0 ? count : -1;\n}\n
coin_change_greedy.rs/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(List<int> coins, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = coins.length - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
coin_change_greedy.c/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfn coin_change_greedy(coins: &[i32], mut amt: i32) -> i32 {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n let mut i = coins.len() - 1;\n let mut count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while amt > 0 {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while i > 0 && coins[i] > amt {\n i -= 1;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count += 1;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n if amt == 0 {\n count\n } else {\n -1\n }\n}\n
coin_change_greedy.zig/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int *coins, int size, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = size - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
"},{"location":"chapter_greedy/greedy_algorithm/#1511","title":"15.1.1 \u00a0 \u8d2a\u5fc3\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"[class]{}-[func]{coinChangeGreedy}\n\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u64cd\u4f5c\u76f4\u63a5\u3001\u5b9e\u73b0\u7b80\u5355\uff0c\u800c\u4e14\u901a\u5e38\u6548\u7387\u4e5f\u5f88\u9ad8\u3002\u5728\u4ee5\u4e0a\u4ee3\u7801\u4e2d\uff0c\u8bb0\u786c\u5e01\u6700\u5c0f\u9762\u503c\u4e3a \\(\\min(coins)\\) \uff0c\u5219\u8d2a\u5fc3\u9009\u62e9\u6700\u591a\u5faa\u73af \\(amt / \\min(coins)\\) \u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(amt / \\min(coins))\\) \u3002\u8fd9\u6bd4\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\times amt)\\) \u63d0\u5347\u4e86\u4e00\u4e2a\u6570\u91cf\u7ea7\u3002
\u7136\u800c\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u9762\u503c\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5e76\u4e0d\u80fd\u627e\u5230\u6700\u4f18\u89e3\u3002\u56fe 15-2 \u7ed9\u51fa\u4e86\u4e24\u4e2a\u793a\u4f8b\u3002
- \u6b63\u4f8b \\(coins = [1, 5, 10, 20, 50, 100]\\)\uff1a\u5728\u8be5\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u7ed9\u5b9a\u4efb\u610f \\(amt\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u90fd\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3\u3002
- \u53cd\u4f8b \\(coins = [1, 20, 50]\\)\uff1a\u5047\u8bbe \\(amt = 60\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 10\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(11\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(20 + 20 + 20\\) \uff0c\u4ec5\u9700 \\(3\\) \u679a\u786c\u5e01\u3002
- \u53cd\u4f8b \\(coins = [1, 49, 50]\\)\uff1a\u5047\u8bbe \\(amt = 98\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 48\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(49\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(49 + 49\\) \uff0c\u4ec5\u9700 \\(2\\) \u679a\u786c\u5e01\u3002
\u56fe 15-2 \u00a0 \u8d2a\u5fc3\u65e0\u6cd5\u627e\u51fa\u6700\u4f18\u89e3\u7684\u793a\u4f8b
\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u4e8e\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u65e0\u6cd5\u4fdd\u8bc1\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3\uff0c\u5e76\u4e14\u6709\u53ef\u80fd\u627e\u5230\u975e\u5e38\u5dee\u7684\u89e3\u3002\u5b83\u66f4\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\u3002
\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u9002\u7528\u60c5\u51b5\u5206\u4ee5\u4e0b\u4e24\u79cd\u3002
- \u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u5f80\u5f80\u662f\u6700\u4f18\u9009\u62e9\uff0c\u56e0\u4e3a\u5b83\u5f80\u5f80\u6bd4\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u66f4\u9ad8\u6548\u3002
- \u53ef\u4ee5\u627e\u5230\u8fd1\u4f3c\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u4e5f\u662f\u53ef\u7528\u7684\u3002\u5bf9\u4e8e\u5f88\u591a\u590d\u6742\u95ee\u9898\u6765\u8bf4\uff0c\u5bfb\u627e\u5168\u5c40\u6700\u4f18\u89e3\u975e\u5e38\u56f0\u96be\uff0c\u80fd\u4ee5\u8f83\u9ad8\u6548\u7387\u627e\u5230\u6b21\u4f18\u89e3\u4e5f\u662f\u975e\u5e38\u4e0d\u9519\u7684\u3002
\u90a3\u4e48\u95ee\u9898\u6765\u4e86\uff0c\u4ec0\u4e48\u6837\u7684\u95ee\u9898\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u5462\uff1f\u6216\u8005\u8bf4\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1f
\u76f8\u8f83\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u4f7f\u7528\u6761\u4ef6\u66f4\u52a0\u82db\u523b\uff0c\u5176\u4e3b\u8981\u5173\u6ce8\u95ee\u9898\u7684\u4e24\u4e2a\u6027\u8d28\u3002
- \u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\uff1a\u53ea\u6709\u5f53\u5c40\u90e8\u6700\u4f18\u9009\u62e9\u59cb\u7ec8\u53ef\u4ee5\u5bfc\u81f4\u5168\u5c40\u6700\u4f18\u89e3\u65f6\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u624d\u80fd\u4fdd\u8bc1\u5f97\u5230\u6700\u4f18\u89e3\u3002
- \u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5305\u542b\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002
\u6700\u4f18\u5b50\u7ed3\u6784\u5df2\u7ecf\u5728\u201c\u52a8\u6001\u89c4\u5212\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u8fd9\u91cc\u4e0d\u518d\u8d58\u8ff0\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e9b\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\u5e76\u4e0d\u660e\u663e\uff0c\u4f46\u4ecd\u7136\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u89e3\u51b3\u3002
\u6211\u4eec\u4e3b\u8981\u63a2\u7a76\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u5224\u65ad\u65b9\u6cd5\u3002\u867d\u7136\u5b83\u7684\u63cf\u8ff0\u770b\u4e0a\u53bb\u6bd4\u8f83\u7b80\u5355\uff0c\u4f46\u5b9e\u9645\u4e0a\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\uff0c\u8bc1\u660e\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u5e76\u975e\u6613\u4e8b\u3002
\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u6211\u4eec\u867d\u7136\u80fd\u591f\u5bb9\u6613\u5730\u4e3e\u51fa\u53cd\u4f8b\uff0c\u5bf9\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u8fdb\u884c\u8bc1\u4f2a\uff0c\u4f46\u8bc1\u5b9e\u7684\u96be\u5ea6\u8f83\u5927\u3002\u5982\u679c\u95ee\uff1a\u6ee1\u8db3\u4ec0\u4e48\u6761\u4ef6\u7684\u786c\u5e01\u7ec4\u5408\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\uff1f\u6211\u4eec\u5f80\u5f80\u53ea\u80fd\u51ed\u501f\u76f4\u89c9\u6216\u4e3e\u4f8b\u5b50\u6765\u7ed9\u51fa\u4e00\u4e2a\u6a21\u68f1\u4e24\u53ef\u7684\u7b54\u6848\uff0c\u800c\u96be\u4ee5\u7ed9\u51fa\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\u3002
Quote
\u6709\u4e00\u7bc7\u8bba\u6587\u7ed9\u51fa\u4e86\u4e00\u4e2a \\(O(n^3)\\) \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u7b97\u6cd5\uff0c\u7528\u4e8e\u5224\u65ad\u4e00\u4e2a\u786c\u5e01\u7ec4\u5408\u80fd\u5426\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u627e\u51fa\u4efb\u610f\u91d1\u989d\u7684\u6700\u4f18\u89e3\u3002
Pearson, David. A polynomial-time algorithm for the change-making problem. Operations Research Letters 33.3 (2005): 231-234.
"},{"location":"chapter_greedy/greedy_algorithm/#1513","title":"15.1.3 \u00a0 \u8d2a\u5fc3\u89e3\u9898\u6b65\u9aa4","text":"\u8d2a\u5fc3\u95ee\u9898\u7684\u89e3\u51b3\u6d41\u7a0b\u5927\u4f53\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e09\u6b65\u3002
- \u95ee\u9898\u5206\u6790\uff1a\u68b3\u7406\u4e0e\u7406\u89e3\u95ee\u9898\u7279\u6027\uff0c\u5305\u62ec\u72b6\u6001\u5b9a\u4e49\u3001\u4f18\u5316\u76ee\u6807\u548c\u7ea6\u675f\u6761\u4ef6\u7b49\u3002\u8fd9\u4e00\u6b65\u5728\u56de\u6eaf\u548c\u52a8\u6001\u89c4\u5212\u4e2d\u90fd\u6709\u6d89\u53ca\u3002
- \u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\uff1a\u786e\u5b9a\u5982\u4f55\u5728\u6bcf\u4e00\u6b65\u4e2d\u505a\u51fa\u8d2a\u5fc3\u9009\u62e9\u3002\u8fd9\u4e2a\u7b56\u7565\u80fd\u591f\u5728\u6bcf\u4e00\u6b65\u51cf\u5c0f\u95ee\u9898\u7684\u89c4\u6a21\uff0c\u5e76\u6700\u7ec8\u89e3\u51b3\u6574\u4e2a\u95ee\u9898\u3002
- \u6b63\u786e\u6027\u8bc1\u660e\uff1a\u901a\u5e38\u9700\u8981\u8bc1\u660e\u95ee\u9898\u5177\u6709\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8fd9\u4e2a\u6b65\u9aa4\u53ef\u80fd\u9700\u8981\u7528\u5230\u6570\u5b66\u8bc1\u660e\uff0c\u4f8b\u5982\u5f52\u7eb3\u6cd5\u6216\u53cd\u8bc1\u6cd5\u7b49\u3002
\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6c42\u89e3\u95ee\u9898\u7684\u6838\u5fc3\u6b65\u9aa4\uff0c\u4f46\u5b9e\u65bd\u8d77\u6765\u53ef\u80fd\u5e76\u4e0d\u5bb9\u6613\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002
- \u4e0d\u540c\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\u7684\u5dee\u5f02\u8f83\u5927\u3002\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\u6765\u8bf4\uff0c\u8d2a\u5fc3\u7b56\u7565\u6bd4\u8f83\u6d45\u663e\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e9b\u5927\u6982\u7684\u601d\u8003\u4e0e\u5c1d\u8bd5\u5c31\u80fd\u5f97\u51fa\u3002\u800c\u5bf9\u4e8e\u4e00\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b56\u7565\u53ef\u80fd\u975e\u5e38\u9690\u853d\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u975e\u5e38\u8003\u9a8c\u4e2a\u4eba\u7684\u89e3\u9898\u7ecf\u9a8c\u4e0e\u7b97\u6cd5\u80fd\u529b\u4e86\u3002
- \u67d0\u4e9b\u8d2a\u5fc3\u7b56\u7565\u5177\u6709\u8f83\u5f3a\u7684\u8ff7\u60d1\u6027\u3002\u5f53\u6211\u4eec\u6ee1\u6000\u4fe1\u5fc3\u8bbe\u8ba1\u597d\u8d2a\u5fc3\u7b56\u7565\uff0c\u5199\u51fa\u89e3\u9898\u4ee3\u7801\u5e76\u63d0\u4ea4\u8fd0\u884c\uff0c\u5f88\u53ef\u80fd\u53d1\u73b0\u90e8\u5206\u6d4b\u8bd5\u6837\u4f8b\u65e0\u6cd5\u901a\u8fc7\u3002\u8fd9\u662f\u56e0\u4e3a\u8bbe\u8ba1\u7684\u8d2a\u5fc3\u7b56\u7565\u53ea\u662f\u201c\u90e8\u5206\u6b63\u786e\u201d\u7684\uff0c\u4e0a\u6587\u4ecb\u7ecd\u7684\u96f6\u94b1\u5151\u6362\u5c31\u662f\u4e00\u4e2a\u5178\u578b\u6848\u4f8b\u3002
\u4e3a\u4e86\u4fdd\u8bc1\u6b63\u786e\u6027\uff0c\u6211\u4eec\u5e94\u8be5\u5bf9\u8d2a\u5fc3\u7b56\u7565\u8fdb\u884c\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\uff0c\u901a\u5e38\u9700\u8981\u7528\u5230\u53cd\u8bc1\u6cd5\u6216\u6570\u5b66\u5f52\u7eb3\u6cd5\u3002
\u7136\u800c\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u4e5f\u5f88\u53ef\u80fd\u4e0d\u662f\u4e00\u4ef6\u6613\u4e8b\u3002\u5982\u82e5\u6ca1\u6709\u5934\u7eea\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u9762\u5411\u6d4b\u8bd5\u7528\u4f8b\u8fdb\u884c\u4ee3\u7801\u8c03\u8bd5\uff0c\u4e00\u6b65\u6b65\u4fee\u6539\u4e0e\u9a8c\u8bc1\u8d2a\u5fc3\u7b56\u7565\u3002
"},{"location":"chapter_greedy/greedy_algorithm/#1514","title":"15.1.4 \u00a0 \u8d2a\u5fc3\u5178\u578b\u4f8b\u9898","text":"\u8d2a\u5fc3\u7b97\u6cd5\u5e38\u5e38\u5e94\u7528\u5728\u6ee1\u8db3\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u4ee5\u4e0b\u5217\u4e3e\u4e86\u4e00\u4e9b\u5178\u578b\u7684\u8d2a\u5fc3\u7b97\u6cd5\u95ee\u9898\u3002
- \u786c\u5e01\u627e\u96f6\u95ee\u9898\uff1a\u5728\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u603b\u662f\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002
- \u533a\u95f4\u8c03\u5ea6\u95ee\u9898\uff1a\u5047\u8bbe\u4f60\u6709\u4e00\u4e9b\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u5728\u4e00\u6bb5\u65f6\u95f4\u5185\u8fdb\u884c\uff0c\u4f60\u7684\u76ee\u6807\u662f\u5b8c\u6210\u5c3d\u53ef\u80fd\u591a\u7684\u4efb\u52a1\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u4efb\u52a1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002
- \u5206\u6570\u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u8f7d\u91cd\u91cf\uff0c\u4f60\u7684\u76ee\u6807\u662f\u9009\u62e9\u4e00\u7ec4\u7269\u54c1\uff0c\u4f7f\u5f97\u603b\u91cd\u91cf\u4e0d\u8d85\u8fc7\u8f7d\u91cd\u91cf\uff0c\u4e14\u603b\u4ef7\u503c\u6700\u5927\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u6027\u4ef7\u6bd4\u6700\u9ad8\uff08\u4ef7\u503c / \u91cd\u91cf\uff09\u7684\u7269\u54c1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4e00\u4e9b\u60c5\u51b5\u4e0b\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002
- \u80a1\u7968\u4e70\u5356\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u80a1\u7968\u7684\u5386\u53f2\u4ef7\u683c\uff0c\u4f60\u53ef\u4ee5\u8fdb\u884c\u591a\u6b21\u4e70\u5356\uff0c\u4f46\u5982\u679c\u4f60\u5df2\u7ecf\u6301\u6709\u80a1\u7968\uff0c\u90a3\u4e48\u5728\u5356\u51fa\u4e4b\u524d\u4e0d\u80fd\u518d\u4e70\uff0c\u76ee\u6807\u662f\u83b7\u53d6\u6700\u5927\u5229\u6da6\u3002
- \u970d\u592b\u66fc\u7f16\u7801\uff1a\u970d\u592b\u66fc\u7f16\u7801\u662f\u4e00\u79cd\u7528\u4e8e\u65e0\u635f\u6570\u636e\u538b\u7f29\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002\u901a\u8fc7\u6784\u5efa\u970d\u592b\u66fc\u6811\uff0c\u6bcf\u6b21\u9009\u62e9\u51fa\u73b0\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u8282\u70b9\u5408\u5e76\uff0c\u6700\u540e\u5f97\u5230\u7684\u970d\u592b\u66fc\u6811\u7684\u5e26\u6743\u8def\u5f84\u957f\u5ea6\uff08\u7f16\u7801\u957f\u5ea6\uff09\u6700\u5c0f\u3002
- Dijkstra \u7b97\u6cd5\uff1a\u5b83\u662f\u4e00\u79cd\u89e3\u51b3\u7ed9\u5b9a\u6e90\u9876\u70b9\u5230\u5176\u4f59\u5404\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84\u95ee\u9898\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002
Question
\u8f93\u5165\u4e00\u4e2a\u6570\u7ec4 \\(ht\\) \uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4ee3\u8868\u4e00\u4e2a\u5782\u76f4\u9694\u677f\u7684\u9ad8\u5ea6\u3002\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9694\u677f\uff0c\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u7a7a\u95f4\u53ef\u4ee5\u7ec4\u6210\u4e00\u4e2a\u5bb9\u5668\u3002
\u5bb9\u5668\u7684\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u548c\u5bbd\u5ea6\u7684\u4e58\u79ef\uff08\u9762\u79ef\uff09\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u8f83\u77ed\u7684\u9694\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u4e2a\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002
\u8bf7\u5728\u6570\u7ec4\u4e2d\u9009\u62e9\u4e24\u4e2a\u9694\u677f\uff0c\u4f7f\u5f97\u7ec4\u6210\u7684\u5bb9\u5668\u7684\u5bb9\u91cf\u6700\u5927\uff0c\u8fd4\u56de\u6700\u5927\u5bb9\u91cf\u3002\u793a\u4f8b\u5982\u56fe 15-7 \u6240\u793a\u3002
\u56fe 15-7 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e
\u5bb9\u5668\u7531\u4efb\u610f\u4e24\u4e2a\u9694\u677f\u56f4\u6210\uff0c\u56e0\u6b64\u672c\u9898\u7684\u72b6\u6001\u4e3a\u4e24\u4e2a\u9694\u677f\u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002
\u6839\u636e\u9898\u610f\uff0c\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u4e58\u4ee5\u5bbd\u5ea6\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002\u8bbe\u5bb9\u91cf\u4e3a \\(cap[i, j]\\) \uff0c\u5219\u53ef\u5f97\u8ba1\u7b97\u516c\u5f0f\uff1a
\\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\]\u8bbe\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u4e24\u4e2a\u9694\u677f\u7684\u7ec4\u5408\u6570\u91cf\uff08\u72b6\u6001\u603b\u6570\uff09\u4e3a \\(C_n^2 = \\frac{n(n - 1)}{2}\\) \u4e2a\u3002\u6700\u76f4\u63a5\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u7a77\u4e3e\u6240\u6709\u72b6\u6001\uff0c\u4ece\u800c\u6c42\u5f97\u6700\u5927\u5bb9\u91cf\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002
"},{"location":"chapter_greedy/max_capacity_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"\u8fd9\u9053\u9898\u8fd8\u6709\u66f4\u9ad8\u6548\u7387\u7684\u89e3\u6cd5\u3002\u5982\u56fe 15-8 \u6240\u793a\uff0c\u73b0\u9009\u53d6\u4e00\u4e2a\u72b6\u6001 \\([i, j]\\) \uff0c\u5176\u6ee1\u8db3\u7d22\u5f15 \\(i < j\\) \u4e14\u9ad8\u5ea6 \\(ht[i] < ht[j]\\) \uff0c\u5373 \\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002
\u56fe 15-8 \u00a0 \u521d\u59cb\u72b6\u6001
\u5982\u56fe 15-9 \u6240\u793a\uff0c\u82e5\u6b64\u65f6\u5c06\u957f\u677f \\(j\\) \u5411\u77ed\u677f \\(i\\) \u9760\u8fd1\uff0c\u5219\u5bb9\u91cf\u4e00\u5b9a\u53d8\u5c0f\u3002
\u8fd9\u662f\u56e0\u4e3a\u5728\u79fb\u52a8\u957f\u677f \\(j\\) \u540e\uff0c\u5bbd\u5ea6 \\(j-i\\) \u80af\u5b9a\u53d8\u5c0f\uff1b\u800c\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u56e0\u6b64\u9ad8\u5ea6\u53ea\u53ef\u80fd\u4e0d\u53d8\uff08 \\(i\\) \u4ecd\u4e3a\u77ed\u677f\uff09\u6216\u53d8\u5c0f\uff08\u79fb\u52a8\u540e\u7684 \\(j\\) \u6210\u4e3a\u77ed\u677f\uff09\u3002
\u56fe 15-9 \u00a0 \u5411\u5185\u79fb\u52a8\u957f\u677f\u540e\u7684\u72b6\u6001
\u53cd\u5411\u601d\u8003\uff0c\u6211\u4eec\u53ea\u6709\u5411\u5185\u6536\u7f29\u77ed\u677f \\(i\\) \uff0c\u624d\u6709\u53ef\u80fd\u4f7f\u5bb9\u91cf\u53d8\u5927\u3002\u56e0\u4e3a\u867d\u7136\u5bbd\u5ea6\u4e00\u5b9a\u53d8\u5c0f\uff0c\u4f46\u9ad8\u5ea6\u53ef\u80fd\u4f1a\u53d8\u5927\uff08\u79fb\u52a8\u540e\u7684\u77ed\u677f \\(i\\) \u53ef\u80fd\u4f1a\u53d8\u957f\uff09\u3002\u4f8b\u5982\u5728\u56fe 15-10 \u4e2d\uff0c\u79fb\u52a8\u77ed\u677f\u540e\u9762\u79ef\u53d8\u5927\u3002
\u56fe 15-10 \u00a0 \u5411\u5185\u79fb\u52a8\u77ed\u677f\u540e\u7684\u72b6\u6001
\u7531\u6b64\u4fbf\u53ef\u63a8\u51fa\u672c\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\uff1a\u521d\u59cb\u5316\u4e24\u6307\u9488\u5206\u5217\u5bb9\u5668\u4e24\u7aef\uff0c\u6bcf\u8f6e\u5411\u5185\u6536\u7f29\u77ed\u677f\u5bf9\u5e94\u7684\u6307\u9488\uff0c\u76f4\u81f3\u4e24\u6307\u9488\u76f8\u9047\u3002
\u56fe 15-11 \u5c55\u793a\u4e86\u8d2a\u5fc3\u7b56\u7565\u7684\u6267\u884c\u8fc7\u7a0b\u3002
- \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u5217\u4e0e\u6570\u7ec4\u4e24\u7aef\u3002
- \u8ba1\u7b97\u5f53\u524d\u72b6\u6001\u7684\u5bb9\u91cf \\(cap[i, j]\\) \uff0c\u5e76\u66f4\u65b0\u6700\u5927\u5bb9\u91cf\u3002
- \u6bd4\u8f83\u677f \\(i\\) \u548c \u677f \\(j\\) \u7684\u9ad8\u5ea6\uff0c\u5e76\u5c06\u77ed\u677f\u5411\u5185\u79fb\u52a8\u4e00\u683c\u3002
- \u5faa\u73af\u6267\u884c\u7b2c
2.\u6b65\u548c\u7b2c3.\u6b65\uff0c\u76f4\u81f3 \\(i\\) \u548c \\(j\\) \u76f8\u9047\u65f6\u7ed3\u675f\u3002
\u56fe 15-11 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u8d2a\u5fc3\u8fc7\u7a0b
"},{"location":"chapter_greedy/max_capacity_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u4ee3\u7801\u5faa\u73af\u6700\u591a \\(n\\) \u8f6e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
\u53d8\u91cf \\(i\\)\u3001\\(j\\)\u3001\\(res\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig max_capacity.py
max_capacity.cppdef max_capacity(ht: list[int]) -> int:\n \"\"\"\u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3\"\"\"\n # \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n i, j = 0, len(ht) - 1\n # \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n res = 0\n # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while i < j:\n # \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n cap = min(ht[i], ht[j]) * (j - i)\n res = max(res, cap)\n # \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if ht[i] < ht[j]:\n i += 1\n else:\n j -= 1\n return res\n
max_capacity.java/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(vector<int> &ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0, j = ht.size() - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int cap = min(ht[i], ht[j]) * (j - i);\n res = max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
max_capacity.cs/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int[] ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0, j = ht.length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int cap = Math.min(ht[i], ht[j]) * (j - i);\n res = Math.max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
max_capacity.go/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint MaxCapacity(int[] ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0, j = ht.Length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int cap = Math.Min(ht[i], ht[j]) * (j - i);\n res = Math.Max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
max_capacity.swift/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht []int) int {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n i, j := 0, len(ht)-1\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n res := 0\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n for i < j {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n res = int(math.Max(float64(res), float64(capacity)))\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if ht[i] < ht[j] {\n i++\n } else {\n j--\n }\n }\n return res\n}\n
max_capacity.js/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht: [Int]) -> Int {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n var i = 0, j = ht.count - 1\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n var res = 0\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while i < j {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n let cap = min(ht[i], ht[j]) * (j - i)\n res = max(res, cap)\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if ht[i] < ht[j] {\n i += 1\n } else {\n j -= 1\n }\n }\n return res\n}\n
max_capacity.ts/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n let i = 0,\n j = ht.length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n let res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n const cap = Math.min(ht[i], ht[j]) * (j - i);\n res = Math.max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i += 1;\n } else {\n j -= 1;\n }\n }\n return res;\n}\n
max_capacity.dart/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht: number[]): number {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n let i = 0,\n j = ht.length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n let res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n res = Math.max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i += 1;\n } else {\n j -= 1;\n }\n }\n return res;\n}\n
max_capacity.rs/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(List<int> ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0, j = ht.length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int cap = min(ht[i], ht[j]) * (j - i);\n res = max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
max_capacity.c/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfn max_capacity(ht: &[i32]) -> i32 {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n let mut i = 0;\n let mut j = ht.len() - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n let mut res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while i < j {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n res = std::cmp::max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if ht[i] < ht[j] {\n i += 1;\n } else {\n j -= 1;\n }\n }\n res\n}\n
max_capacity.zig/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int ht[], int htLength) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0;\n int j = htLength - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int capacity = myMin(ht[i], ht[j]) * (j - i);\n res = myMax(res, capacity);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
"},{"location":"chapter_greedy/max_capacity_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"[class]{}-[func]{maxCapacity}\n\u4e4b\u6240\u4ee5\u8d2a\u5fc3\u6bd4\u7a77\u4e3e\u66f4\u5feb\uff0c\u662f\u56e0\u4e3a\u6bcf\u8f6e\u7684\u8d2a\u5fc3\u9009\u62e9\u90fd\u4f1a\u201c\u8df3\u8fc7\u201d\u4e00\u4e9b\u72b6\u6001\u3002
\u6bd4\u5982\u5728\u72b6\u6001 \\(cap[i, j]\\) \u4e0b\uff0c\\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002\u82e5\u8d2a\u5fc3\u5730\u5c06\u77ed\u677f \\(i\\) \u5411\u5185\u79fb\u52a8\u4e00\u683c\uff0c\u4f1a\u5bfc\u81f4\u56fe 15-12 \u6240\u793a\u7684\u72b6\u6001\u88ab\u201c\u8df3\u8fc7\u201d\u3002\u8fd9\u610f\u5473\u7740\u4e4b\u540e\u65e0\u6cd5\u9a8c\u8bc1\u8fd9\u4e9b\u72b6\u6001\u7684\u5bb9\u91cf\u5927\u5c0f\u3002
\\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\]\u56fe 15-12 \u00a0 \u79fb\u52a8\u77ed\u677f\u5bfc\u81f4\u88ab\u8df3\u8fc7\u7684\u72b6\u6001
\u89c2\u5bdf\u53d1\u73b0\uff0c\u8fd9\u4e9b\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u5b9e\u9645\u4e0a\u5c31\u662f\u5c06\u957f\u677f \\(j\\) \u5411\u5185\u79fb\u52a8\u7684\u6240\u6709\u72b6\u6001\u3002\u524d\u9762\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u5185\u79fb\u957f\u677f\u4e00\u5b9a\u4f1a\u5bfc\u81f4\u5bb9\u91cf\u53d8\u5c0f\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u90fd\u4e0d\u53ef\u80fd\u662f\u6700\u4f18\u89e3\uff0c\u8df3\u8fc7\u5b83\u4eec\u4e0d\u4f1a\u5bfc\u81f4\u9519\u8fc7\u6700\u4f18\u89e3\u3002
\u4ee5\u4e0a\u5206\u6790\u8bf4\u660e\uff0c\u79fb\u52a8\u77ed\u677f\u7684\u64cd\u4f5c\u662f\u201c\u5b89\u5168\u201d\u7684\uff0c\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002
"},{"location":"chapter_greedy/max_product_cutting_problem/","title":"15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5c06\u5176\u5207\u5206\u4e3a\u81f3\u5c11\u4e24\u4e2a\u6b63\u6574\u6570\u7684\u548c\uff0c\u6c42\u5207\u5206\u540e\u6240\u6709\u6574\u6570\u7684\u4e58\u79ef\u6700\u5927\u662f\u591a\u5c11\uff0c\u5982\u56fe 15-13 \u6240\u793a\u3002
\u56fe 15-13 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u95ee\u9898\u5b9a\u4e49
\u5047\u8bbe\u6211\u4eec\u5c06 \\(n\\) \u5207\u5206\u4e3a \\(m\\) \u4e2a\u6574\u6570\u56e0\u5b50\uff0c\u5176\u4e2d\u7b2c \\(i\\) \u4e2a\u56e0\u5b50\u8bb0\u4e3a \\(n_i\\) \uff0c\u5373
\\[ n = \\sum_{i=1}^{m}n_i \\]\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u5f97\u6240\u6709\u6574\u6570\u56e0\u5b50\u7684\u6700\u5927\u4e58\u79ef\uff0c\u5373
\\[ \\max(\\prod_{i=1}^{m}n_i) \\]\u6211\u4eec\u9700\u8981\u601d\u8003\u7684\u662f\uff1a\u5207\u5206\u6570\u91cf \\(m\\) \u5e94\u8be5\u591a\u5927\uff0c\u6bcf\u4e2a \\(n_i\\) \u5e94\u8be5\u662f\u591a\u5c11\uff1f
"},{"location":"chapter_greedy/max_product_cutting_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"\u6839\u636e\u7ecf\u9a8c\uff0c\u4e24\u4e2a\u6574\u6570\u7684\u4e58\u79ef\u5f80\u5f80\u6bd4\u5b83\u4eec\u7684\u52a0\u548c\u66f4\u5927\u3002\u5047\u8bbe\u4ece \\(n\\) \u4e2d\u5206\u51fa\u4e00\u4e2a\u56e0\u5b50 \\(2\\) \uff0c\u5219\u5b83\u4eec\u7684\u4e58\u79ef\u4e3a \\(2(n-2)\\) \u3002\u6211\u4eec\u5c06\u8be5\u4e58\u79ef\u4e0e \\(n\\) \u4f5c\u6bd4\u8f83\uff1a
\\[ \\begin{aligned} 2(n-2) & \\geq n \\newline 2n - n - 4 & \\geq 0 \\newline n & \\geq 4 \\end{aligned} \\]\u5982\u56fe 15-14 \u6240\u793a\uff0c\u5f53 \\(n \\geq 4\\) \u65f6\uff0c\u5207\u5206\u51fa\u4e00\u4e2a \\(2\\) \u540e\u4e58\u79ef\u4f1a\u53d8\u5927\uff0c\u8fd9\u8bf4\u660e\u5927\u4e8e\u7b49\u4e8e \\(4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u88ab\u5207\u5206\u3002
\u8d2a\u5fc3\u7b56\u7565\u4e00\uff1a\u5982\u679c\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b \\(\\geq 4\\) \u7684\u56e0\u5b50\uff0c\u90a3\u4e48\u5b83\u5c31\u5e94\u8be5\u88ab\u7ee7\u7eed\u5207\u5206\u3002\u6700\u7ec8\u7684\u5207\u5206\u65b9\u6848\u53ea\u5e94\u51fa\u73b0 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u79cd\u56e0\u5b50\u3002
\u56fe 15-14 \u00a0 \u5207\u5206\u5bfc\u81f4\u4e58\u79ef\u53d8\u5927
\u63a5\u4e0b\u6765\u601d\u8003\u54ea\u4e2a\u56e0\u5b50\u662f\u6700\u4f18\u7684\u3002\u5728 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u4e2a\u56e0\u5b50\u4e2d\uff0c\u663e\u7136 \\(1\\) \u662f\u6700\u5dee\u7684\uff0c\u56e0\u4e3a \\(1 \\times (n-1) < n\\) \u6052\u6210\u7acb\uff0c\u5373\u5207\u5206\u51fa \\(1\\) \u53cd\u800c\u4f1a\u5bfc\u81f4\u4e58\u79ef\u51cf\u5c0f\u3002
\u5982\u56fe 15-15 \u6240\u793a\uff0c\u5f53 \\(n = 6\\) \u65f6\uff0c\u6709 \\(3 \\times 3 > 2 \\times 2 \\times 2\\) \u3002\u8fd9\u610f\u5473\u7740\u5207\u5206\u51fa \\(3\\) \u6bd4\u5207\u5206\u51fa \\(2\\) \u66f4\u4f18\u3002
\u8d2a\u5fc3\u7b56\u7565\u4e8c\uff1a\u5728\u5207\u5206\u65b9\u6848\u4e2d\uff0c\u6700\u591a\u53ea\u5e94\u5b58\u5728\u4e24\u4e2a \\(2\\) \u3002\u56e0\u4e3a\u4e09\u4e2a \\(2\\) \u603b\u662f\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002
\u56fe 15-15 \u00a0 \u6700\u4f18\u5207\u5206\u56e0\u5b50
\u7efc\u4e0a\u6240\u8ff0\uff0c\u53ef\u63a8\u7406\u51fa\u4ee5\u4e0b\u8d2a\u5fc3\u7b56\u7565\u3002
- \u8f93\u5165\u6574\u6570 \\(n\\) \uff0c\u4ece\u5176\u4e0d\u65ad\u5730\u5207\u5206\u51fa\u56e0\u5b50 \\(3\\) \uff0c\u76f4\u81f3\u4f59\u6570\u4e3a \\(0\\)\u3001\\(1\\)\u3001\\(2\\) \u3002
- \u5f53\u4f59\u6570\u4e3a \\(0\\) \u65f6\uff0c\u4ee3\u8868 \\(n\\) \u662f \\(3\\) \u7684\u500d\u6570\uff0c\u56e0\u6b64\u4e0d\u505a\u4efb\u4f55\u5904\u7406\u3002
- \u5f53\u4f59\u6570\u4e3a \\(2\\) \u65f6\uff0c\u4e0d\u7ee7\u7eed\u5212\u5206\uff0c\u4fdd\u7559\u4e4b\u3002
- \u5f53\u4f59\u6570\u4e3a \\(1\\) \u65f6\uff0c\u7531\u4e8e \\(2 \\times 2 > 1 \\times 3\\) \uff0c\u56e0\u6b64\u5e94\u5c06\u6700\u540e\u4e00\u4e2a \\(3\\) \u66ff\u6362\u4e3a \\(2\\) \u3002
\u5982\u56fe 15-16 \u6240\u793a\uff0c\u6211\u4eec\u65e0\u987b\u901a\u8fc7\u5faa\u73af\u6765\u5207\u5206\u6574\u6570\uff0c\u800c\u53ef\u4ee5\u5229\u7528\u5411\u4e0b\u6574\u9664\u8fd0\u7b97\u5f97\u5230 \\(3\\) \u7684\u4e2a\u6570 \\(a\\) \uff0c\u7528\u53d6\u6a21\u8fd0\u7b97\u5f97\u5230\u4f59\u6570 \\(b\\) \uff0c\u6b64\u65f6\u6709\uff1a
\\[ n = 3 a + b \\]\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e \\(n \\leq 3\\) \u7684\u8fb9\u754c\u60c5\u51b5\uff0c\u5fc5\u987b\u62c6\u5206\u51fa\u4e00\u4e2a \\(1\\) \uff0c\u4e58\u79ef\u4e3a \\(1 \\times (n - 1)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig max_product_cutting.py
max_product_cutting.cppdef max_product_cutting(n: int) -> int:\n \"\"\"\u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3\"\"\"\n # \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if n <= 3:\n return 1 * (n - 1)\n # \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n a, b = n // 3, n % 3\n if b == 1:\n # \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return int(math.pow(3, a - 1)) * 2 * 2\n if b == 2:\n # \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return int(math.pow(3, a)) * 2\n # \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return int(math.pow(3, a))\n
max_product_cutting.java/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n / 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return (int)pow(3, a - 1) * 2 * 2;\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int)pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int)pow(3, a);\n}\n
max_product_cutting.cs/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n / 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return (int) Math.pow(3, a - 1) * 2 * 2;\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int) Math.pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int) Math.pow(3, a);\n}\n
max_product_cutting.go/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint MaxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n / 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return (int)Math.Pow(3, a - 1) * 2 * 2;\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int)Math.Pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int)Math.Pow(3, a);\n}\n
max_product_cutting.swift/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n int) int {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if n <= 3 {\n return 1 * (n - 1)\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n a := n / 3\n b := n % 3\n if b == 1 {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return int(math.Pow(3, float64(a-1))) * 2 * 2\n }\n if b == 2 {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return int(math.Pow(3, float64(a))) * 2\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return int(math.Pow(3, float64(a)))\n}\n
max_product_cutting.js/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n: Int) -> Int {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if n <= 3 {\n return 1 * (n - 1)\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n let a = n / 3\n let b = n % 3\n if b == 1 {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return pow(3, a - 1) * 2 * 2\n }\n if b == 2 {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a) * 2\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a)\n}\n
max_product_cutting.ts/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n let a = Math.floor(n / 3);\n let b = n % 3;\n if (b === 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return Math.pow(3, a - 1) * 2 * 2;\n }\n if (b === 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return Math.pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return Math.pow(3, a);\n}\n
max_product_cutting.dart/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n: number): number {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n let a: number = Math.floor(n / 3);\n let b: number = n % 3;\n if (b === 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return Math.pow(3, a - 1) * 2 * 2;\n }\n if (b === 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return Math.pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return Math.pow(3, a);\n}\n
max_product_cutting.rs/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n ~/ 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return (pow(3, a - 1) * 2 * 2).toInt();\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (pow(3, a) * 2).toInt();\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a).toInt();\n}\n
max_product_cutting.c/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfn max_product_cutting(n: i32) -> i32 {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if n <= 3 {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n let a = n / 3;\n let b = n % 3;\n if b == 1 {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n 3_i32.pow(a as u32 - 1) * 2 * 2\n } else if b == 2 {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n 3_i32.pow(a as u32) * 2\n } else {\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n 3_i32.pow(a as u32)\n }\n}\n
max_product_cutting.zig/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n / 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return pow(3, a - 1) * 2 * 2;\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a);\n}\n[class]{}-[func]{maxProductCutting}\n\u56fe 15-16 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u8ba1\u7b97\u65b9\u6cd5
\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u5e42\u8fd0\u7b97\u7684\u5b9e\u73b0\u65b9\u6cd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5e38\u7528\u7684\u5e42\u8ba1\u7b97\u51fd\u6570\u6709\u4e09\u79cd\u3002
- \u8fd0\u7b97\u7b26
**\u548c\u51fd\u6570pow()\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log\u2061 a)\\) \u3002 - \u51fd\u6570
math.pow()\u5185\u90e8\u8c03\u7528 C \u8bed\u8a00\u5e93\u7684pow()\u51fd\u6570\uff0c\u5176\u6267\u884c\u6d6e\u70b9\u53d6\u5e42\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
\u53d8\u91cf \\(a\\) \u548c \\(b\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
"},{"location":"chapter_greedy/max_product_cutting_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"\u4f7f\u7528\u53cd\u8bc1\u6cd5\uff0c\u53ea\u5206\u6790 \\(n \\geq 3\\) \u7684\u60c5\u51b5\u3002
- \u6240\u6709\u56e0\u5b50 \\(\\leq 3\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728 \\(\\geq 4\\) \u7684\u56e0\u5b50 \\(x\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u5c06\u5176\u7ee7\u7eed\u5212\u5206\u4e3a \\(2(x-2)\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002
- \u5207\u5206\u65b9\u6848\u4e0d\u5305\u542b \\(1\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728\u4e00\u4e2a\u56e0\u5b50 \\(1\\) \uff0c\u90a3\u4e48\u5b83\u4e00\u5b9a\u53ef\u4ee5\u5408\u5e76\u5165\u53e6\u5916\u4e00\u4e2a\u56e0\u5b50\u4e2d\uff0c\u4ee5\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002
- \u5207\u5206\u65b9\u6848\u6700\u591a\u5305\u542b\u4e24\u4e2a \\(2\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b\u4e09\u4e2a \\(2\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4e58\u79ef\u66f4\u5927\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002
- \u8d2a\u5fc3\u7b97\u6cd5\u901a\u5e38\u7528\u4e8e\u89e3\u51b3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5176\u539f\u7406\u662f\u5728\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\u90fd\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002
- \u8d2a\u5fc3\u7b97\u6cd5\u4f1a\u8fed\u4ee3\u5730\u505a\u51fa\u4e00\u4e2a\u53c8\u4e00\u4e2a\u7684\u8d2a\u5fc3\u9009\u62e9\uff0c\u6bcf\u8f6e\u90fd\u5c06\u95ee\u9898\u8f6c\u5316\u6210\u4e00\u4e2a\u89c4\u6a21\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u95ee\u9898\u88ab\u89e3\u51b3\u3002
- \u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u5b9e\u73b0\u7b80\u5355\uff0c\u8fd8\u5177\u6709\u5f88\u9ad8\u7684\u89e3\u9898\u6548\u7387\u3002\u76f8\u6bd4\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u66f4\u4f4e\u3002
- \u5728\u96f6\u94b1\u5151\u6362\u95ee\u9898\u4e2d\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1b\u5bf9\u4e8e\u53e6\u5916\u4e00\u4e9b\u786c\u5e01\u7ec4\u5408\u5219\u4e0d\u7136\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u80fd\u627e\u5230\u5f88\u5dee\u7684\u89e3\u3002
- \u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u7684\u95ee\u9898\u5177\u6709\u4e24\u5927\u6027\u8d28\uff1a\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u4ee3\u8868\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002
- \u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u8bc1\u660e\u5e76\u4e0d\u7b80\u5355\u3002\u76f8\u5bf9\u6765\u8bf4\uff0c\u8bc1\u4f2a\u66f4\u52a0\u5bb9\u6613\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002
- \u6c42\u89e3\u8d2a\u5fc3\u95ee\u9898\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\uff1a\u95ee\u9898\u5206\u6790\u3001\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u3001\u6b63\u786e\u6027\u8bc1\u660e\u3002\u5176\u4e2d\uff0c\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6838\u5fc3\u6b65\u9aa4\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u5f80\u5f80\u662f\u96be\u70b9\u3002
- \u5206\u6570\u80cc\u5305\u95ee\u9898\u5728 0-1 \u80cc\u5305\u7684\u57fa\u7840\u4e0a\uff0c\u5141\u8bb8\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u56e0\u6b64\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u3002\u8d2a\u5fc3\u7b56\u7565\u7684\u6b63\u786e\u6027\u53ef\u4ee5\u4f7f\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\u3002
- \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u53ef\u4f7f\u7528\u7a77\u4e3e\u6cd5\u6c42\u89e3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u901a\u8fc7\u8bbe\u8ba1\u8d2a\u5fc3\u7b56\u7565\uff0c\u6bcf\u8f6e\u5411\u5185\u79fb\u52a8\u77ed\u677f\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \u3002
- \u5728\u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u63a8\u7406\u51fa\u4e24\u4e2a\u8d2a\u5fc3\u7b56\u7565\uff1a\\(\\geq 4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u7ee7\u7eed\u5207\u5206\u3001\u6700\u4f18\u5207\u5206\u56e0\u5b50\u4e3a \\(3\\) \u3002\u4ee3\u7801\u4e2d\u5305\u542b\u5e42\u8fd0\u7b97\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5e42\u8fd0\u7b97\u5b9e\u73b0\u65b9\u6cd5\uff0c\u901a\u5e38\u4e3a \\(O(1)\\) \u6216 \\(O(\\log n)\\) \u3002
Abstract
\u5728\u8ba1\u7b97\u673a\u4e16\u754c\u4e2d\uff0c\u54c8\u5e0c\u8868\u5982\u540c\u4e00\u4f4d\u806a\u6167\u7684\u56fe\u4e66\u7ba1\u7406\u5458\u3002
\u4ed6\u77e5\u9053\u5982\u4f55\u8ba1\u7b97\u7d22\u4e66\u53f7\uff0c\u4ece\u800c\u53ef\u4ee5\u5feb\u901f\u627e\u5230\u76ee\u6807\u56fe\u4e66\u3002
"},{"location":"chapter_hashing/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 6.1 \u00a0 \u54c8\u5e0c\u8868
- 6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81
- 6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5
- 6.4 \u00a0 \u5c0f\u7ed3
\u524d\u4e24\u8282\u4ecb\u7ecd\u4e86\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u539f\u7406\u548c\u54c8\u5e0c\u51b2\u7a81\u7684\u5904\u7406\u65b9\u6cd5\u3002\u7136\u800c\u65e0\u8bba\u662f\u5f00\u653e\u5bfb\u5740\u8fd8\u662f\u94fe\u5f0f\u5730\u5740\uff0c\u5b83\u4eec\u53ea\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u53d1\u751f\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\uff0c\u800c\u65e0\u6cd5\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u3002
\u5982\u679c\u54c8\u5e0c\u51b2\u7a81\u8fc7\u4e8e\u9891\u7e41\uff0c\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u5219\u4f1a\u6025\u5267\u52a3\u5316\u3002\u5982\u56fe 6-8 \u6240\u793a\uff0c\u5bf9\u4e8e\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\uff0c\u7406\u60f3\u60c5\u51b5\u4e0b\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u5404\u4e2a\u6876\u4e2d\uff0c\u8fbe\u5230\u6700\u4f73\u67e5\u8be2\u6548\u7387\uff1b\u6700\u5dee\u60c5\u51b5\u4e0b\u6240\u6709\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002
\u56fe 6-8 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u7684\u6700\u4f73\u60c5\u51b5\u4e0e\u6700\u5dee\u60c5\u51b5
\u952e\u503c\u5bf9\u7684\u5206\u5e03\u60c5\u51b5\u7531\u54c8\u5e0c\u51fd\u6570\u51b3\u5b9a\u3002\u56de\u5fc6\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u6b65\u9aa4\uff0c\u5148\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u518d\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\uff1a
index = hash(key) % capacity\n\u89c2\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf
capacity\u56fa\u5b9a\u65f6\uff0c\u54c8\u5e0c\u7b97\u6cd5hash()\u51b3\u5b9a\u4e86\u8f93\u51fa\u503c\uff0c\u8fdb\u800c\u51b3\u5b9a\u4e86\u952e\u503c\u5bf9\u5728\u54c8\u5e0c\u8868\u4e2d\u7684\u5206\u5e03\u60c5\u51b5\u3002\u8fd9\u610f\u5473\u7740\uff0c\u4e3a\u4e86\u964d\u4f4e\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u6982\u7387\uff0c\u6211\u4eec\u5e94\u5f53\u5c06\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u54c8\u5e0c\u7b97\u6cd5
"},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u76ee\u6807","text":"hash()\u7684\u8bbe\u8ba1\u4e0a\u3002\u4e3a\u4e86\u5b9e\u73b0\u201c\u65e2\u5feb\u53c8\u7a33\u201d\u7684\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u5177\u5907\u4ee5\u4e0b\u7279\u70b9\u3002
- \u786e\u5b9a\u6027\uff1a\u5bf9\u4e8e\u76f8\u540c\u7684\u8f93\u5165\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u59cb\u7ec8\u4ea7\u751f\u76f8\u540c\u7684\u8f93\u51fa\u3002\u8fd9\u6837\u624d\u80fd\u786e\u4fdd\u54c8\u5e0c\u8868\u662f\u53ef\u9760\u7684\u3002
- \u6548\u7387\u9ad8\uff1a\u8ba1\u7b97\u54c8\u5e0c\u503c\u7684\u8fc7\u7a0b\u5e94\u8be5\u8db3\u591f\u5feb\u3002\u8ba1\u7b97\u5f00\u9500\u8d8a\u5c0f\uff0c\u54c8\u5e0c\u8868\u7684\u5b9e\u7528\u6027\u8d8a\u9ad8\u3002
- \u5747\u5300\u5206\u5e03\uff1a\u54c8\u5e0c\u7b97\u6cd5\u5e94\u4f7f\u5f97\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u54c8\u5e0c\u8868\u4e2d\u3002\u5206\u5e03\u8d8a\u5747\u5300\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002
\u5b9e\u9645\u4e0a\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u54c8\u5e0c\u8868\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5176\u4ed6\u9886\u57df\u4e2d\u3002
- \u5bc6\u7801\u5b58\u50a8\uff1a\u4e3a\u4e86\u4fdd\u62a4\u7528\u6237\u5bc6\u7801\u7684\u5b89\u5168\uff0c\u7cfb\u7edf\u901a\u5e38\u4e0d\u4f1a\u76f4\u63a5\u5b58\u50a8\u7528\u6237\u7684\u660e\u6587\u5bc6\u7801\uff0c\u800c\u662f\u5b58\u50a8\u5bc6\u7801\u7684\u54c8\u5e0c\u503c\u3002\u5f53\u7528\u6237\u8f93\u5165\u5bc6\u7801\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5bf9\u8f93\u5165\u7684\u5bc6\u7801\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u7136\u540e\u4e0e\u5b58\u50a8\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u5bc6\u7801\u5c31\u88ab\u89c6\u4e3a\u6b63\u786e\u3002
- \u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5\uff1a\u6570\u636e\u53d1\u9001\u65b9\u53ef\u4ee5\u8ba1\u7b97\u6570\u636e\u7684\u54c8\u5e0c\u503c\u5e76\u5c06\u5176\u4e00\u540c\u53d1\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8ba1\u7b97\u63a5\u6536\u5230\u7684\u6570\u636e\u7684\u54c8\u5e0c\u503c\uff0c\u5e76\u4e0e\u63a5\u6536\u5230\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u6570\u636e\u5c31\u88ab\u89c6\u4e3a\u5b8c\u6574\u3002
\u5bf9\u4e8e\u5bc6\u7801\u5b66\u7684\u76f8\u5173\u5e94\u7528\uff0c\u4e3a\u4e86\u9632\u6b62\u4ece\u54c8\u5e0c\u503c\u63a8\u5bfc\u51fa\u539f\u59cb\u5bc6\u7801\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9700\u8981\u5177\u5907\u66f4\u9ad8\u7b49\u7ea7\u7684\u5b89\u5168\u7279\u6027\u3002
- \u5355\u5411\u6027\uff1a\u65e0\u6cd5\u901a\u8fc7\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u5173\u4e8e\u8f93\u5165\u6570\u636e\u7684\u4efb\u4f55\u4fe1\u606f\u3002
- \u6297\u78b0\u649e\u6027\uff1a\u5e94\u5f53\u6781\u96be\u627e\u5230\u4e24\u4e2a\u4e0d\u540c\u7684\u8f93\u5165\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u54c8\u5e0c\u503c\u76f8\u540c\u3002
- \u96ea\u5d29\u6548\u5e94\uff1a\u8f93\u5165\u7684\u5fae\u5c0f\u53d8\u5316\u5e94\u5f53\u5bfc\u81f4\u8f93\u51fa\u7684\u663e\u8457\u4e14\u4e0d\u53ef\u9884\u6d4b\u7684\u53d8\u5316\u3002
\u8bf7\u6ce8\u610f\uff0c\u201c\u5747\u5300\u5206\u5e03\u201d\u4e0e\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u4e24\u4e2a\u72ec\u7acb\u7684\u6982\u5ff5\uff0c\u6ee1\u8db3\u5747\u5300\u5206\u5e03\u4e0d\u4e00\u5b9a\u6ee1\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u968f\u673a\u8f93\u5165
"},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1","text":"key\u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570key % 100\u53ef\u4ee5\u4ea7\u751f\u5747\u5300\u5206\u5e03\u7684\u8f93\u51fa\u3002\u7136\u800c\u8be5\u54c8\u5e0c\u7b97\u6cd5\u8fc7\u4e8e\u7b80\u5355\uff0c\u6240\u6709\u540e\u4e24\u4f4d\u76f8\u7b49\u7684key\u7684\u8f93\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4ece\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684key\uff0c\u4ece\u800c\u7834\u89e3\u5bc6\u7801\u3002\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u9700\u8981\u8003\u8651\u8bb8\u591a\u56e0\u7d20\u7684\u590d\u6742\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u67d0\u4e9b\u8981\u6c42\u4e0d\u9ad8\u7684\u573a\u666f\uff0c\u6211\u4eec\u4e5f\u80fd\u8bbe\u8ba1\u4e00\u4e9b\u7b80\u5355\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002
- \u52a0\u6cd5\u54c8\u5e0c\uff1a\u5bf9\u8f93\u5165\u7684\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u8fdb\u884c\u76f8\u52a0\uff0c\u5c06\u5f97\u5230\u7684\u603b\u548c\u4f5c\u4e3a\u54c8\u5e0c\u503c\u3002
- \u4e58\u6cd5\u54c8\u5e0c\uff1a\u5229\u7528\u4e58\u6cd5\u7684\u4e0d\u76f8\u5173\u6027\uff0c\u6bcf\u8f6e\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5404\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u54c8\u5e0c\u503c\u4e2d\u3002
- \u5f02\u6216\u54c8\u5e0c\uff1a\u5c06\u8f93\u5165\u6570\u636e\u7684\u6bcf\u4e2a\u5143\u7d20\u901a\u8fc7\u5f02\u6216\u64cd\u4f5c\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\u3002
- \u65cb\u8f6c\u54c8\u5e0c\uff1a\u5c06\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u79ef\u4e4b\u524d\u90fd\u4f1a\u5bf9\u54c8\u5e0c\u503c\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u3002
simple_hash.cppdef add_hash(key: str) -> int:\n \"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\n hash = 0\n modulus = 1000000007\n for c in key:\n hash += ord(c)\n return hash % modulus\n\ndef mul_hash(key: str) -> int:\n \"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\n hash = 0\n modulus = 1000000007\n for c in key:\n hash = 31 * hash + ord(c)\n return hash % modulus\n\ndef xor_hash(key: str) -> int:\n \"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\n hash = 0\n modulus = 1000000007\n for c in key:\n hash ^= ord(c)\n return hash % modulus\n\ndef rot_hash(key: str) -> int:\n \"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\n hash = 0\n modulus = 1000000007\n for c in key:\n hash = (hash << 4) ^ (hash >> 28) ^ ord(c)\n return hash % modulus\n
simple_hash.java/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (unsigned char c : key) {\n hash = (hash + (int)c) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (unsigned char c : key) {\n hash = (31 * hash + (int)c) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\n int hash = 0;\n const int MODULUS = 1000000007;\n for (unsigned char c : key) {\n hash ^= (int)c;\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (unsigned char c : key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ (int)c) % MODULUS;\n }\n return (int)hash;\n}\n
simple_hash.cs/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n long hash = 0;\n final int MODULUS = 1000000007;\n for (char c : key.toCharArray()) {\n hash = (hash + (int) c) % MODULUS;\n }\n return (int) hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n long hash = 0;\n final int MODULUS = 1000000007;\n for (char c : key.toCharArray()) {\n hash = (31 * hash + (int) c) % MODULUS;\n }\n return (int) hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (char c : key.toCharArray()) {\n hash ^= (int) c;\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n long hash = 0;\n final int MODULUS = 1000000007;\n for (char c : key.toCharArray()) {\n hash = ((hash << 4) ^ (hash >> 28) ^ (int) c) % MODULUS;\n }\n return (int) hash;\n}\n
simple_hash.go/* \u52a0\u6cd5\u54c8\u5e0c */\nint AddHash(string key) {\n long hash = 0;\n const int MODULUS = 1000000007;\n foreach (char c in key) {\n hash = (hash + c) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint MulHash(string key) {\n long hash = 0;\n const int MODULUS = 1000000007;\n foreach (char c in key) {\n hash = (31 * hash + c) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint XorHash(string key) {\n int hash = 0;\n const int MODULUS = 1000000007;\n foreach (char c in key) {\n hash ^= c;\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint RotHash(string key) {\n long hash = 0;\n const int MODULUS = 1000000007;\n foreach (char c in key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ c) % MODULUS;\n }\n return (int)hash;\n}\n
simple_hash.swift/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\n var hash int64\n var modulus int64\n\n modulus = 1000000007\n for _, b := range []byte(key) {\n hash = (hash + int64(b)) % modulus\n }\n return int(hash)\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\n var hash int64\n var modulus int64\n\n modulus = 1000000007\n for _, b := range []byte(key) {\n hash = (31*hash + int64(b)) % modulus\n }\n return int(hash)\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\n hash := 0\n modulus := 1000000007\n for _, b := range []byte(key) {\n fmt.Println(int(b))\n hash ^= int(b)\n hash = (31*hash + int(b)) % modulus\n }\n return hash & modulus\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\n var hash int64\n var modulus int64\n\n modulus = 1000000007\n for _, b := range []byte(key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ int64(b)) % modulus\n }\n return int(hash)\n}\n
simple_hash.js/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -> Int {\n var hash = 0\n let MODULUS = 1_000_000_007\n for c in key {\n for scalar in c.unicodeScalars {\n hash = (hash + Int(scalar.value)) % MODULUS\n }\n }\n return hash\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -> Int {\n var hash = 0\n let MODULUS = 1_000_000_007\n for c in key {\n for scalar in c.unicodeScalars {\n hash = (31 * hash + Int(scalar.value)) % MODULUS\n }\n }\n return hash\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -> Int {\n var hash = 0\n let MODULUS = 1_000_000_007\n for c in key {\n for scalar in c.unicodeScalars {\n hash ^= Int(scalar.value)\n }\n }\n return hash & MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -> Int {\n var hash = 0\n let MODULUS = 1_000_000_007\n for c in key {\n for scalar in c.unicodeScalars {\n hash = ((hash << 4) ^ (hash >> 28) ^ Int(scalar.value)) % MODULUS\n }\n }\n return hash\n}\n
simple_hash.ts/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key) {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = (hash + c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key) {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key) {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash ^= c.charCodeAt(0);\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key) {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n
simple_hash.dart/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key: string): number {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = (hash + c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key: string): number {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key: string): number {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash ^= c.charCodeAt(0);\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key: string): number {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n
simple_hash.rs/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (int i = 0; i < key.length; i++) {\n hash = (hash + key.codeUnitAt(i)) % MODULUS;\n }\n return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (int i = 0; i < key.length; i++) {\n hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n }\n return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (int i = 0; i < key.length; i++) {\n hash ^= key.codeUnitAt(i);\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (int i = 0; i < key.length; i++) {\n hash = ((hash << 4) ^ (hash >> 28) ^ key.codeUnitAt(i)) % MODULUS;\n }\n return hash;\n}\n
simple_hash.c/* \u52a0\u6cd5\u54c8\u5e0c */\nfn add_hash(key: &str) -> i32 {\n let mut hash = 0_i64;\n const MODULUS: i64 = 1000000007;\n\n for c in key.chars() {\n hash = (hash + c as i64) % MODULUS;\n }\n\n hash as i32\n} \n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfn mul_hash(key: &str) -> i32 {\n let mut hash = 0_i64;\n const MODULUS: i64 = 1000000007;\n\n for c in key.chars() {\n hash = (31 * hash + c as i64) % MODULUS;\n }\n\n hash as i32\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfn xor_hash(key: &str) -> i32 {\n let mut hash = 0_i64;\n const MODULUS: i64 = 1000000007;\n\n for c in key.chars() {\n hash ^= c as i64;\n }\n\n (hash & MODULUS) as i32\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfn rot_hash(key: &str) -> i32 {\n let mut hash = 0_i64;\n const MODULUS: i64 = 1000000007;\n\n for c in key.chars() {\n hash = ((hash << 4) ^ (hash >> 28) ^ c as i64) % MODULUS;\n }\n\n hash as i32\n}\n
simple_hash.zig/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(char *key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (int i = 0; i < strlen(key); i++) {\n hash = (hash + (unsigned char)key[i]) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(char *key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (int i = 0; i < strlen(key); i++) {\n hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(char *key) {\n int hash = 0;\n const int MODULUS = 1000000007;\n\n for (int i = 0; i < strlen(key); i++) {\n hash ^= (unsigned char)key[i];\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(char *key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (int i = 0; i < strlen(key); i++) {\n hash = ((hash << 4) ^ (hash >> 28) ^ (unsigned char)key[i]) % MODULUS;\n }\n\n return (int)hash;\n}\n[class]{}-[func]{addHash}\n\n[class]{}-[func]{mulHash}\n\n[class]{}-[func]{xorHash}\n\n[class]{}-[func]{rotHash}\n\u89c2\u5bdf\u53d1\u73b0\uff0c\u6bcf\u79cd\u54c8\u5e0c\u7b97\u6cd5\u7684\u6700\u540e\u4e00\u6b65\u90fd\u662f\u5bf9\u5927\u8d28\u6570 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u786e\u4fdd\u54c8\u5e0c\u503c\u5728\u5408\u9002\u7684\u8303\u56f4\u5185\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u8981\u5f3a\u8c03\u5bf9\u8d28\u6570\u53d6\u6a21\uff0c\u6216\u8005\u8bf4\u5bf9\u5408\u6570\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u4e48\uff1f\u8fd9\u662f\u4e00\u4e2a\u6709\u8da3\u7684\u95ee\u9898\u3002
\u5148\u629b\u51fa\u7ed3\u8bba\uff1a\u4f7f\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\u3002\u56e0\u4e3a\u8d28\u6570\u4e0d\u4e0e\u5176\u4ed6\u6570\u5b57\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u53ef\u4ee5\u51cf\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u4ea7\u751f\u7684\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u4ece\u800c\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u3002
\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u5408\u6570 \\(9\\) \u4f5c\u4e3a\u6a21\u6570\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\uff0c\u90a3\u4e48\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684
\\[ \\begin{aligned} \\text{modulus} & = 9 \\newline \\text{key} & = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} & = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\]key\u90fd\u4f1a\u88ab\u6620\u5c04\u5230 \\(0\\)\u3001\\(3\\)\u3001\\(6\\) \u8fd9\u4e09\u4e2a\u54c8\u5e0c\u503c\u3002\u5982\u679c\u8f93\u5165
\\[ \\begin{aligned} \\text{modulus} & = 13 \\newline \\text{key} & = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} & = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\]key\u6070\u597d\u6ee1\u8db3\u8fd9\u79cd\u7b49\u5dee\u6570\u5217\u7684\u6570\u636e\u5206\u5e03\uff0c\u90a3\u4e48\u54c8\u5e0c\u503c\u5c31\u4f1a\u51fa\u73b0\u805a\u5806\uff0c\u4ece\u800c\u52a0\u91cd\u54c8\u5e0c\u51b2\u7a81\u3002\u73b0\u5728\uff0c\u5047\u8bbe\u5c06modulus\u66ff\u6362\u4e3a\u8d28\u6570 \\(13\\) \uff0c\u7531\u4e8ekey\u548cmodulus\u4e4b\u95f4\u4e0d\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u56e0\u6b64\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u7684\u5747\u5300\u6027\u4f1a\u660e\u663e\u63d0\u5347\u3002\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5982\u679c\u80fd\u591f\u4fdd\u8bc1
key\u662f\u968f\u673a\u5747\u5300\u5206\u5e03\u7684\uff0c\u90a3\u4e48\u9009\u62e9\u8d28\u6570\u6216\u8005\u5408\u6570\u4f5c\u4e3a\u6a21\u6570\u90fd\u53ef\u4ee5\uff0c\u5b83\u4eec\u90fd\u80fd\u8f93\u51fa\u5747\u5300\u5206\u5e03\u7684\u54c8\u5e0c\u503c\u3002\u800c\u5f53key\u7684\u5206\u5e03\u5b58\u5728\u67d0\u79cd\u5468\u671f\u6027\u65f6\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73b0\u805a\u96c6\u73b0\u8c61\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u901a\u5e38\u9009\u53d6\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u5e76\u4e14\u8fd9\u4e2a\u8d28\u6570\u6700\u597d\u8db3\u591f\u5927\uff0c\u4ee5\u5c3d\u53ef\u80fd\u6d88\u9664\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002
"},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3 \u00a0 \u5e38\u89c1\u54c8\u5e0c\u7b97\u6cd5","text":"\u4e0d\u96be\u53d1\u73b0\uff0c\u4ee5\u4e0a\u4ecb\u7ecd\u7684\u7b80\u5355\u54c8\u5e0c\u7b97\u6cd5\u90fd\u6bd4\u8f83\u201c\u8106\u5f31\u201d\uff0c\u8fdc\u8fdc\u6ca1\u6709\u8fbe\u5230\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u3002\u4f8b\u5982\uff0c\u7531\u4e8e\u52a0\u6cd5\u548c\u5f02\u6216\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u54c8\u5e0c\u548c\u5f02\u6216\u54c8\u5e0c\u65e0\u6cd5\u533a\u5206\u5185\u5bb9\u76f8\u540c\u4f46\u987a\u5e8f\u4e0d\u540c\u7684\u5b57\u7b26\u4e32\uff0c\u8fd9\u53ef\u80fd\u4f1a\u52a0\u5267\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e76\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u95ee\u9898\u3002
\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u7528\u4e00\u4e9b\u6807\u51c6\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5\u3001SHA-1\u3001SHA-2\u3001SHA-3 \u7b49\u3002\u5b83\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u957f\u5ea6\u7684\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u6052\u5b9a\u957f\u5ea6\u7684\u54c8\u5e0c\u503c\u3002
\u8fd1\u4e00\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5904\u5728\u4e0d\u65ad\u5347\u7ea7\u4e0e\u4f18\u5316\u7684\u8fc7\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u52aa\u529b\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u6027\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u548c\u9ed1\u5ba2\u5219\u81f4\u529b\u4e8e\u5bfb\u627e\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u95ee\u9898\u3002\u8868 6-2 \u5c55\u793a\u4e86\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002
- MD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u51fb\uff0c\u56e0\u6b64\u5b83\u4eec\u88ab\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u5f03\u7528\u3002
- SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u54c8\u5e0c\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73b0\u6210\u529f\u7684\u653b\u51fb\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u7528\u5728\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u4e2d\u3002
- SHA-3 \u76f8\u8f83 SHA-2 \u7684\u5b9e\u73b0\u5f00\u9500\u66f4\u4f4e\u3001\u8ba1\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u76d6\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002
\u8868 6-2 \u00a0 \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5
MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u65f6\u95f4 1992 1995 2002 2008 \u8f93\u51fa\u957f\u5ea6 128 bits 160 bits 256/512 bits 224/256/384/512 bits \u54c8\u5e0c\u51b2\u7a81 \u8f83\u591a \u8f83\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7ea7 \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u9ad8 \u9ad8 \u5e94\u7528 \u5df2\u88ab\u5f03\u7528\uff0c\u4ecd\u7528\u4e8e\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5 \u5df2\u88ab\u5f03\u7528 \u52a0\u5bc6\u8d27\u5e01\u4ea4\u6613\u9a8c\u8bc1\u3001\u6570\u5b57\u7b7e\u540d\u7b49 \u53ef\u7528\u4e8e\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u54c8\u5e0c\u503c","text":"\u6211\u4eec\u77e5\u9053\uff0c\u54c8\u5e0c\u8868\u7684
key\u53ef\u4ee5\u662f\u6574\u6570\u3001\u5c0f\u6570\u6216\u5b57\u7b26\u4e32\u7b49\u6570\u636e\u7c7b\u578b\u3002\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u8fd9\u4e9b\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u8c03\u7528hash()\u51fd\u6570\u6765\u8ba1\u7b97\u5404\u79cd\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u3002- \u6574\u6570\u548c\u5e03\u5c14\u91cf\u7684\u54c8\u5e0c\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002
- \u6d6e\u70b9\u6570\u548c\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u8f83\u4e3a\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8bf7\u81ea\u884c\u5b66\u4e60\u3002
- \u5143\u7ec4\u7684\u54c8\u5e0c\u503c\u662f\u5bf9\u5176\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u54c8\u5e0c\uff0c\u7136\u540e\u5c06\u8fd9\u4e9b\u54c8\u5e0c\u503c\u7ec4\u5408\u8d77\u6765\uff0c\u5f97\u5230\u5355\u4e00\u7684\u54c8\u5e0c\u503c\u3002
- \u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u57fa\u4e8e\u5176\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u901a\u8fc7\u91cd\u5199\u5bf9\u8c61\u7684\u54c8\u5e0c\u65b9\u6cd5\uff0c\u53ef\u5b9e\u73b0\u57fa\u4e8e\u5185\u5bb9\u751f\u6210\u54c8\u5e0c\u503c\u3002
Tip
\u8bf7\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u54c8\u5e0c\u503c\u8ba1\u7b97\u51fd\u6570\u7684\u5b9a\u4e49\u548c\u65b9\u6cd5\u4e0d\u540c\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig built_in_hash.py
built_in_hash.cppnum = 3\nhash_num = hash(num)\n# \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u5c14\u91cf True \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 326484311674566659\n\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 4617003410720528961\n\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7ec4 (12836, '\u5c0f\u54c8') \u7684\u54c8\u5e0c\u503c\u4e3a 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u8282\u70b9\u5bf9\u8c61 <ListNode object at 0x1058fd810> \u7684\u54c8\u5e0c\u503c\u4e3a 274267521\n
built_in_hash.javaint num = 3;\nsize_t hashNum = hash<int>()(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbool bol = true;\nsize_t hashBol = hash<bool>()(bol);\n// \u5e03\u5c14\u91cf 1 \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash<double>()(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 4614256650576692846\n\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash<string>()(str);\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 15466937326284535026\n\n// \u5728 C++ \u4e2d\uff0c\u5185\u7f6e std:hash() \u4ec5\u63d0\u4f9b\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\n// \u6570\u7ec4\u3001\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u9700\u8981\u81ea\u884c\u5b9e\u73b0\n
built_in_hash.csint num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -727081396\n\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@7dc5e7b4 \u7684\u54c8\u5e0c\u503c\u4e3a 2110121908\n
built_in_hash.goint num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729;\n\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -586107568;\n\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// \u8282\u70b9\u5bf9\u8c61 0 \u7684\u54c8\u5e0c\u503c\u4e3a 39053774;\n
built_in_hash.swift// Go \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n
built_in_hash.jslet num = 3\nlet hashNum = num.hashValue\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -2465384235396674631\n\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u6570\u7ec4 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u54c8\u5e0c\u503c\u4e3a -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode \u7684\u54c8\u5e0c\u503c\u4e3a -2434780518035996159\n
built_in_hash.ts// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n
built_in_hash.dart// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n
built_in_hash.rsint num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u5c14\u503c true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2570631074981783\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 468167534\n\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u8282\u70b9\u5bf9\u8c61 Instance of 'ListNode' \u7684\u54c8\u5e0c\u503c\u4e3a 1033450432\n
built_in_hash.cuse std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&mut num_hasher);\nlet hash_num = num_hasher.finish();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\nprintln!(\"\u5c0f\u6570 {} \u7684\u54c8\u5e0c\u503c\u4e3a {}\", dec, hash_dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2566941990314602357\n\nlet str = \"Hello \u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&mut str_hasher);\nlet hash_str = str_hasher.finish();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 16092673739211250988\n\nlet arr = (&12836, &\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// \u5143\u7ec4 (12836, \"\u5c0f\u54c8\") \u7684\u54c8\u5e0c\u503c\u4e3a 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&mut hasher);\nlet hash = hasher.finish();\n// \u8282\u70b9\u5bf9\u8c61 RefCell { value: ListNode { val: 42, next: None } } \u7684\u54c8\u5e0c\u503c\u4e3a15387811073369036852\n
built_in_hash.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n\n\u5728\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u624d\u53ef\u4f5c\u4e3a\u54c8\u5e0c\u8868\u7684
key\u3002\u5047\u5982\u6211\u4eec\u5c06\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4f5c\u4e3akey\uff0c\u5f53\u5217\u8868\u7684\u5185\u5bb9\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u5b83\u7684\u54c8\u5e0c\u503c\u4e5f\u968f\u4e4b\u6539\u53d8\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5230\u539f\u5148\u7684value\u4e86\u3002\u867d\u7136\u81ea\u5b9a\u4e49\u5bf9\u8c61\uff08\u6bd4\u5982\u94fe\u8868\u8282\u70b9\uff09\u7684\u6210\u5458\u53d8\u91cf\u662f\u53ef\u53d8\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u54c8\u5e0c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u901a\u5e38\u662f\u57fa\u4e8e\u5185\u5b58\u5730\u5740\u751f\u6210\u7684\uff0c\u5373\u4f7f\u5bf9\u8c61\u7684\u5185\u5bb9\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u4f46\u5b83\u7684\u5185\u5b58\u5730\u5740\u4e0d\u53d8\uff0c\u54c8\u5e0c\u503c\u4ecd\u7136\u662f\u4e0d\u53d8\u7684\u3002
\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u53d1\u73b0\u5728\u4e0d\u540c\u63a7\u5236\u53f0\u4e2d\u8fd0\u884c\u7a0b\u5e8f\u65f6\uff0c\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u662f\u4e0d\u540c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a Python \u89e3\u91ca\u5668\u5728\u6bcf\u6b21\u542f\u52a8\u65f6\uff0c\u90fd\u4f1a\u4e3a\u5b57\u7b26\u4e32\u54c8\u5e0c\u51fd\u6570\u52a0\u5165\u4e00\u4e2a\u968f\u673a\u7684\u76d0\uff08Salt\uff09\u503c\u3002\u8fd9\u79cd\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u51fb\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002
"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"\u4e0a\u4e00\u8282\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u7406\u8bba\u4e0a\u54c8\u5e0c\u51b2\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u6bd4\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u4e3a\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u4e3a\u6570\u7ec4\u5bb9\u91cf\u5927\u5c0f\uff0c\u5219\u5fc5\u7136\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u81f3\u540c\u4e00\u6876\u7d22\u5f15\u3002
\u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002\u4e3a\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u6bcf\u5f53\u9047\u5230\u54c8\u5e0c\u51b2\u7a81\u5c31\u8fdb\u884c\u54c8\u5e0c\u8868\u6269\u5bb9\uff0c\u76f4\u81f3\u51b2\u7a81\u6d88\u5931\u4e3a\u6b62\u3002\u6b64\u65b9\u6cd5\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u6570\u636e\u642c\u8fd0\u4e0e\u54c8\u5e0c\u503c\u8ba1\u7b97\u3002\u4e3a\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u7b56\u7565\u3002
- \u6539\u826f\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u4f7f\u5f97\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\u3002
- \u4ec5\u5728\u5fc5\u8981\u65f6\uff0c\u5373\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u624d\u6267\u884c\u6269\u5bb9\u64cd\u4f5c\u3002
\u54c8\u5e0c\u8868\u7684\u7ed3\u6784\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u201c\u94fe\u5f0f\u5730\u5740\u201d\u548c\u201c\u5f00\u653e\u5bfb\u5740\u201d\u3002
"},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1 \u00a0 \u94fe\u5f0f\u5730\u5740","text":"\u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u4ec5\u80fd\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u300c\u94fe\u5f0f\u5730\u5740 separate chaining\u300d\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u6362\u4e3a\u94fe\u8868\uff0c\u5c06\u952e\u503c\u5bf9\u4f5c\u4e3a\u94fe\u8868\u8282\u70b9\uff0c\u5c06\u6240\u6709\u53d1\u751f\u51b2\u7a81\u7684\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5728\u540c\u4e00\u94fe\u8868\u4e2d\u3002\u56fe 6-5 \u5c55\u793a\u4e86\u4e00\u4e2a\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u4f8b\u5b50\u3002
\u56fe 6-5 \u00a0 \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868
\u57fa\u4e8e\u94fe\u5f0f\u5730\u5740\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u53d1\u751f\u4e86\u4ee5\u4e0b\u53d8\u5316\u3002
- \u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165
key\uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6876\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4key\u4ee5\u67e5\u627e\u76ee\u6807\u952e\u503c\u5bf9\u3002 - \u6dfb\u52a0\u5143\u7d20\uff1a\u9996\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u5c06\u8282\u70b9\uff08\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u4e2d\u3002
- \u5220\u9664\u5143\u7d20\uff1a\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7684\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u63a5\u7740\u904d\u5386\u94fe\u8868\u4ee5\u67e5\u627e\u76ee\u6807\u8282\u70b9\u5e76\u5c06\u5176\u5220\u9664\u3002
\u94fe\u5f0f\u5730\u5740\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002
- \u5360\u7528\u7a7a\u95f4\u589e\u5927\uff0c\u94fe\u8868\u5305\u542b\u8282\u70b9\u6307\u9488\uff0c\u5b83\u76f8\u6bd4\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002
- \u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff0c\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\u3002
\u4ee5\u4e0b\u4ee3\u7801\u7ed9\u51fa\u4e86\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u9700\u8981\u6ce8\u610f\u4e24\u70b9\u3002
- \u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4ee3\u66ff\u94fe\u8868\uff0c\u4ece\u800c\u7b80\u5316\u4ee3\u7801\u3002\u5728\u8fd9\u79cd\u8bbe\u5b9a\u4e0b\uff0c\u54c8\u5e0c\u8868\uff08\u6570\u7ec4\uff09\u5305\u542b\u591a\u4e2a\u6876\uff0c\u6bcf\u4e2a\u6876\u90fd\u662f\u4e00\u4e2a\u5217\u8868\u3002
- \u4ee5\u4e0b\u5b9e\u73b0\u5305\u542b\u54c8\u5e0c\u8868\u6269\u5bb9\u65b9\u6cd5\u3002\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(\\frac{2}{3}\\) \u65f6\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002
hash_map_chaining.cppclass HashMapChaining:\n \"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self.size = 0 # \u952e\u503c\u5bf9\u6570\u91cf\n self.capacity = 4 # \u54c8\u5e0c\u8868\u5bb9\u91cf\n self.load_thres = 2.0 / 3.0 # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n self.extend_ratio = 2 # \u6269\u5bb9\u500d\u6570\n self.buckets = [[] for _ in range(self.capacity)] # \u6876\u6570\u7ec4\n\n def hash_func(self, key: int) -> int:\n \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n return key % self.capacity\n\n def load_factor(self) -> float:\n \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n return self.size / self.capacity\n\n def get(self, key: int) -> str | None:\n \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n index = self.hash_func(key)\n bucket = self.buckets[index]\n # \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for pair in bucket:\n if pair.key == key:\n return pair.val\n # \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\n return None\n\n def put(self, key: int, val: str):\n \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if self.load_factor() > self.load_thres:\n self.extend()\n index = self.hash_func(key)\n bucket = self.buckets[index]\n # \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for pair in bucket:\n if pair.key == key:\n pair.val = val\n return\n # \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n pair = Pair(key, val)\n bucket.append(pair)\n self.size += 1\n\n def remove(self, key: int):\n \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n index = self.hash_func(key)\n bucket = self.buckets[index]\n # \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for pair in bucket:\n if pair.key == key:\n bucket.remove(pair)\n self.size -= 1\n break\n\n def extend(self):\n \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n buckets = self.buckets\n # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n self.capacity *= self.extend_ratio\n self.buckets = [[] for _ in range(self.capacity)]\n self.size = 0\n # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for bucket in buckets:\n for pair in bucket:\n self.put(pair.key, pair.val)\n\n def print(self):\n \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n for bucket in self.buckets:\n res = []\n for pair in bucket:\n res.append(str(pair.key) + \" -> \" + pair.val)\n print(res)\n
hash_map_chaining.java/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n private:\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n vector<vector<Pair *>> buckets; // \u6876\u6570\u7ec4\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n buckets.resize(capacity);\n }\n\n /* \u6790\u6784\u65b9\u6cd5 */\n ~HashMapChaining() {\n for (auto &bucket : buckets) {\n for (Pair *pair : bucket) {\n // \u91ca\u653e\u5185\u5b58\n delete pair;\n }\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return (double)size / (double)capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n string get(int key) {\n int index = hashFunc(key);\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (Pair *pair : buckets[index]) {\n if (pair->key == key) {\n return pair->val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\";\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, string val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n int index = hashFunc(key);\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (Pair *pair : buckets[index]) {\n if (pair->key == key) {\n pair->val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n buckets[index].push_back(new Pair(key, val));\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n int index = hashFunc(key);\n auto &bucket = buckets[index];\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (int i = 0; i < bucket.size(); i++) {\n if (bucket[i]->key == key) {\n Pair *tmp = bucket[i];\n bucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n delete tmp; // \u91ca\u653e\u5185\u5b58\n size--;\n return;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n vector<vector<Pair *>> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets.clear();\n buckets.resize(capacity);\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (auto &bucket : bucketsTmp) {\n for (Pair *pair : bucket) {\n put(pair->key, pair->val);\n // \u91ca\u653e\u5185\u5b58\n delete pair;\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void print() {\n for (auto &bucket : buckets) {\n cout << \"[\";\n for (Pair *pair : bucket) {\n cout << pair->key << \" -> \" << pair->val << \", \";\n }\n cout << \"]\\n\";\n }\n }\n};\n
hash_map_chaining.cs/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n List<List<Pair>> buckets; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public HashMapChaining() {\n size = 0;\n capacity = 4;\n loadThres = 2.0 / 3.0;\n extendRatio = 2;\n buckets = new ArrayList<>(capacity);\n for (int i = 0; i < capacity; i++) {\n buckets.add(new ArrayList<>());\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return (double) size / capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n String get(int key) {\n int index = hashFunc(key);\n List<Pair> bucket = buckets.get(index);\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (Pair pair : bucket) {\n if (pair.key == key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, String val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n int index = hashFunc(key);\n List<Pair> bucket = buckets.get(index);\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (Pair pair : bucket) {\n if (pair.key == key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n Pair pair = new Pair(key, val);\n bucket.add(pair);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n int index = hashFunc(key);\n List<Pair> bucket = buckets.get(index);\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (Pair pair : bucket) {\n if (pair.key == key) {\n bucket.remove(pair);\n size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n List<List<Pair>> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = new ArrayList<>(capacity);\n for (int i = 0; i < capacity; i++) {\n buckets.add(new ArrayList<>());\n }\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (List<Pair> bucket : bucketsTmp) {\n for (Pair pair : bucket) {\n put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void print() {\n for (List<Pair> bucket : buckets) {\n List<String> res = new ArrayList<>();\n for (Pair pair : bucket) {\n res.add(pair.key + \" -> \" + pair.val);\n }\n System.out.println(res);\n }\n }\n}\n
hash_map_chaining.go/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n List<List<Pair>> buckets; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public HashMapChaining() {\n size = 0;\n capacity = 4;\n loadThres = 2.0 / 3.0;\n extendRatio = 2;\n buckets = new List<List<Pair>>(capacity);\n for (int i = 0; i < capacity; i++) {\n buckets.Add([]);\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int HashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double LoadFactor() {\n return (double)size / capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public string? Get(int key) {\n int index = HashFunc(key);\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n foreach (Pair pair in buckets[index]) {\n if (pair.key == key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void Put(int key, string val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (LoadFactor() > loadThres) {\n Extend();\n }\n int index = HashFunc(key);\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n foreach (Pair pair in buckets[index]) {\n if (pair.key == key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n buckets[index].Add(new Pair(key, val));\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void Remove(int key) {\n int index = HashFunc(key);\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n foreach (Pair pair in buckets[index].ToList()) {\n if (pair.key == key) {\n buckets[index].Remove(pair);\n size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void Extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n List<List<Pair>> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = new List<List<Pair>>(capacity);\n for (int i = 0; i < capacity; i++) {\n buckets.Add([]);\n }\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n foreach (List<Pair> bucket in bucketsTmp) {\n foreach (Pair pair in bucket) {\n Put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void Print() {\n foreach (List<Pair> bucket in buckets) {\n List<string> res = [];\n foreach (Pair pair in bucket) {\n res.Add(pair.key + \" -> \" + pair.val);\n }\n foreach (string kv in res) {\n Console.WriteLine(kv);\n }\n }\n }\n}\n
hash_map_chaining.swift/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\n size int // \u952e\u503c\u5bf9\u6570\u91cf\n capacity int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n loadThres float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n extendRatio int // \u6269\u5bb9\u500d\u6570\n buckets [][]pair // \u6876\u6570\u7ec4\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\n buckets := make([][]pair, 4)\n for i := 0; i < 4; i++ {\n buckets[i] = make([]pair, 0)\n }\n return &hashMapChaining{\n size: 0,\n capacity: 4,\n loadThres: 2.0 / 3.0,\n extendRatio: 2,\n buckets: buckets,\n }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\n idx := m.hashFunc(key)\n bucket := m.buckets[idx]\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for _, p := range bucket {\n if p.key == key {\n return p.val\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if m.loadFactor() > m.loadThres {\n m.extend()\n }\n idx := m.hashFunc(key)\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for i := range m.buckets[idx] {\n if m.buckets[idx][i].key == key {\n m.buckets[idx][i].val = val\n return\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n p := pair{\n key: key,\n val: val,\n }\n m.buckets[idx] = append(m.buckets[idx], p)\n m.size += 1\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\n idx := m.hashFunc(key)\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for i, p := range m.buckets[idx] {\n if p.key == key {\n // \u5207\u7247\u5220\u9664\n m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n m.size -= 1\n break\n }\n }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n tmpBuckets := make([][]pair, len(m.buckets))\n for i := 0; i < len(m.buckets); i++ {\n tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n copy(tmpBuckets[i], m.buckets[i])\n }\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n m.capacity *= m.extendRatio\n m.buckets = make([][]pair, m.capacity)\n for i := 0; i < m.capacity; i++ {\n m.buckets[i] = make([]pair, 0)\n }\n m.size = 0\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for _, bucket := range tmpBuckets {\n for _, p := range bucket {\n m.put(p.key, p.val)\n }\n }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\n var builder strings.Builder\n\n for _, bucket := range m.buckets {\n builder.WriteString(\"[\")\n for _, p := range bucket {\n builder.WriteString(strconv.Itoa(p.key) + \" -> \" + p.val + \" \")\n }\n builder.WriteString(\"]\")\n fmt.Println(builder.String())\n builder.Reset()\n }\n}\n
hash_map_chaining.js/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n var buckets: [[Pair]] // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init() {\n size = 0\n capacity = 4\n loadThres = 2.0 / 3.0\n extendRatio = 2\n buckets = Array(repeating: [], count: capacity)\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n func hashFunc(key: Int) -> Int {\n key % capacity\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n func loadFactor() -> Double {\n Double(size / capacity)\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n func get(key: Int) -> String? {\n let index = hashFunc(key: key)\n let bucket = buckets[index]\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for pair in bucket {\n if pair.key == key {\n return pair.val\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de nil\n return nil\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n func put(key: Int, val: String) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if loadFactor() > loadThres {\n extend()\n }\n let index = hashFunc(key: key)\n let bucket = buckets[index]\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for pair in bucket {\n if pair.key == key {\n pair.val = val\n return\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n let pair = Pair(key: key, val: val)\n buckets[index].append(pair)\n size += 1\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n func remove(key: Int) {\n let index = hashFunc(key: key)\n let bucket = buckets[index]\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (pairIndex, pair) in bucket.enumerated() {\n if pair.key == key {\n buckets[index].remove(at: pairIndex)\n }\n }\n size -= 1\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n func extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n let bucketsTmp = buckets\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio\n buckets = Array(repeating: [], count: capacity)\n size = 0\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for bucket in bucketsTmp {\n for pair in bucket {\n put(key: pair.key, val: pair.val)\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n func print() {\n for bucket in buckets {\n let res = bucket.map { \"\\($0.key) -> \\($0.val)\" }\n Swift.print(res)\n }\n }\n}\n
hash_map_chaining.ts/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n #size; // \u952e\u503c\u5bf9\u6570\u91cf\n #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n #extendRatio; // \u6269\u5bb9\u500d\u6570\n #buckets; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.#size = 0;\n this.#capacity = 4;\n this.#loadThres = 2.0 / 3.0;\n this.#extendRatio = 2;\n this.#buckets = new Array(this.#capacity).fill(null).map((x) => []);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n #hashFunc(key) {\n return key % this.#capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n #loadFactor() {\n return this.#size / this.#capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key) {\n const index = this.#hashFunc(key);\n const bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (const pair of bucket) {\n if (pair.key === key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n put(key, val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (this.#loadFactor() > this.#loadThres) {\n this.#extend();\n }\n const index = this.#hashFunc(key);\n const bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (const pair of bucket) {\n if (pair.key === key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n const pair = new Pair(key, val);\n bucket.push(pair);\n this.#size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n remove(key) {\n const index = this.#hashFunc(key);\n let bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (let i = 0; i < bucket.length; i++) {\n if (bucket[i].key === key) {\n bucket.splice(i, 1);\n this.#size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n #extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n const bucketsTmp = this.#buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n this.#capacity *= this.#extendRatio;\n this.#buckets = new Array(this.#capacity).fill(null).map((x) => []);\n this.#size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (const bucket of bucketsTmp) {\n for (const pair of bucket) {\n this.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print() {\n for (const bucket of this.#buckets) {\n let res = [];\n for (const pair of bucket) {\n res.push(pair.key + ' -> ' + pair.val);\n }\n console.log(res);\n }\n }\n}\n
hash_map_chaining.dart/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n #size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n #capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n #loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n #extendRatio: number; // \u6269\u5bb9\u500d\u6570\n #buckets: Pair[][]; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.#size = 0;\n this.#capacity = 4;\n this.#loadThres = 2.0 / 3.0;\n this.#extendRatio = 2;\n this.#buckets = new Array(this.#capacity).fill(null).map((x) => []);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n #hashFunc(key: number): number {\n return key % this.#capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n #loadFactor(): number {\n return this.#size / this.#capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key: number): string | null {\n const index = this.#hashFunc(key);\n const bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (const pair of bucket) {\n if (pair.key === key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n put(key: number, val: string): void {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (this.#loadFactor() > this.#loadThres) {\n this.#extend();\n }\n const index = this.#hashFunc(key);\n const bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (const pair of bucket) {\n if (pair.key === key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n const pair = new Pair(key, val);\n bucket.push(pair);\n this.#size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n remove(key: number): void {\n const index = this.#hashFunc(key);\n let bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (let i = 0; i < bucket.length; i++) {\n if (bucket[i].key === key) {\n bucket.splice(i, 1);\n this.#size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n #extend(): void {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n const bucketsTmp = this.#buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n this.#capacity *= this.#extendRatio;\n this.#buckets = new Array(this.#capacity).fill(null).map((x) => []);\n this.#size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (const bucket of bucketsTmp) {\n for (const pair of bucket) {\n this.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print(): void {\n for (const bucket of this.#buckets) {\n let res = [];\n for (const pair of bucket) {\n res.push(pair.key + ' -> ' + pair.val);\n }\n console.log(res);\n }\n }\n}\n
hash_map_chaining.rs/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n late int size; // \u952e\u503c\u5bf9\u6570\u91cf\n late int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n late double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n late int extendRatio; // \u6269\u5bb9\u500d\u6570\n late List<List<Pair>> buckets; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n HashMapChaining() {\n size = 0;\n capacity = 4;\n loadThres = 2.0 / 3.0;\n extendRatio = 2;\n buckets = List.generate(capacity, (_) => []);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return size / capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n String? get(int key) {\n int index = hashFunc(key);\n List<Pair> bucket = buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (Pair pair in bucket) {\n if (pair.key == key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, String val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n int index = hashFunc(key);\n List<Pair> bucket = buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (Pair pair in bucket) {\n if (pair.key == key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n Pair pair = Pair(key, val);\n bucket.add(pair);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n int index = hashFunc(key);\n List<Pair> bucket = buckets[index];\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (Pair pair in bucket) {\n if (pair.key == key) {\n bucket.remove(pair);\n size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n List<List<Pair>> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = List.generate(capacity, (_) => []);\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (List<Pair> bucket in bucketsTmp) {\n for (Pair pair in bucket) {\n put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void printHashMap() {\n for (List<Pair> bucket in buckets) {\n List<String> res = [];\n for (Pair pair in bucket) {\n res.add(\"${pair.key} -> ${pair.val}\");\n }\n print(res);\n }\n }\n}\n
hash_map_chaining.c/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapChaining {\n size: i32,\n capacity: i32,\n load_thres: f32,\n extend_ratio: i32,\n buckets: Vec<Vec<Pair>>,\n}\n\nimpl HashMapChaining {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new() -> Self {\n Self {\n size: 0,\n capacity: 4,\n load_thres: 2.0 / 3.0,\n extend_ratio: 2,\n buckets: vec![vec![]; 4],\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n fn hash_func(&self, key: i32) -> usize {\n key as usize % self.capacity as usize\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n fn load_factor(&self) -> f32 {\n self.size as f32 / self.capacity as f32\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n fn remove(&mut self, key: i32) -> Option<String> {\n let index = self.hash_func(key);\n let bucket = &mut self.buckets[index];\n\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for i in 0..bucket.len() {\n if bucket[i].key == key {\n let pair = bucket.remove(i);\n self.size -= 1;\n return Some(pair.val);\n }\n }\n\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\n None\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n fn extend(&mut self) {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n let buckets_tmp = std::mem::replace(&mut self.buckets, vec![]);\n\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n self.capacity *= self.extend_ratio;\n self.buckets = vec![Vec::new(); self.capacity as usize];\n self.size = 0;\n\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for bucket in buckets_tmp {\n for pair in bucket {\n self.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n fn print(&self) {\n for bucket in &self.buckets {\n let mut res = Vec::new();\n for pair in bucket {\n res.push(format!(\"{} -> {}\", pair.key, pair.val));\n }\n println!(\"{:?}\", res);\n }\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n fn put(&mut self, key: i32, val: String) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if self.load_factor() > self.load_thres {\n self.extend();\n }\n\n let index = self.hash_func(key);\n let bucket = &mut self.buckets[index];\n\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for pair in bucket {\n if pair.key == key {\n pair.val = val.clone();\n return;\n }\n }\n let bucket = &mut self.buckets[index];\n\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n let pair = Pair {\n key,\n val: val.clone(),\n };\n bucket.push(pair);\n self.size += 1;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n fn get(&self, key: i32) -> Option<&str> {\n let index = self.hash_func(key);\n let bucket = &self.buckets[index];\n\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for pair in bucket {\n if pair.key == key {\n return Some(&pair.val);\n }\n }\n\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\n None\n }\n}\n
hash_map_chaining.zig/* \u94fe\u8868\u8282\u70b9 */\ntypedef struct Node {\n Pair *pair;\n struct Node *next;\n} Node;\n\n/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n Node **buckets; // \u6876\u6570\u7ec4\n} HashMapChaining;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapChaining *newHashMapChaining() {\n HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n hashMap->size = 0;\n hashMap->capacity = 4;\n hashMap->loadThres = 2.0 / 3.0;\n hashMap->extendRatio = 2;\n hashMap->buckets = (Node **)malloc(hashMap->capacity * sizeof(Node *));\n for (int i = 0; i < hashMap->capacity; i++) {\n hashMap->buckets[i] = NULL;\n }\n return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n for (int i = 0; i < hashMap->capacity; i++) {\n Node *cur = hashMap->buckets[i];\n while (cur) {\n Node *tmp = cur;\n cur = cur->next;\n free(tmp->pair);\n free(tmp);\n }\n }\n free(hashMap->buckets);\n free(hashMap);\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapChaining *hashMap, int key) {\n return key % hashMap->capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\n return (double)hashMap->size / (double)hashMap->capacity;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\n int index = hashFunc(hashMap, key);\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n Node *cur = hashMap->buckets[index];\n while (cur) {\n if (cur->pair->key == key) {\n return cur->pair->val;\n }\n cur = cur->next;\n }\n return \"\"; // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor(hashMap) > hashMap->loadThres) {\n extend(hashMap);\n }\n int index = hashFunc(hashMap, key);\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n Node *cur = hashMap->buckets[index];\n while (cur) {\n if (cur->pair->key == key) {\n strcpy(cur->pair->val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n return;\n }\n cur = cur->next;\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n Pair *newPair = (Pair *)malloc(sizeof(Pair));\n newPair->key = key;\n strcpy(newPair->val, val);\n Node *newNode = (Node *)malloc(sizeof(Node));\n newNode->pair = newPair;\n newNode->next = hashMap->buckets[index];\n hashMap->buckets[index] = newNode;\n hashMap->size++;\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n int oldCapacity = hashMap->capacity;\n Node **oldBuckets = hashMap->buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n hashMap->capacity *= hashMap->extendRatio;\n hashMap->buckets = (Node **)malloc(hashMap->capacity * sizeof(Node *));\n for (int i = 0; i < hashMap->capacity; i++) {\n hashMap->buckets[i] = NULL;\n }\n hashMap->size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (int i = 0; i < oldCapacity; i++) {\n Node *cur = oldBuckets[i];\n while (cur) {\n put(hashMap, cur->pair->key, cur->pair->val);\n Node *temp = cur;\n cur = cur->next;\n // \u91ca\u653e\u5185\u5b58\n free(temp->pair);\n free(temp);\n }\n }\n\n free(oldBuckets);\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n int index = hashFunc(hashMap, key);\n Node *cur = hashMap->buckets[index];\n Node *pre = NULL;\n while (cur) {\n if (cur->pair->key == key) {\n // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n if (pre) {\n pre->next = cur->next;\n } else {\n hashMap->buckets[index] = cur->next;\n }\n // \u91ca\u653e\u5185\u5b58\n free(cur->pair);\n free(cur);\n hashMap->size--;\n return;\n }\n pre = cur;\n cur = cur->next;\n }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapChaining *hashMap) {\n for (int i = 0; i < hashMap->capacity; i++) {\n Node *cur = hashMap->buckets[i];\n printf(\"[\");\n while (cur) {\n printf(\"%d -> %s, \", cur->pair->key, cur->pair->val);\n cur = cur->next;\n }\n printf(\"]\\n\");\n }\n}\n[class]{HashMapChaining}-[func]{}\n\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5f53\u94fe\u8868\u5f88\u957f\u65f6\uff0c\u67e5\u8be2\u6548\u7387 \\(O(n)\\) \u5f88\u5dee\u3002\u6b64\u65f6\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u201cAVL \u6811\u201d\u6216\u201c\u7ea2\u9ed1\u6811\u201d\uff0c\u4ece\u800c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002
"},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2 \u00a0 \u5f00\u653e\u5bfb\u5740","text":"\u300c\u5f00\u653e\u5bfb\u5740 open addressing\u300d\u4e0d\u5f15\u5165\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\uff0c\u63a2\u6d4b\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u3001\u591a\u6b21\u54c8\u5e0c\u7b49\u3002
\u4e0b\u9762\u4ee5\u7ebf\u6027\u63a2\u6d4b\u4e3a\u4f8b\uff0c\u4ecb\u7ecd\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u673a\u5236\u3002
"},{"location":"chapter_hashing/hash_collision/#1","title":"1. \u00a0 \u7ebf\u6027\u63a2\u6d4b","text":"\u7ebf\u6027\u63a2\u6d4b\u91c7\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u641c\u7d22\u6765\u8fdb\u884c\u63a2\u6d4b\uff0c\u5176\u64cd\u4f5c\u65b9\u6cd5\u4e0e\u666e\u901a\u54c8\u5e0c\u8868\u6709\u6240\u4e0d\u540c\u3002
- \u63d2\u5165\u5143\u7d20\uff1a\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u82e5\u53d1\u73b0\u6876\u5185\u5df2\u6709\u5143\u7d20\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u901a\u5e38\u4e3a \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u6876\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002
- \u67e5\u627e\u5143\u7d20\uff1a\u82e5\u53d1\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u5411\u540e\u7ebf\u6027\u904d\u5386\uff0c\u76f4\u5230\u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de
value\u5373\u53ef\uff1b\u5982\u679c\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u76ee\u6807\u5143\u7d20\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8fd4\u56de \\(\\text{None}\\) \u3002
\u56fe 6-6 \u5c55\u793a\u4e86\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u952e\u503c\u5bf9\u5206\u5e03\u3002\u6839\u636e\u6b64\u54c8\u5e0c\u51fd\u6570\uff0c\u6700\u540e\u4e24\u4f4d\u76f8\u540c\u7684
key\u90fd\u4f1a\u88ab\u6620\u5c04\u5230\u76f8\u540c\u7684\u6876\u3002\u800c\u901a\u8fc7\u7ebf\u6027\u63a2\u6d4b\uff0c\u5b83\u4eec\u88ab\u4f9d\u6b21\u5b58\u50a8\u5728\u8be5\u6876\u4ee5\u53ca\u4e4b\u4e0b\u7684\u6876\u4e2d\u3002\u56fe 6-6 \u00a0 \u5f00\u653e\u5bfb\u5740\u548c\u7ebf\u6027\u63a2\u6d4b
\u7136\u800c\uff0c\u7ebf\u6027\u63a2\u6d4b\u5bb9\u6613\u4ea7\u751f\u201c\u805a\u96c6\u73b0\u8c61\u201d\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6570\u7ec4\u4e2d\u8fde\u7eed\u88ab\u5360\u7528\u7684\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u4ece\u800c\u8fdb\u4e00\u6b65\u4fc3\u4f7f\u8be5\u4f4d\u7f6e\u7684\u805a\u5806\u751f\u957f\uff0c\u5f62\u6210\u6076\u6027\u5faa\u73af\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u52a3\u5316\u3002
\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6211\u4eec\u4e0d\u80fd\u5728\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u4e2d\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u8fd9\u662f\u56e0\u4e3a\u5220\u9664\u5143\u7d20\u4f1a\u5728\u6570\u7ec4\u5185\u4ea7\u751f\u4e00\u4e2a\u7a7a\u6876 \\(\\text{None}\\) \uff0c\u800c\u5f53\u67e5\u8be2\u5143\u7d20\u65f6\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230\u8be5\u7a7a\u6876\u5c31\u4f1a\u8fd4\u56de\uff0c\u56e0\u6b64\u5728\u8be5\u7a7a\u6876\u4e4b\u4e0b\u7684\u5143\u7d20\u90fd\u65e0\u6cd5\u518d\u88ab\u8bbf\u95ee\u5230\uff0c\u7a0b\u5e8f\u53ef\u80fd\u8bef\u5224\u8fd9\u4e9b\u5143\u7d20\u4e0d\u5b58\u5728\u3002
\u56fe 6-7 \u00a0 \u5728\u5f00\u653e\u5bfb\u5740\u4e2d\u5220\u9664\u5143\u7d20\u5bfc\u81f4\u7684\u67e5\u8be2\u95ee\u9898
\u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u300c\u61d2\u5220\u9664 lazy deletion\u300d\u673a\u5236\uff1a\u5b83\u4e0d\u76f4\u63a5\u4ece\u54c8\u5e0c\u8868\u4e2d\u79fb\u9664\u5143\u7d20\uff0c\u800c\u662f\u5229\u7528\u4e00\u4e2a\u5e38\u91cf
TOMBSTONE\u6765\u6807\u8bb0\u8fd9\u4e2a\u6876\u3002\u5728\u8be5\u673a\u5236\u4e0b\uff0c\\(\\text{None}\\) \u548cTOMBSTONE\u90fd\u4ee3\u8868\u7a7a\u6876\uff0c\u90fd\u53ef\u4ee5\u653e\u7f6e\u952e\u503c\u5bf9\u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230TOMBSTONE\u65f6\u5e94\u8be5\u7ee7\u7eed\u904d\u5386\uff0c\u56e0\u4e3a\u5176\u4e4b\u4e0b\u53ef\u80fd\u8fd8\u5b58\u5728\u952e\u503c\u5bf9\u3002\u7136\u800c\uff0c\u61d2\u5220\u9664\u53ef\u80fd\u4f1a\u52a0\u901f\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u9000\u5316\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6b21\u5220\u9664\u64cd\u4f5c\u90fd\u4f1a\u4ea7\u751f\u4e00\u4e2a\u5220\u9664\u6807\u8bb0\uff0c\u968f\u7740
TOMBSTONE\u7684\u589e\u52a0\uff0c\u641c\u7d22\u65f6\u95f4\u4e5f\u4f1a\u589e\u52a0\uff0c\u56e0\u4e3a\u7ebf\u6027\u63a2\u6d4b\u53ef\u80fd\u9700\u8981\u8df3\u8fc7\u591a\u4e2aTOMBSTONE\u624d\u80fd\u627e\u5230\u76ee\u6807\u5143\u7d20\u3002\u4e3a\u6b64\uff0c\u8003\u8651\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a
TOMBSTONE\u7684\u7d22\u5f15\uff0c\u5e76\u5c06\u641c\u7d22\u5230\u7684\u76ee\u6807\u5143\u7d20\u4e0e\u8be5TOMBSTONE\u4ea4\u6362\u4f4d\u7f6e\u3002\u8fd9\u6837\u505a\u7684\u597d\u5904\u662f\u5f53\u6bcf\u6b21\u67e5\u8be2\u6216\u6dfb\u52a0\u5143\u7d20\u65f6\uff0c\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u81f3\u8ddd\u79bb\u7406\u60f3\u4f4d\u7f6e\uff08\u63a2\u6d4b\u8d77\u59cb\u70b9\uff09\u66f4\u8fd1\u7684\u6876\uff0c\u4ece\u800c\u4f18\u5316\u67e5\u8be2\u6548\u7387\u3002\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u5305\u542b\u61d2\u5220\u9664\u7684\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u3002\u4e3a\u4e86\u66f4\u52a0\u5145\u5206\u5730\u4f7f\u7528\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u770b\u4f5c\u4e00\u4e2a\u201c\u73af\u5f62\u6570\u7ec4\u201d\uff0c\u5f53\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map_open_addressing.py
hash_map_open_addressing.cppclass HashMapOpenAddressing:\n \"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self.size = 0 # \u952e\u503c\u5bf9\u6570\u91cf\n self.capacity = 4 # \u54c8\u5e0c\u8868\u5bb9\u91cf\n self.load_thres = 2.0 / 3.0 # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n self.extend_ratio = 2 # \u6269\u5bb9\u500d\u6570\n self.buckets: list[Pair | None] = [None] * self.capacity # \u6876\u6570\u7ec4\n self.TOMBSTONE = Pair(-1, \"-1\") # \u5220\u9664\u6807\u8bb0\n\n def hash_func(self, key: int) -> int:\n \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n return key % self.capacity\n\n def load_factor(self) -> float:\n \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n return self.size / self.capacity\n\n def find_bucket(self, key: int) -> int:\n \"\"\"\u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\"\"\"\n index = self.hash_func(key)\n first_tombstone = -1\n # \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while self.buckets[index] is not None:\n # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if self.buckets[index].key == key:\n # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if first_tombstone != -1:\n self.buckets[first_tombstone] = self.buckets[index]\n self.buckets[index] = self.TOMBSTONE\n return first_tombstone # \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n return index # \u8fd4\u56de\u6876\u7d22\u5f15\n # \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n first_tombstone = index\n # \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % self.capacity\n # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return index if first_tombstone == -1 else first_tombstone\n\n def get(self, key: int) -> str:\n \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n index = self.find_bucket(key)\n # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if self.buckets[index] not in [None, self.TOMBSTONE]:\n return self.buckets[index].val\n # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de None\n return None\n\n def put(self, key: int, val: str):\n \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if self.load_factor() > self.load_thres:\n self.extend()\n # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n index = self.find_bucket(key)\n # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if self.buckets[index] not in [None, self.TOMBSTONE]:\n self.buckets[index].val = val\n return\n # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n self.buckets[index] = Pair(key, val)\n self.size += 1\n\n def remove(self, key: int):\n \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n index = self.find_bucket(key)\n # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if self.buckets[index] not in [None, self.TOMBSTONE]:\n self.buckets[index] = self.TOMBSTONE\n self.size -= 1\n\n def extend(self):\n \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n buckets_tmp = self.buckets\n # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n self.capacity *= self.extend_ratio\n self.buckets = [None] * self.capacity\n self.size = 0\n # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for pair in buckets_tmp:\n if pair not in [None, self.TOMBSTONE]:\n self.put(pair.key, pair.val)\n\n def print(self):\n \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n for pair in self.buckets:\n if pair is None:\n print(\"None\")\n elif pair is self.TOMBSTONE:\n print(\"TOMBSTONE\")\n else:\n print(pair.key, \"->\", pair.val)\n
hash_map_open_addressing.java/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n private:\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n const double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n const int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n vector<Pair *> buckets; // \u6876\u6570\u7ec4\n Pair *TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n }\n\n /* \u6790\u6784\u65b9\u6cd5 */\n ~HashMapOpenAddressing() {\n for (Pair *pair : buckets) {\n if (pair != nullptr && pair != TOMBSTONE) {\n delete pair;\n }\n }\n delete TOMBSTONE;\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return (double)size / capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n int findBucket(int key) {\n int index = hashFunc(key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (buckets[index] != nullptr) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (buckets[index]->key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n buckets[firstTombstone] = buckets[index];\n buckets[index] = TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && buckets[index] == TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n string get(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) {\n return buckets[index]->val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\";\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, string val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) {\n buckets[index]->val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n buckets[index] = new Pair(key, val);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) {\n delete buckets[index];\n buckets[index] = TOMBSTONE;\n size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n vector<Pair *> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = vector<Pair *>(capacity, nullptr);\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (Pair *pair : bucketsTmp) {\n if (pair != nullptr && pair != TOMBSTONE) {\n put(pair->key, pair->val);\n delete pair;\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void print() {\n for (Pair *pair : buckets) {\n if (pair == nullptr) {\n cout << \"nullptr\" << endl;\n } else if (pair == TOMBSTONE) {\n cout << \"TOMBSTONE\" << endl;\n } else {\n cout << pair->key << \" -> \" << pair->val << endl;\n }\n }\n }\n};\n
hash_map_open_addressing.cs/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n private int size; // \u952e\u503c\u5bf9\u6570\u91cf\n private int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n private final double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n private final int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n private Pair[] buckets; // \u6876\u6570\u7ec4\n private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public HashMapOpenAddressing() {\n size = 0;\n buckets = new Pair[capacity];\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n private double loadFactor() {\n return (double) size / capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n private int findBucket(int key) {\n int index = hashFunc(key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (buckets[index] != null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (buckets[index].key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n buckets[firstTombstone] = buckets[index];\n buckets[index] = TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && buckets[index] == TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public String get(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n return buckets[index].val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void put(int key, String val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n buckets[index].val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n buckets[index] = new Pair(key, val);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void remove(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n buckets[index] = TOMBSTONE;\n size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n private void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n Pair[] bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = new Pair[capacity];\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (Pair pair : bucketsTmp) {\n if (pair != null && pair != TOMBSTONE) {\n put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void print() {\n for (Pair pair : buckets) {\n if (pair == null) {\n System.out.println(\"null\");\n } else if (pair == TOMBSTONE) {\n System.out.println(\"TOMBSTONE\");\n } else {\n System.out.println(pair.key + \" -> \" + pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.go/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n Pair[] buckets; // \u6876\u6570\u7ec4\n Pair TOMBSTONE = new(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public HashMapOpenAddressing() {\n size = 0;\n buckets = new Pair[capacity];\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int HashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double LoadFactor() {\n return (double)size / capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n int FindBucket(int key) {\n int index = HashFunc(key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (buckets[index] != null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (buckets[index].key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n buckets[firstTombstone] = buckets[index];\n buckets[index] = TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && buckets[index] == TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public string? Get(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = FindBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n return buckets[index].val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void Put(int key, string val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (LoadFactor() > loadThres) {\n Extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = FindBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n buckets[index].val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n buckets[index] = new Pair(key, val);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void Remove(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = FindBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n buckets[index] = TOMBSTONE;\n size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void Extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n Pair[] bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = new Pair[capacity];\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n foreach (Pair pair in bucketsTmp) {\n if (pair != null && pair != TOMBSTONE) {\n Put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void Print() {\n foreach (Pair pair in buckets) {\n if (pair == null) {\n Console.WriteLine(\"null\");\n } else if (pair == TOMBSTONE) {\n Console.WriteLine(\"TOMBSTONE\");\n } else {\n Console.WriteLine(pair.key + \" -> \" + pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.swift/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\n size int // \u952e\u503c\u5bf9\u6570\u91cf\n capacity int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n loadThres float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n extendRatio int // \u6269\u5bb9\u500d\u6570\n buckets []pair // \u6876\u6570\u7ec4\n removed pair // \u5220\u9664\u6807\u8bb0\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n buckets := make([]pair, 4)\n return &hashMapOpenAddressing{\n size: 0,\n capacity: 4,\n loadThres: 2.0 / 3.0,\n extendRatio: 2,\n buckets: buckets,\n removed: pair{\n key: -1,\n val: \"-1\",\n },\n }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapOpenAddressing) hashFunc(key int) int {\n return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapOpenAddressing) loadFactor() float64 {\n return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) get(key int) string {\n idx := m.hashFunc(key)\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\n for i := 0; i < m.capacity; i++ {\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n j := (idx + i) % m.capacity\n // \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\n if m.buckets[j] == (pair{}) {\n return \"\"\n }\n // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if m.buckets[j].key == key && m.buckets[j] != m.removed {\n return m.buckets[j].val\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) put(key int, val string) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if m.loadFactor() > m.loadThres {\n m.extend()\n }\n idx := m.hashFunc(key)\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\n for i := 0; i < m.capacity; i++ {\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n j := (idx + i) % m.capacity\n // \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\n if m.buckets[j] == (pair{}) || m.buckets[j] == m.removed {\n m.buckets[j] = pair{\n key: key,\n val: val,\n }\n m.size += 1\n return\n }\n // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\n if m.buckets[j].key == key {\n m.buckets[j].val = val\n return\n }\n }\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) remove(key int) {\n idx := m.hashFunc(key)\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\n for i := 0; i < m.capacity; i++ {\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n j := (idx + i) % m.capacity\n // \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if m.buckets[j] == (pair{}) {\n return\n }\n // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\n if m.buckets[j].key == key {\n m.buckets[j] = m.removed\n m.size -= 1\n }\n }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n tmpBuckets := make([]pair, len(m.buckets))\n copy(tmpBuckets, m.buckets)\n\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n m.capacity *= m.extendRatio\n m.buckets = make([]pair, m.capacity)\n m.size = 0\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for _, p := range tmpBuckets {\n if p != (pair{}) && p != m.removed {\n m.put(p.key, p.val)\n }\n }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) print() {\n for _, p := range m.buckets {\n if p != (pair{}) {\n fmt.Println(strconv.Itoa(p.key) + \" -> \" + p.val)\n } else {\n fmt.Println(\"nil\")\n }\n }\n}\n
hash_map_open_addressing.js/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n var buckets: [Pair?] // \u6876\u6570\u7ec4\n var TOMBSTONE: Pair // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init() {\n size = 0\n capacity = 4\n loadThres = 2.0 / 3.0\n extendRatio = 2\n buckets = Array(repeating: nil, count: capacity)\n TOMBSTONE = Pair(key: -1, val: \"-1\")\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n func hashFunc(key: Int) -> Int {\n key % capacity\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n func loadFactor() -> Double {\n Double(size / capacity)\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n func findBucket(key: Int) -> Int {\n var index = hashFunc(key: key)\n var firstTombstone = -1\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while buckets[index] != nil {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if buckets[index]!.key == key {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if firstTombstone != -1 {\n buckets[firstTombstone] = buckets[index]\n buckets[index] = TOMBSTONE\n return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if firstTombstone == -1 && buckets[index] == TOMBSTONE {\n firstTombstone = index\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % capacity\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n func get(key: Int) -> String? {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = findBucket(key: key)\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if buckets[index] != nil, buckets[index] != TOMBSTONE {\n return buckets[index]!.val\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return nil\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n func put(key: Int, val: String) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if loadFactor() > loadThres {\n extend()\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = findBucket(key: key)\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if buckets[index] != nil, buckets[index] != TOMBSTONE {\n buckets[index]!.val = val\n return\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n buckets[index] = Pair(key: key, val: val)\n size += 1\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n func remove(key: Int) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = findBucket(key: key)\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if buckets[index] != nil, buckets[index] != TOMBSTONE {\n buckets[index] = TOMBSTONE\n size -= 1\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n func extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n let bucketsTmp = buckets\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio\n buckets = Array(repeating: nil, count: capacity)\n size = 0\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for pair in bucketsTmp {\n if let pair, pair != TOMBSTONE {\n put(key: pair.key, val: pair.val)\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n func print() {\n for pair in buckets {\n if pair == nil {\n Swift.print(\"null\")\n } else if pair == TOMBSTONE {\n Swift.print(\"TOMBSTONE\")\n } else {\n Swift.print(\"\\(pair!.key) -> \\(pair!.val)\")\n }\n }\n }\n}\n
hash_map_open_addressing.ts/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n #size; // \u952e\u503c\u5bf9\u6570\u91cf\n #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n #extendRatio; // \u6269\u5bb9\u500d\u6570\n #buckets; // \u6876\u6570\u7ec4\n #TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.#size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n this.#capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n this.#loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n this.#extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n this.#buckets = Array(this.#capacity).fill(null); // \u6876\u6570\u7ec4\n this.#TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n #hashFunc(key) {\n return key % this.#capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n #loadFactor() {\n return this.#size / this.#capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n #findBucket(key) {\n let index = this.#hashFunc(key);\n let firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (this.#buckets[index] !== null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (this.#buckets[index].key === key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone !== -1) {\n this.#buckets[firstTombstone] = this.#buckets[index];\n this.#buckets[index] = this.#TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (\n firstTombstone === -1 &&\n this.#buckets[index] === this.#TOMBSTONE\n ) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % this.#capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone === -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.#findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (\n this.#buckets[index] !== null &&\n this.#buckets[index] !== this.#TOMBSTONE\n ) {\n return this.#buckets[index].val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n put(key, val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (this.#loadFactor() > this.#loadThres) {\n this.#extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.#findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (\n this.#buckets[index] !== null &&\n this.#buckets[index] !== this.#TOMBSTONE\n ) {\n this.#buckets[index].val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n this.#buckets[index] = new Pair(key, val);\n this.#size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n remove(key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.#findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (\n this.#buckets[index] !== null &&\n this.#buckets[index] !== this.#TOMBSTONE\n ) {\n this.#buckets[index] = this.#TOMBSTONE;\n this.#size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n #extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n const bucketsTmp = this.#buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n this.#capacity *= this.#extendRatio;\n this.#buckets = Array(this.#capacity).fill(null);\n this.#size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (const pair of bucketsTmp) {\n if (pair !== null && pair !== this.#TOMBSTONE) {\n this.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print() {\n for (const pair of this.#buckets) {\n if (pair === null) {\n console.log('null');\n } else if (pair === this.#TOMBSTONE) {\n console.log('TOMBSTONE');\n } else {\n console.log(pair.key + ' -> ' + pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.dart/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n private size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n private capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n private loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n private extendRatio: number; // \u6269\u5bb9\u500d\u6570\n private buckets: Array<Pair | null>; // \u6876\u6570\u7ec4\n private TOMBSTONE: Pair; // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n this.capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n this.loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n this.extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n this.buckets = Array(this.capacity).fill(null); // \u6876\u6570\u7ec4\n this.TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private hashFunc(key: number): number {\n return key % this.capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n private loadFactor(): number {\n return this.size / this.capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n private findBucket(key: number): number {\n let index = this.hashFunc(key);\n let firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (this.buckets[index] !== null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (this.buckets[index]!.key === key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone !== -1) {\n this.buckets[firstTombstone] = this.buckets[index];\n this.buckets[index] = this.TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (\n firstTombstone === -1 &&\n this.buckets[index] === this.TOMBSTONE\n ) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % this.capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone === -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key: number): string | null {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (\n this.buckets[index] !== null &&\n this.buckets[index] !== this.TOMBSTONE\n ) {\n return this.buckets[index]!.val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n put(key: number, val: string): void {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (this.loadFactor() > this.loadThres) {\n this.extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (\n this.buckets[index] !== null &&\n this.buckets[index] !== this.TOMBSTONE\n ) {\n this.buckets[index]!.val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n this.buckets[index] = new Pair(key, val);\n this.size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n remove(key: number): void {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (\n this.buckets[index] !== null &&\n this.buckets[index] !== this.TOMBSTONE\n ) {\n this.buckets[index] = this.TOMBSTONE;\n this.size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n private extend(): void {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n const bucketsTmp = this.buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n this.capacity *= this.extendRatio;\n this.buckets = Array(this.capacity).fill(null);\n this.size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (const pair of bucketsTmp) {\n if (pair !== null && pair !== this.TOMBSTONE) {\n this.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print(): void {\n for (const pair of this.buckets) {\n if (pair === null) {\n console.log('null');\n } else if (pair === this.TOMBSTONE) {\n console.log('TOMBSTONE');\n } else {\n console.log(pair.key + ' -> ' + pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.rs/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n late int _size; // \u952e\u503c\u5bf9\u6570\u91cf\n int _capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double _loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int _extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n late List<Pair?> _buckets; // \u6876\u6570\u7ec4\n Pair _TOMBSTONE = Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n HashMapOpenAddressing() {\n _size = 0;\n _buckets = List.generate(_capacity, (index) => null);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % _capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return _size / _capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n int findBucket(int key) {\n int index = hashFunc(key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (_buckets[index] != null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (_buckets[index]!.key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n _buckets[firstTombstone] = _buckets[index];\n _buckets[index] = _TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && _buckets[index] == _TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % _capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n String? get(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (_buckets[index] != null && _buckets[index] != _TOMBSTONE) {\n return _buckets[index]!.val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, String val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > _loadThres) {\n extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (_buckets[index] != null && _buckets[index] != _TOMBSTONE) {\n _buckets[index]!.val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n _buckets[index] = new Pair(key, val);\n _size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (_buckets[index] != null && _buckets[index] != _TOMBSTONE) {\n _buckets[index] = _TOMBSTONE;\n _size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n List<Pair?> bucketsTmp = _buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n _capacity *= _extendRatio;\n _buckets = List.generate(_capacity, (index) => null);\n _size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (Pair? pair in bucketsTmp) {\n if (pair != null && pair != _TOMBSTONE) {\n put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void printHashMap() {\n for (Pair? pair in _buckets) {\n if (pair == null) {\n print(\"null\");\n } else if (pair == _TOMBSTONE) {\n print(\"TOMBSTONE\");\n } else {\n print(\"${pair.key} -> ${pair.val}\");\n }\n }\n }\n}\n
hash_map_open_addressing.c/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapOpenAddressing {\n size: usize, // \u952e\u503c\u5bf9\u6570\u91cf\n capacity: usize, // \u54c8\u5e0c\u8868\u5bb9\u91cf\n load_thres: f64, // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n extend_ratio: usize, // \u6269\u5bb9\u500d\u6570\n buckets: Vec<Option<Pair>>, // \u6876\u6570\u7ec4\n TOMBSTONE: Option<Pair>, // \u5220\u9664\u6807\u8bb0\n}\n\n\nimpl HashMapOpenAddressing {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new() -> Self {\n Self {\n size: 0,\n capacity: 4,\n load_thres: 2.0 / 3.0,\n extend_ratio: 2,\n buckets: vec![None; 4],\n TOMBSTONE: Some(Pair {key: -1, val: \"-1\".to_string()}),\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n fn hash_func(&self, key: i32) -> usize {\n (key % self.capacity as i32) as usize\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n fn load_factor(&self) -> f64 {\n self.size as f64 / self.capacity as f64\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n fn find_bucket(&mut self, key: i32) -> usize {\n let mut index = self.hash_func(key);\n let mut first_tombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while self.buckets[index].is_some() {\n // \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n if self.buckets[index].as_ref().unwrap().key == key {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u5efa\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if first_tombstone != -1 {\n self.buckets[first_tombstone as usize] = self.buckets[index].take();\n self.buckets[index] = self.TOMBSTONE.clone();\n return first_tombstone as usize; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if first_tombstone == -1 && self.buckets[index] == self.TOMBSTONE {\n first_tombstone = index as i32;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % self.capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n if first_tombstone == -1 { index } else { first_tombstone as usize }\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n fn get(&mut self, key: i32) -> Option<&str> {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = self.find_bucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if self.buckets[index].is_some() && self.buckets[index] != self.TOMBSTONE {\n return self.buckets[index].as_ref().map(|pair| &pair.val as &str);\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n None\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n fn put(&mut self, key: i32, val: String) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if self.load_factor() > self.load_thres {\n self.extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = self.find_bucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if self.buckets[index].is_some() && self.buckets[index] != self.TOMBSTONE {\n self.buckets[index].as_mut().unwrap().val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n self.buckets[index] = Some(Pair { key, val });\n self.size += 1;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n fn remove(&mut self, key: i32) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = self.find_bucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if self.buckets[index].is_some() && self.buckets[index] != self.TOMBSTONE {\n self.buckets[index] = self.TOMBSTONE.clone();\n self.size -= 1;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n fn extend(&mut self) {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n let buckets_tmp = self.buckets.clone();\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n self.capacity *= self.extend_ratio;\n self.buckets = vec![None; self.capacity];\n self.size = 0;\n\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for pair in buckets_tmp {\n if pair.is_none() || pair == self.TOMBSTONE {\n continue;\n }\n let pair = pair.unwrap();\n\n self.put(pair.key, pair.val);\n }\n }\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n fn print(&self) {\n for pair in &self.buckets {\n if pair.is_none() {\n println!(\"null\");\n } else if pair == &self.TOMBSTONE {\n println!(\"TOMBSTONE\");\n } else {\n let pair = pair.as_ref().unwrap();\n println!(\"{} -> {}\", pair.key, pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.zig/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n Pair **buckets; // \u6876\u6570\u7ec4\n Pair *TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n} HashMapOpenAddressing;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n hashMap->size = 0;\n hashMap->capacity = 4;\n hashMap->loadThres = 2.0 / 3.0;\n hashMap->extendRatio = 2;\n hashMap->buckets = (Pair **)malloc(sizeof(Pair *) * hashMap->capacity);\n hashMap->TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n hashMap->TOMBSTONE->key = -1;\n hashMap->TOMBSTONE->val = \"-1\";\n\n return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n for (int i = 0; i < hashMap->capacity; i++) {\n Pair *pair = hashMap->buckets[i];\n if (pair != NULL && pair != hashMap->TOMBSTONE) {\n free(pair->val);\n free(pair);\n }\n }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n return key % hashMap->capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n return (double)hashMap->size / (double)hashMap->capacity;\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n int index = hashFunc(hashMap, key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (hashMap->buckets[index] != NULL) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (hashMap->buckets[index]->key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n hashMap->buckets[firstTombstone] = hashMap->buckets[index];\n hashMap->buckets[index] = hashMap->TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && hashMap->buckets[index] == hashMap->TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % hashMap->capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(hashMap, key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (hashMap->buckets[index] != NULL && hashMap->buckets[index] != hashMap->TOMBSTONE) {\n return hashMap->buckets[index]->val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\";\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor(hashMap) > hashMap->loadThres) {\n extend(hashMap);\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(hashMap, key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (hashMap->buckets[index] != NULL && hashMap->buckets[index] != hashMap->TOMBSTONE) {\n free(hashMap->buckets[index]->val);\n hashMap->buckets[index]->val = (char *)malloc(sizeof(strlen(val + 1)));\n strcpy(hashMap->buckets[index]->val, val);\n hashMap->buckets[index]->val[strlen(val)] = '\\0';\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n Pair *pair = (Pair *)malloc(sizeof(Pair));\n pair->key = key;\n pair->val = (char *)malloc(sizeof(strlen(val + 1)));\n strcpy(pair->val, val);\n pair->val[strlen(val)] = '\\0';\n\n hashMap->buckets[index] = pair;\n hashMap->size++;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(hashMap, key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (hashMap->buckets[index] != NULL && hashMap->buckets[index] != hashMap->TOMBSTONE) {\n Pair *pair = hashMap->buckets[index];\n free(pair->val);\n free(pair);\n hashMap->buckets[index] = hashMap->TOMBSTONE;\n hashMap->size--;\n }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n Pair **bucketsTmp = hashMap->buckets;\n int oldCapacity = hashMap->capacity;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n hashMap->capacity *= hashMap->extendRatio;\n hashMap->buckets = (Pair **)malloc(sizeof(Pair *) * hashMap->capacity);\n hashMap->size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (int i = 0; i < oldCapacity; i++) {\n Pair *pair = bucketsTmp[i];\n if (pair != NULL && pair != hashMap->TOMBSTONE) {\n put(hashMap, pair->key, pair->val);\n free(pair->val);\n free(pair);\n }\n }\n free(bucketsTmp);\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\n for (int i = 0; i < hashMap->capacity; i++) {\n Pair *pair = hashMap->buckets[i];\n if (pair == NULL) {\n printf(\"NULL\\n\");\n } else if (pair == hashMap->TOMBSTONE) {\n printf(\"TOMBSTONE\\n\");\n } else {\n printf(\"%d -> %s\\n\", pair->key, pair->val);\n }\n }\n}\n
"},{"location":"chapter_hashing/hash_collision/#2","title":"2. \u00a0 \u5e73\u65b9\u63a2\u6d4b","text":"[class]{HashMapOpenAddressing}-[func]{}\n\u5e73\u65b9\u63a2\u6d4b\u4e0e\u7ebf\u6027\u63a2\u6d4b\u7c7b\u4f3c\uff0c\u90fd\u662f\u5f00\u653e\u5bfb\u5740\u7684\u5e38\u89c1\u7b56\u7565\u4e4b\u4e00\u3002\u5f53\u53d1\u751f\u51b2\u7a81\u65f6\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e0d\u662f\u7b80\u5355\u5730\u8df3\u8fc7\u4e00\u4e2a\u56fa\u5b9a\u7684\u6b65\u6570\uff0c\u800c\u662f\u8df3\u8fc7\u201c\u63a2\u6d4b\u6b21\u6570\u7684\u5e73\u65b9\u201d\u7684\u6b65\u6570\uff0c\u5373 \\(1, 4, 9, \\dots\\) \u6b65\u3002
\u5e73\u65b9\u63a2\u6d4b\u4e3b\u8981\u5177\u6709\u4ee5\u4e0b\u4f18\u52bf\u3002
- \u5e73\u65b9\u63a2\u6d4b\u901a\u8fc7\u8df3\u8fc7\u63a2\u6d4b\u6b21\u6570\u5e73\u65b9\u7684\u8ddd\u79bb\uff0c\u8bd5\u56fe\u7f13\u89e3\u7ebf\u6027\u63a2\u6d4b\u7684\u805a\u96c6\u6548\u5e94\u3002
- \u5e73\u65b9\u63a2\u6d4b\u4f1a\u8df3\u8fc7\u66f4\u5927\u7684\u8ddd\u79bb\u6765\u5bfb\u627e\u7a7a\u4f4d\u7f6e\uff0c\u6709\u52a9\u4e8e\u6570\u636e\u5206\u5e03\u5f97\u66f4\u52a0\u5747\u5300\u3002
\u7136\u800c\uff0c\u5e73\u65b9\u63a2\u6d4b\u5e76\u4e0d\u662f\u5b8c\u7f8e\u7684\u3002
- \u4ecd\u7136\u5b58\u5728\u805a\u96c6\u73b0\u8c61\uff0c\u5373\u67d0\u4e9b\u4f4d\u7f6e\u6bd4\u5176\u4ed6\u4f4d\u7f6e\u66f4\u5bb9\u6613\u88ab\u5360\u7528\u3002
- \u7531\u4e8e\u5e73\u65b9\u7684\u589e\u957f\uff0c\u5e73\u65b9\u63a2\u6d4b\u53ef\u80fd\u4e0d\u4f1a\u63a2\u6d4b\u6574\u4e2a\u54c8\u5e0c\u8868\uff0c\u8fd9\u610f\u5473\u7740\u5373\u4f7f\u54c8\u5e0c\u8868\u4e2d\u6709\u7a7a\u6876\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e5f\u53ef\u80fd\u65e0\u6cd5\u8bbf\u95ee\u5230\u5b83\u3002
\u987e\u540d\u601d\u4e49\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\)\u3001\\(f_2(x)\\)\u3001\\(f_3(x)\\)\u3001\\(\\dots\\) \u8fdb\u884c\u63a2\u6d4b\u3002
- \u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u6876\u540e\u63d2\u5165\u5143\u7d20\u3002
- \u67e5\u627e\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u4e0b\u8fdb\u884c\u67e5\u627e\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u65f6\u8fd4\u56de\uff1b\u82e5\u9047\u5230\u7a7a\u6876\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u4e0d\u5b58\u5728\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(\\text{None}\\) \u3002
\u4e0e\u7ebf\u6027\u63a2\u6d4b\u76f8\u6bd4\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u4f1a\u5e26\u6765\u989d\u5916\u7684\u8ba1\u7b97\u91cf\u3002
Tip
\u8bf7\u6ce8\u610f\uff0c\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u548c\u591a\u6b21\u54c8\u5e0c\uff09\u54c8\u5e0c\u8868\u90fd\u5b58\u5728\u201c\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u201d\u7684\u95ee\u9898\u3002
"},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u9009\u62e9","text":"\u5404\u79cd\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u7b56\u7565\uff0c\u4e0b\u9762\u4e3e\u51e0\u4e2a\u4f8b\u5b50\u3002
- Python \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002\u5b57\u5178 dict \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002
- Java \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002\u81ea JDK 1.8 \u4ee5\u6765\uff0c\u5f53 HashMap \u5185\u6570\u7ec4\u957f\u5ea6\u8fbe\u5230 64 \u4e14\u94fe\u8868\u957f\u5ea6\u8fbe\u5230 8 \u65f6\uff0c\u94fe\u8868\u4f1a\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002
- Go \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\u3002\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u786e\u4fdd\u6027\u80fd\u3002
\u300c\u54c8\u5e0c\u8868 hash table\u300d\uff0c\u53c8\u79f0\u300c\u6563\u5217\u8868\u300d\uff0c\u5176\u901a\u8fc7\u5efa\u7acb\u952e
key\u4e0e\u503cvalue\u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8be2\u3002\u5177\u4f53\u800c\u8a00\uff0c\u6211\u4eec\u5411\u54c8\u5e0c\u8868\u8f93\u5165\u4e00\u4e2a\u952ekey\uff0c\u5219\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5bf9\u5e94\u7684\u503cvalue\u3002\u5982\u56fe 6-1 \u6240\u793a\uff0c\u7ed9\u5b9a \\(n\\) \u4e2a\u5b66\u751f\uff0c\u6bcf\u4e2a\u5b66\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b66\u53f7\u201d\u4e24\u9879\u6570\u636e\u3002\u5047\u5982\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u201c\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\u201d\u7684\u67e5\u8be2\u529f\u80fd\uff0c\u5219\u53ef\u4ee5\u91c7\u7528\u56fe 6-1 \u6240\u793a\u7684\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002
\u56fe 6-1 \u00a0 \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a
\u9664\u54c8\u5e0c\u8868\u5916\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u4e5f\u53ef\u4ee5\u5b9e\u73b0\u67e5\u8be2\u529f\u80fd\uff0c\u5b83\u4eec\u7684\u6548\u7387\u5bf9\u6bd4\u5982\u8868 6-1 \u6240\u793a\u3002
- \u6dfb\u52a0\u5143\u7d20\uff1a\u4ec5\u9700\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u6570\u7ec4\uff08\u94fe\u8868\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
- \u67e5\u8be2\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\uff08\u94fe\u8868\uff09\u662f\u4e71\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u904d\u5386\u5176\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
- \u5220\u9664\u5143\u7d20\uff1a\u9700\u8981\u5148\u67e5\u8be2\u5230\u5143\u7d20\uff0c\u518d\u4ece\u6570\u7ec4\uff08\u94fe\u8868\uff09\u4e2d\u5220\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
\u8868 6-1 \u00a0 \u5143\u7d20\u67e5\u8be2\u6548\u7387\u5bf9\u6bd4
\u6570\u7ec4 \u94fe\u8868 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\)\u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u8fdb\u884c\u589e\u5220\u67e5\u6539\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
"},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1 \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"\u54c8\u5e0c\u8868\u7684\u5e38\u89c1\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u548c\u5220\u9664\u952e\u503c\u5bf9\u7b49\uff0c\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map.py
hash_map.cpp# \u521d\u59cb\u5316\u54c8\u5e0c\u8868\nhmap: dict = {}\n\n# \u6dfb\u52a0\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n# \u67e5\u8be2\u64cd\u4f5c\n# \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = hmap[15937]\n\n# \u5220\u9664\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nhmap.pop(10583)\n
hash_map.java/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map<int, string> map;\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n
hash_map.cs/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<Integer, String> map = new HashMap<>();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\"); \nmap.put(15937, \"\u5c0f\u5570\"); \nmap.put(16750, \"\u5c0f\u7b97\"); \nmap.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n
hash_map_test.go/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary<int, string> map = new() {\n /* \u6dfb\u52a0\u64cd\u4f5c */\n // \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\n { 12836, \"\u5c0f\u54c8\" },\n { 15937, \"\u5c0f\u5570\" },\n { 16750, \"\u5c0f\u7b97\" },\n { 13276, \"\u5c0f\u6cd5\" },\n { 10583, \"\u5c0f\u9e2d\" }\n};\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n
hash_map.swift/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nhmap := make(map[int]string)\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := hmap[15937]\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(hmap, 10583)\n
hash_map.js/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n
hash_map.ts/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n
hash_map.dart/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map<number, string>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n
hash_map.rs/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<int, String> map = {};\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n
hash_map.cuse std::collections::HashMap;\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nlet mut map: HashMap<i32, String> = HashMap::new();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.insert(12836, \"\u5c0f\u54c8\".to_string());\nmap.insert(15937, \"\u5c0f\u5570\".to_string());\nmap.insert(16750, \"\u5c0f\u7b97\".to_string());\nmap.insert(13279, \"\u5c0f\u6cd5\".to_string());\nmap.insert(10583, \"\u5c0f\u9e2d\".to_string());\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet _name: Option<&String> = map.get(&15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nlet _removed_value: Option<String> = map.remove(&10583);\n
hash_map.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n\n\u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u5f0f\uff1a\u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u548c\u904d\u5386\u503c\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map.py
hash_map.cpp# \u904d\u5386\u54c8\u5e0c\u8868\n# \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value in hmap.items():\n print(key, \"->\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in hmap.keys():\n print(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in hmap.values():\n print(value)\n
hash_map.java/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (auto kv: map) {\n cout << kv.first << \" -> \" << kv.second << endl;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u5668\u904d\u5386 key->value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n cout << iter->first << \"->\" << iter->second << endl;\n}\n
hash_map.cs/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (Map.Entry <Integer, String> kv: map.entrySet()) {\n System.out.println(kv.getKey() + \" -> \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\n System.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\n System.out.println(val);\n}\n
hash_map_test.go/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nforeach (var kv in map) {\n Console.WriteLine(kv.Key + \" -> \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys) {\n Console.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (string val in map.Values) {\n Console.WriteLine(val);\n}\n
hash_map.swift/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value := range hmap {\n fmt.Println(key, \"->\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range hmap {\n fmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range hmap {\n fmt.Println(value)\n}\n
hash_map.js/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nfor (key, value) in map {\n print(\"\\(key) -> \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\n print(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\n print(value)\n}\n
hash_map.ts/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\n console.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n console.info(v);\n}\n
hash_map.dart/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\n console.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n console.info(v);\n}\n
hash_map.rs/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nmap.forEach((key, value) {\n print('$key -> $value');\n});\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nmap.keys.forEach((key) {\n print(key);\n});\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nmap.values.forEach((value) {\n print(value);\n});\n
hash_map.c/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nfor (key, value) in &map {\n println!(\"{key} -> {value}\");\n}\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys() {\n println!(\"{key}\"); \n}\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values() {\n println!(\"{value}\");\n}\n
hash_map.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n
"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2 \u00a0 \u54c8\u5e0c\u8868\u7b80\u5355\u5b9e\u73b0","text":"\n\u6211\u4eec\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u300c\u6876 bucket\u300d\uff0c\u6bcf\u4e2a\u6876\u53ef\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u56e0\u6b64\uff0c\u67e5\u8be2\u64cd\u4f5c\u5c31\u662f\u627e\u5230
key\u5bf9\u5e94\u7684\u6876\uff0c\u5e76\u5728\u6876\u4e2d\u83b7\u53d6value\u3002\u90a3\u4e48\uff0c\u5982\u4f55\u57fa\u4e8e
key\u5b9a\u4f4d\u5bf9\u5e94\u7684\u6876\u5462\uff1f\u8fd9\u662f\u901a\u8fc7\u300c\u54c8\u5e0c\u51fd\u6570 hash function\u300d\u5b9e\u73b0\u7684\u3002\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u8f93\u51fa\u7a7a\u95f4\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u6240\u6709key\uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u6240\u6709\u6876\uff08\u6570\u7ec4\u7d22\u5f15\uff09\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8f93\u5165\u4e00\u4e2akey\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u8be5key\u5bf9\u5e94\u7684\u952e\u503c\u5bf9\u5728\u6570\u7ec4\u4e2d\u7684\u5b58\u50a8\u4f4d\u7f6e\u3002\u8f93\u5165\u4e00\u4e2a
key\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u8fc7\u7a0b\u5206\u4e3a\u4ee5\u4e0b\u4e24\u6b65\u3002- \u901a\u8fc7\u67d0\u79cd\u54c8\u5e0c\u7b97\u6cd5
hash()\u8ba1\u7b97\u5f97\u5230\u54c8\u5e0c\u503c\u3002 - \u5c06\u54c8\u5e0c\u503c\u5bf9\u6876\u6570\u91cf\uff08\u6570\u7ec4\u957f\u5ea6\uff09
capacity\u53d6\u6a21\uff0c\u4ece\u800c\u83b7\u53d6\u8be5key\u5bf9\u5e94\u7684\u6570\u7ec4\u7d22\u5f15index\u3002
index = hash(key) % capacity\n\u968f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528
index\u5728\u54c8\u5e0c\u8868\u4e2d\u8bbf\u95ee\u5bf9\u5e94\u7684\u6876\uff0c\u4ece\u800c\u83b7\u53d6value\u3002\u8bbe\u6570\u7ec4\u957f\u5ea6
capacity = 100\u3001\u54c8\u5e0c\u7b97\u6cd5hash(key) = key\uff0c\u6613\u5f97\u54c8\u5e0c\u51fd\u6570\u4e3akey % 100\u3002\u56fe 6-2 \u4ee5key\u5b66\u53f7\u548cvalue\u59d3\u540d\u4e3a\u4f8b\uff0c\u5c55\u793a\u4e86\u54c8\u5e0c\u51fd\u6570\u7684\u5de5\u4f5c\u539f\u7406\u3002\u56fe 6-2 \u00a0 \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406
\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u7b80\u5355\u54c8\u5e0c\u8868\u3002\u5176\u4e2d\uff0c\u6211\u4eec\u5c06
PythonC++JavaC#GoSwiftJSTSDartRustCZig array_hash_map.pykey\u548cvalue\u5c01\u88c5\u6210\u4e00\u4e2a\u7c7bPair\uff0c\u4ee5\u8868\u793a\u952e\u503c\u5bf9\u3002
array_hash_map.cppclass Pair:\n \"\"\"\u952e\u503c\u5bf9\"\"\"\n\n def __init__(self, key: int, val: str):\n self.key = key\n self.val = val\n\nclass ArrayHashMap:\n \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n # \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n self.buckets: list[Pair | None] = [None] * 100\n\n def hash_func(self, key: int) -> int:\n \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n index = key % 100\n return index\n\n def get(self, key: int) -> str:\n \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n index: int = self.hash_func(key)\n pair: Pair = self.buckets[index]\n if pair is None:\n return None\n return pair.val\n\n def put(self, key: int, val: str):\n \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n pair = Pair(key, val)\n index: int = self.hash_func(key)\n self.buckets[index] = pair\n\n def remove(self, key: int):\n \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n index: int = self.hash_func(key)\n # \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n self.buckets[index] = None\n\n def entry_set(self) -> list[Pair]:\n \"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\n result: list[Pair] = []\n for pair in self.buckets:\n if pair is not None:\n result.append(pair)\n return result\n\n def key_set(self) -> list[int]:\n \"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\n result = []\n for pair in self.buckets:\n if pair is not None:\n result.append(pair.key)\n return result\n\n def value_set(self) -> list[str]:\n \"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\n result = []\n for pair in self.buckets:\n if pair is not None:\n result.append(pair.val)\n return result\n\n def print(self):\n \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n for pair in self.buckets:\n if pair is not None:\n print(pair.key, \"->\", pair.val)\n
array_hash_map.java/* \u952e\u503c\u5bf9 */\nstruct Pair {\n public:\n int key;\n string val;\n Pair(int key, string val) {\n this->key = key;\n this->val = val;\n }\n};\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n private:\n vector<Pair *> buckets;\n\n public:\n ArrayHashMap() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n buckets = vector<Pair *>(100);\n }\n\n ~ArrayHashMap() {\n // \u91ca\u653e\u5185\u5b58\n for (const auto &bucket : buckets) {\n delete bucket;\n }\n buckets.clear();\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n int index = key % 100;\n return index;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n string get(int key) {\n int index = hashFunc(key);\n Pair *pair = buckets[index];\n if (pair == nullptr)\n return \"\";\n return pair->val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, string val) {\n Pair *pair = new Pair(key, val);\n int index = hashFunc(key);\n buckets[index] = pair;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n int index = hashFunc(key);\n // \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\n delete buckets[index];\n buckets[index] = nullptr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n vector<Pair *> pairSet() {\n vector<Pair *> pairSet;\n for (Pair *pair : buckets) {\n if (pair != nullptr) {\n pairSet.push_back(pair);\n }\n }\n return pairSet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n vector<int> keySet() {\n vector<int> keySet;\n for (Pair *pair : buckets) {\n if (pair != nullptr) {\n keySet.push_back(pair->key);\n }\n }\n return keySet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n vector<string> valueSet() {\n vector<string> valueSet;\n for (Pair *pair : buckets) {\n if (pair != nullptr) {\n valueSet.push_back(pair->val);\n }\n }\n return valueSet;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void print() {\n for (Pair *kv : pairSet()) {\n cout << kv->key << \" -> \" << kv->val << endl;\n }\n }\n};\n
array_hash_map.cs/* \u952e\u503c\u5bf9 */\nclass Pair {\n public int key;\n public String val;\n\n public Pair(int key, String val) {\n this.key = key;\n this.val = val;\n }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n private List<Pair> buckets;\n\n public ArrayHashMap() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n buckets = new ArrayList<>();\n for (int i = 0; i < 100; i++) {\n buckets.add(null);\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private int hashFunc(int key) {\n int index = key % 100;\n return index;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public String get(int key) {\n int index = hashFunc(key);\n Pair pair = buckets.get(index);\n if (pair == null)\n return null;\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void put(int key, String val) {\n Pair pair = new Pair(key, val);\n int index = hashFunc(key);\n buckets.set(index, pair);\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void remove(int key) {\n int index = hashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n buckets.set(index, null);\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n public List<Pair> pairSet() {\n List<Pair> pairSet = new ArrayList<>();\n for (Pair pair : buckets) {\n if (pair != null)\n pairSet.add(pair);\n }\n return pairSet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n public List<Integer> keySet() {\n List<Integer> keySet = new ArrayList<>();\n for (Pair pair : buckets) {\n if (pair != null)\n keySet.add(pair.key);\n }\n return keySet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n public List<String> valueSet() {\n List<String> valueSet = new ArrayList<>();\n for (Pair pair : buckets) {\n if (pair != null)\n valueSet.add(pair.val);\n }\n return valueSet;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void print() {\n for (Pair kv : pairSet()) {\n System.out.println(kv.key + \" -> \" + kv.val);\n }\n }\n}\n
array_hash_map.go/* \u952e\u503c\u5bf9 int->string */\nclass Pair(int key, string val) {\n public int key = key;\n public string val = val;\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n List<Pair?> buckets;\n public ArrayHashMap() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n buckets = [];\n for (int i = 0; i < 100; i++) {\n buckets.Add(null);\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int HashFunc(int key) {\n int index = key % 100;\n return index;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public string? Get(int key) {\n int index = HashFunc(key);\n Pair? pair = buckets[index];\n if (pair == null) return null;\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void Put(int key, string val) {\n Pair pair = new(key, val);\n int index = HashFunc(key);\n buckets[index] = pair;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void Remove(int key) {\n int index = HashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n buckets[index] = null;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n public List<Pair> PairSet() {\n List<Pair> pairSet = [];\n foreach (Pair? pair in buckets) {\n if (pair != null)\n pairSet.Add(pair);\n }\n return pairSet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n public List<int> KeySet() {\n List<int> keySet = [];\n foreach (Pair? pair in buckets) {\n if (pair != null)\n keySet.Add(pair.key);\n }\n return keySet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n public List<string> ValueSet() {\n List<string> valueSet = [];\n foreach (Pair? pair in buckets) {\n if (pair != null)\n valueSet.Add(pair.val);\n }\n return valueSet;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void Print() {\n foreach (Pair kv in PairSet()) {\n Console.WriteLine(kv.key + \" -> \" + kv.val);\n }\n }\n}\n
array_hash_map.swift/* \u952e\u503c\u5bf9 */\ntype pair struct {\n key int\n val string\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\n buckets []*pair\n}\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n buckets := make([]*pair, 100)\n return &arrayHashMap{buckets: buckets}\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n index := key % 100\n return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\n index := a.hashFunc(key)\n pair := a.buckets[index]\n if pair == nil {\n return \"Not Found\"\n }\n return pair.val\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\n pair := &pair{key: key, val: val}\n index := a.hashFunc(key)\n a.buckets[index] = pair\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\n index := a.hashFunc(key)\n // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n a.buckets[index] = nil\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\n var pairs []*pair\n for _, pair := range a.buckets {\n if pair != nil {\n pairs = append(pairs, pair)\n }\n }\n return pairs\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\n var keys []int\n for _, pair := range a.buckets {\n if pair != nil {\n keys = append(keys, pair.key)\n }\n }\n return keys\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\n var values []string\n for _, pair := range a.buckets {\n if pair != nil {\n values = append(values, pair.val)\n }\n }\n return values\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\n for _, pair := range a.buckets {\n if pair != nil {\n fmt.Println(pair.key, \"->\", pair.val)\n }\n }\n}\n
array_hash_map.js/* \u952e\u503c\u5bf9 */\nclass Pair: Equatable {\n public var key: Int\n public var val: String\n\n public init(key: Int, val: String) {\n self.key = key\n self.val = val\n }\n\n public static func == (lhs: Pair, rhs: Pair) -> Bool {\n lhs.key == rhs.key && lhs.val == rhs.val\n }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n private var buckets: [Pair?] = []\n\n init() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n for _ in 0 ..< 100 {\n buckets.append(nil)\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private func hashFunc(key: Int) -> Int {\n let index = key % 100\n return index\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n func get(key: Int) -> String? {\n let index = hashFunc(key: key)\n let pair = buckets[index]\n return pair?.val\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n func put(key: Int, val: String) {\n let pair = Pair(key: key, val: val)\n let index = hashFunc(key: key)\n buckets[index] = pair\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n func remove(key: Int) {\n let index = hashFunc(key: key)\n // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n buckets[index] = nil\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n func pairSet() -> [Pair] {\n var pairSet: [Pair] = []\n for pair in buckets {\n if let pair = pair {\n pairSet.append(pair)\n }\n }\n return pairSet\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n func keySet() -> [Int] {\n var keySet: [Int] = []\n for pair in buckets {\n if let pair = pair {\n keySet.append(pair.key)\n }\n }\n return keySet\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n func valueSet() -> [String] {\n var valueSet: [String] = []\n for pair in buckets {\n if let pair = pair {\n valueSet.append(pair.val)\n }\n }\n return valueSet\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n func print() {\n for pair in pairSet() {\n Swift.print(\"\\(pair.key) -> \\(pair.val)\")\n }\n }\n}\n
array_hash_map.ts/* \u952e\u503c\u5bf9 Number -> String */\nclass Pair {\n constructor(key, val) {\n this.key = key;\n this.val = val;\n }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n #buckets;\n constructor() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n this.#buckets = new Array(100).fill(null);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n #hashFunc(key) {\n return key % 100;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key) {\n let index = this.#hashFunc(key);\n let pair = this.#buckets[index];\n if (pair === null) return null;\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n set(key, val) {\n let index = this.#hashFunc(key);\n this.#buckets[index] = new Pair(key, val);\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n delete(key) {\n let index = this.#hashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n this.#buckets[index] = null;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n entries() {\n let arr = [];\n for (let i = 0; i < this.#buckets.length; i++) {\n if (this.#buckets[i]) {\n arr.push(this.#buckets[i]);\n }\n }\n return arr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n keys() {\n let arr = [];\n for (let i = 0; i < this.#buckets.length; i++) {\n if (this.#buckets[i]) {\n arr.push(this.#buckets[i].key);\n }\n }\n return arr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n values() {\n let arr = [];\n for (let i = 0; i < this.#buckets.length; i++) {\n if (this.#buckets[i]) {\n arr.push(this.#buckets[i].val);\n }\n }\n return arr;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print() {\n let pairSet = this.entries();\n for (const pair of pairSet) {\n console.info(`${pair.key} -> ${pair.val}`);\n }\n }\n}\n
array_hash_map.dart/* \u952e\u503c\u5bf9 Number -> String */\nclass Pair {\n public key: number;\n public val: string;\n\n constructor(key: number, val: string) {\n this.key = key;\n this.val = val;\n }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n private readonly buckets: (Pair | null)[];\n\n constructor() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n this.buckets = new Array(100).fill(null);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private hashFunc(key: number): number {\n return key % 100;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public get(key: number): string | null {\n let index = this.hashFunc(key);\n let pair = this.buckets[index];\n if (pair === null) return null;\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public set(key: number, val: string) {\n let index = this.hashFunc(key);\n this.buckets[index] = new Pair(key, val);\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public delete(key: number) {\n let index = this.hashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n this.buckets[index] = null;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n public entries(): (Pair | null)[] {\n let arr: (Pair | null)[] = [];\n for (let i = 0; i < this.buckets.length; i++) {\n if (this.buckets[i]) {\n arr.push(this.buckets[i]);\n }\n }\n return arr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n public keys(): (number | undefined)[] {\n let arr: (number | undefined)[] = [];\n for (let i = 0; i < this.buckets.length; i++) {\n if (this.buckets[i]) {\n arr.push(this.buckets[i].key);\n }\n }\n return arr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n public values(): (string | undefined)[] {\n let arr: (string | undefined)[] = [];\n for (let i = 0; i < this.buckets.length; i++) {\n if (this.buckets[i]) {\n arr.push(this.buckets[i].val);\n }\n }\n return arr;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public print() {\n let pairSet = this.entries();\n for (const pair of pairSet) {\n console.info(`${pair.key} -> ${pair.val}`);\n }\n }\n}\n
array_hash_map.rs/* \u952e\u503c\u5bf9 */\nclass Pair {\n int key;\n String val;\n Pair(this.key, this.val);\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n late List<Pair?> _buckets;\n\n ArrayHashMap() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n _buckets = List.filled(100, null);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int _hashFunc(int key) {\n final int index = key % 100;\n return index;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n String? get(int key) {\n final int index = _hashFunc(key);\n final Pair? pair = _buckets[index];\n if (pair == null) {\n return null;\n }\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, String val) {\n final Pair pair = Pair(key, val);\n final int index = _hashFunc(key);\n _buckets[index] = pair;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n final int index = _hashFunc(key);\n _buckets[index] = null;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n List<Pair> pairSet() {\n List<Pair> pairSet = [];\n for (final Pair? pair in _buckets) {\n if (pair != null) {\n pairSet.add(pair);\n }\n }\n return pairSet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n List<int> keySet() {\n List<int> keySet = [];\n for (final Pair? pair in _buckets) {\n if (pair != null) {\n keySet.add(pair.key);\n }\n }\n return keySet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n List<String> values() {\n List<String> valueSet = [];\n for (final Pair? pair in _buckets) {\n if (pair != null) {\n valueSet.add(pair.val);\n }\n }\n return valueSet;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void printHashMap() {\n for (final Pair kv in pairSet()) {\n print(\"${kv.key} -> ${kv.val}\");\n }\n }\n}\n
array_hash_map.c/* \u952e\u503c\u5bf9 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n pub key: i32,\n pub val: String,\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\npub struct ArrayHashMap {\n buckets: Vec<Option<Pair>>\n}\n\nimpl ArrayHashMap {\n pub fn new() -> ArrayHashMap {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n Self { buckets: vec![None; 100] }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n fn hash_func(&self, key: i32) -> usize {\n key as usize % 100\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n pub fn get(&self, key: i32) -> Option<&String> {\n let index = self.hash_func(key);\n self.buckets[index].as_ref().map(|pair| &pair.val)\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n pub fn put(&mut self, key: i32, val: &str) {\n let index = self.hash_func(key);\n self.buckets[index] = Some(Pair {\n key,\n val: val.to_string(),\n });\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n pub fn remove(&mut self, key: i32) {\n let index = self.hash_func(key);\n // \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n self.buckets[index] = None;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n pub fn entry_set(&self) -> Vec<&Pair> {\n self.buckets.iter().filter_map(|pair| pair.as_ref()).collect()\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n pub fn key_set(&self) -> Vec<&i32> {\n self.buckets.iter().filter_map(|pair| pair.as_ref().map(|pair| &pair.key)).collect()\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n pub fn value_set(&self) -> Vec<&String> {\n self.buckets.iter().filter_map(|pair| pair.as_ref().map(|pair| &pair.val)).collect()\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n pub fn print(&self) {\n for pair in self.entry_set() {\n println!(\"{} -> {}\", pair.key, pair.val);\n }\n }\n}\n
array_hash_map.zig/* \u952e\u503c\u5bf9 int->string */\ntypedef struct {\n int key;\n char *val;\n} Pair;\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntypedef struct {\n Pair *buckets[HASHTABLE_CAPACITY];\n} ArrayHashMap;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayHashMap *newArrayHashMap() {\n ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n return hmap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n for (int i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n free(hmap->buckets[i]->val);\n free(hmap->buckets[i]);\n }\n }\n free(hmap);\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n Pair *Pair = malloc(sizeof(Pair));\n Pair->key = key;\n Pair->val = malloc(strlen(val) + 1);\n strcpy(Pair->val, val);\n\n int index = hashFunc(key);\n hmap->buckets[index] = Pair;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n int index = hashFunc(key);\n free(hmap->buckets[index]->val);\n free(hmap->buckets[index]);\n hmap->buckets[index] = NULL;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n Pair *entries;\n int i = 0, index = 0;\n int total = 0;\n /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n total++;\n }\n }\n entries = malloc(sizeof(Pair) * total);\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n entries[index].key = hmap->buckets[i]->key;\n entries[index].val = malloc(strlen(hmap->buckets[i]->val) + 1);\n strcpy(entries[index].val, hmap->buckets[i]->val);\n index++;\n }\n }\n set->set = entries;\n set->len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n int *keys;\n int i = 0, index = 0;\n int total = 0;\n /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n total++;\n }\n }\n keys = malloc(total * sizeof(int));\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n keys[index] = hmap->buckets[i]->key;\n index++;\n }\n }\n set->set = keys;\n set->len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n char **vals;\n int i = 0, index = 0;\n int total = 0;\n /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n total++;\n }\n }\n vals = malloc(total * sizeof(char *));\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n vals[index] = hmap->buckets[i]->val;\n index++;\n }\n }\n set->set = vals;\n set->len = total;\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(ArrayHashMap *hmap) {\n int i;\n MapSet set;\n pairSet(hmap, &set);\n Pair *entries = (Pair *)set.set;\n for (i = 0; i < set.len; i++) {\n printf(\"%d -> %s\\n\", entries[i].key, entries[i].val);\n }\n free(set.set);\n}\n
"},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u4e0e\u6269\u5bb9","text":"// \u952e\u503c\u5bf9\nconst Pair = struct {\n key: usize = undefined,\n val: []const u8 = undefined,\n\n pub fn init(key: usize, val: []const u8) Pair {\n return Pair {\n .key = key,\n .val = val,\n };\n }\n};\n\n// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\n return struct {\n bucket: ?std.ArrayList(?T) = null,\n mem_allocator: std.mem.Allocator = undefined,\n\n const Self = @This();\n\n // \u6784\u9020\u51fd\u6570\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n self.mem_allocator = allocator;\n // \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\n self.bucket = std.ArrayList(?T).init(self.mem_allocator);\n var i: i32 = 0;\n while (i < 100) : (i += 1) {\n try self.bucket.?.append(null);\n }\n }\n\n // \u6790\u6784\u51fd\u6570\n pub fn deinit(self: *Self) void {\n if (self.bucket != null) self.bucket.?.deinit();\n }\n\n // \u54c8\u5e0c\u51fd\u6570\n fn hashFunc(key: usize) usize {\n var index = key % 100;\n return index;\n }\n\n // \u67e5\u8be2\u64cd\u4f5c\n pub fn get(self: *Self, key: usize) []const u8 {\n var index = hashFunc(key);\n var pair = self.bucket.?.items[index];\n return pair.?.val;\n }\n\n // \u6dfb\u52a0\u64cd\u4f5c\n pub fn put(self: *Self, key: usize, val: []const u8) !void {\n var pair = Pair.init(key, val);\n var index = hashFunc(key);\n self.bucket.?.items[index] = pair;\n }\n\n // \u5220\u9664\u64cd\u4f5c\n pub fn remove(self: *Self, key: usize) !void {\n var index = hashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n self.bucket.?.items[index] = null;\n } \n\n // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\n pub fn pairSet(self: *Self) !std.ArrayList(T) {\n var entry_set = std.ArrayList(T).init(self.mem_allocator);\n for (self.bucket.?.items) |item| {\n if (item == null) continue;\n try entry_set.append(item.?);\n }\n return entry_set;\n } \n\n // \u83b7\u53d6\u6240\u6709\u952e\n pub fn keySet(self: *Self) !std.ArrayList(usize) {\n var key_set = std.ArrayList(usize).init(self.mem_allocator);\n for (self.bucket.?.items) |item| {\n if (item == null) continue;\n try key_set.append(item.?.key);\n }\n return key_set;\n } \n\n // \u83b7\u53d6\u6240\u6709\u503c\n pub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\n var value_set = std.ArrayList([]const u8).init(self.mem_allocator);\n for (self.bucket.?.items) |item| {\n if (item == null) continue;\n try value_set.append(item.?.val);\n }\n return value_set;\n }\n\n // \u6253\u5370\u54c8\u5e0c\u8868\n pub fn print(self: *Self) !void {\n var entry_set = try self.pairSet();\n defer entry_set.deinit();\n for (entry_set.items) |item| {\n std.debug.print(\"{} -> {s}\\n\", .{item.key, item.val});\n }\n }\n };\n}\n\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u6240\u6709
key\u6784\u6210\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u6570\u7ec4\u6240\u6709\u7d22\u5f15\u6784\u6210\u7684\u8f93\u51fa\u7a7a\u95f4\uff0c\u800c\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7406\u8bba\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u76f8\u540c\u8f93\u51fa\u201d\u7684\u60c5\u51b5\u3002\u5bf9\u4e8e\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u54c8\u5e0c\u51fd\u6570\uff0c\u5f53\u8f93\u5165\u7684
key\u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u51fa\u7ed3\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8be2\u5b66\u53f7\u4e3a 12836 \u548c 20336 \u7684\u4e24\u4e2a\u5b66\u751f\u65f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a12836 % 100 = 36\n20336 % 100 = 36\n\u5982\u56fe 6-3 \u6240\u793a\uff0c\u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002\u6211\u4eec\u5c06\u8fd9\u79cd\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u540c\u4e00\u8f93\u51fa\u7684\u60c5\u51b5\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 hash collision\u300d\u3002
\u56fe 6-3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b
\u5bb9\u6613\u60f3\u5230\uff0c\u54c8\u5e0c\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u4e2a
key\u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002\u5982\u56fe 6-4 \u6240\u793a\uff0c\u6269\u5bb9\u524d\u952e\u503c\u5bf9
(136, A)\u548c(236, D)\u53d1\u751f\u51b2\u7a81\uff0c\u6269\u5bb9\u540e\u51b2\u7a81\u6d88\u5931\u3002\u56fe 6-4 \u00a0 \u54c8\u5e0c\u8868\u6269\u5bb9
\u7c7b\u4f3c\u4e8e\u6570\u7ec4\u6269\u5bb9\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u8fc1\u79fb\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u975e\u5e38\u8017\u65f6\uff1b\u5e76\u4e14\u7531\u4e8e\u54c8\u5e0c\u8868\u5bb9\u91cf
capacity\u6539\u53d8\uff0c\u6211\u4eec\u9700\u8981\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u6765\u91cd\u65b0\u8ba1\u7b97\u6240\u6709\u952e\u503c\u5bf9\u7684\u5b58\u50a8\u4f4d\u7f6e\uff0c\u8fd9\u8fdb\u4e00\u6b65\u63d0\u9ad8\u4e86\u6269\u5bb9\u8fc7\u7a0b\u7684\u8ba1\u7b97\u5f00\u9500\u3002\u4e3a\u6b64\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u9884\u7559\u8db3\u591f\u5927\u7684\u54c8\u5e0c\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u9891\u7e41\u6269\u5bb9\u3002\u300c\u8d1f\u8f7d\u56e0\u5b50 load factor\u300d\u662f\u54c8\u5e0c\u8868\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u7684\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u7528\u4e8e\u8861\u91cf\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u4f5c\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002
"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_hashing/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"- \u8f93\u5165
key\uff0c\u54c8\u5e0c\u8868\u80fd\u591f\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u67e5\u8be2\u5230value\uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002 - \u5e38\u89c1\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u3001\u5220\u9664\u952e\u503c\u5bf9\u548c\u904d\u5386\u54c8\u5e0c\u8868\u7b49\u3002
- \u54c8\u5e0c\u51fd\u6570\u5c06
key\u6620\u5c04\u4e3a\u6570\u7ec4\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5bf9\u5e94\u6876\u5e76\u83b7\u53d6value\u3002 - \u4e24\u4e2a\u4e0d\u540c\u7684
key\u53ef\u80fd\u5728\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u540e\u5f97\u5230\u76f8\u540c\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u51fa\u9519\uff0c\u8fd9\u79cd\u73b0\u8c61\u88ab\u79f0\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002 - \u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u7684\u5f00\u9500\u5f88\u5927\u3002
- \u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u53cd\u6620\u4e86\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89e6\u53d1\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u6761\u4ef6\u3002
- \u94fe\u5f0f\u5730\u5740\u901a\u8fc7\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u4e3a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u5b58\u50a8\u5728\u540c\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002\u7136\u800c\uff0c\u94fe\u8868\u8fc7\u957f\u4f1a\u964d\u4f4e\u67e5\u8be2\u6548\u7387\uff0c\u53ef\u4ee5\u901a\u8fc7\u8fdb\u4e00\u6b65\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6765\u63d0\u9ad8\u6548\u7387\u3002
- \u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u8f83\u7ebf\u6027\u63a2\u6d4b\u66f4\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002
- \u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u4f8b\u5982\uff0cJava \u7684
HashMap\u4f7f\u7528\u94fe\u5f0f\u5730\u5740\uff0c\u800c Python \u7684Dict\u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002 - \u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u54c8\u5e0c\u7b97\u6cd5\u5177\u6709\u786e\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u5300\u5206\u5e03\u7684\u7279\u70b9\u3002\u5728\u5bc6\u7801\u5b66\u4e2d\uff0c\u54c8\u5e0c\u7b97\u6cd5\u8fd8\u5e94\u8be5\u5177\u5907\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u5e94\u3002
- \u54c8\u5e0c\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u5747\u5300\u5206\u5e03\uff0c\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002
- \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u5305\u62ec MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002MD5 \u5e38\u7528\u4e8e\u6821\u9a8c\u6587\u4ef6\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u4e8e\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u3002
- \u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u662f\u53ef\u54c8\u5e0c\u7684\u3002
\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f \\(O(n)\\) \uff1f
\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u5f53\u54c8\u5e0c\u51fd\u6570\u8bbe\u8ba1\u5f97\u6bd4\u8f83\u597d\u3001\u5bb9\u91cf\u8bbe\u7f6e\u6bd4\u8f83\u5408\u7406\u3001\u51b2\u7a81\u6bd4\u8f83\u5e73\u5747\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u4eec\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u54c8\u5e0c\u8868\u65f6\uff0c\u901a\u5e38\u8ba4\u4e3a\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002
\u4e3a\u4ec0\u4e48\u4e0d\u4f7f\u7528\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x\\) \u5462\uff1f\u8fd9\u6837\u5c31\u4e0d\u4f1a\u6709\u51b2\u7a81\u4e86
\u5728 \\(f(x) = x\\) \u54c8\u5e0c\u51fd\u6570\u4e0b\uff0c\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u8fd9\u4e0e\u6570\u7ec4\u7b49\u4ef7\u3002\u7136\u800c\uff0c\u8f93\u5165\u7a7a\u95f4\u901a\u5e38\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u957f\u5ea6\uff09\uff0c\u56e0\u6b64\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\u7684\u76ee\u6807\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u72b6\u6001\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u7a7a\u95f4\uff0c\u5e76\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8be2\u6548\u7387\u3002
\u54c8\u5e0c\u8868\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff0c\u4f46\u4e3a\u4ec0\u4e48\u6548\u7387\u53ef\u4ee5\u6bd4\u5b83\u4eec\u66f4\u9ad8\u5462\uff1f
\u9996\u5148\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\uff0c\u4f46\u7a7a\u95f4\u6548\u7387\u53d8\u4f4e\u4e86\u3002\u54c8\u5e0c\u8868\u6709\u76f8\u5f53\u4e00\u90e8\u5206\u5185\u5b58\u672a\u4f7f\u7528\u3002
\u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u573a\u666f\u4e0b\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u4e2a\u529f\u80fd\u80fd\u591f\u5728\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\uff0c\u90a3\u4e48\u901a\u5e38\u6bd4\u54c8\u5e0c\u8868\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u9700\u8981\u5f00\u9500\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u6570\u9879\u66f4\u5927\u3002
\u6700\u540e\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u80fd\u53d1\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u94fe\u5f0f\u5730\u5740\u4e2d\uff0c\u6211\u4eec\u91c7\u53d6\u5728\u94fe\u8868\u6216\u7ea2\u9ed1\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u65f6\u95f4\u7684\u98ce\u9669\u3002
\u591a\u6b21\u54c8\u5e0c\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\u5417\uff1f\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u8fd8\u80fd\u518d\u6b21\u4f7f\u7528\u5417\uff1f
\u591a\u6b21\u54c8\u5e0c\u662f\u5f00\u653e\u5bfb\u5740\u7684\u4e00\u79cd\uff0c\u5f00\u653e\u5bfb\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u901a\u8fc7\u6807\u8bb0\u5220\u9664\u3002\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u53ef\u4ee5\u518d\u6b21\u4f7f\u7528\u3002\u5f53\u5c06\u65b0\u5143\u7d20\u63d2\u5165\u54c8\u5e0c\u8868\uff0c\u5e76\u4e14\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u4f4d\u7f6e\u65f6\uff0c\u8be5\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u5143\u7d20\u4f7f\u7528\u3002\u8fd9\u6837\u505a\u65e2\u80fd\u4fdd\u6301\u54c8\u5e0c\u8868\u7684\u63a2\u6d4b\u5e8f\u5217\u4e0d\u53d8\uff0c\u53c8\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\u4f7f\u7528\u7387\u3002
\u4e3a\u4ec0\u4e48\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\uff0c\u67e5\u627e\u5143\u7d20\u7684\u65f6\u5019\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f
\u67e5\u627e\u7684\u65f6\u5019\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u5bf9\u5e94\u7684\u6876\u548c\u952e\u503c\u5bf9\uff0c\u53d1\u73b0
key\u4e0d\u5339\u914d\uff0c\u8fd9\u5c31\u4ee3\u8868\u6709\u54c8\u5e0c\u51b2\u7a81\u3002\u56e0\u6b64\uff0c\u7ebf\u6027\u63a2\u6d4b\u6cd5\u4f1a\u6839\u636e\u9884\u5148\u8bbe\u5b9a\u7684\u6b65\u957f\u4f9d\u6b21\u5411\u4e0b\u67e5\u627e\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u786e\u7684\u952e\u503c\u5bf9\u6216\u65e0\u6cd5\u627e\u5230\u8df3\u51fa\u4e3a\u6b62\u3002\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u6269\u5bb9\u80fd\u591f\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff1f
\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d6\u4f59\uff0c\u8ba9\u8f93\u51fa\u503c\u843d\u5728\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\u5185\uff1b\u5728\u6269\u5bb9\u540e\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d1\u751f\u53d8\u5316\uff0c\u800c
"},{"location":"chapter_heap/","title":"\u7b2c 8 \u7ae0 \u00a0 \u5806","text":"key\u5bf9\u5e94\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u4e2a\u6876\u7684\u591a\u4e2akey\uff0c\u5728\u6269\u5bb9\u540e\u53ef\u80fd\u4f1a\u88ab\u5206\u914d\u5230\u591a\u4e2a\u6876\u4e2d\uff0c\u4ece\u800c\u5b9e\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u7f13\u89e3\u3002Abstract
\u5806\u5c31\u50cf\u662f\u5c71\u5cb3\u5cf0\u5ce6\uff0c\u5c42\u53e0\u8d77\u4f0f\u3001\u5f62\u6001\u5404\u5f02\u3002
\u5ea7\u5ea7\u5c71\u5cf0\u9ad8\u4f4e\u9519\u843d\uff0c\u800c\u6700\u9ad8\u7684\u5c71\u5cf0\u603b\u662f\u6700\u5148\u6620\u5165\u773c\u5e18\u3002
"},{"location":"chapter_heap/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 8.1 \u00a0 \u5806
- 8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c
- 8.3 \u00a0 Top-K \u95ee\u9898
- 8.4 \u00a0 \u5c0f\u7ed3
\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u6765\u6784\u5efa\u4e00\u4e2a\u5806\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u3002
"},{"location":"chapter_heap/build_heap/#821","title":"8.2.1 \u00a0 \u501f\u52a9\u5165\u5806\u64cd\u4f5c\u5b9e\u73b0","text":"\u6211\u4eec\u9996\u5148\u521b\u5efa\u4e00\u4e2a\u7a7a\u5806\uff0c\u7136\u540e\u904d\u5386\u5217\u8868\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u5143\u7d20\u6267\u884c\u201c\u5165\u5806\u64cd\u4f5c\u201d\uff0c\u5373\u5148\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u5806\u7684\u5c3e\u90e8\uff0c\u518d\u5bf9\u8be5\u5143\u7d20\u6267\u884c\u201c\u4ece\u5e95\u81f3\u9876\u201d\u5806\u5316\u3002
\u6bcf\u5f53\u4e00\u4e2a\u5143\u7d20\u5165\u5806\uff0c\u5806\u7684\u957f\u5ea6\u5c31\u52a0\u4e00\u3002\u7531\u4e8e\u8282\u70b9\u662f\u4ece\u9876\u5230\u5e95\u4f9d\u6b21\u88ab\u6dfb\u52a0\u8fdb\u4e8c\u53c9\u6811\u7684\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u6784\u5efa\u7684\u3002
\u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6bcf\u4e2a\u5143\u7d20\u7684\u5165\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(\\log{n})\\) \u65f6\u95f4\uff0c\u56e0\u6b64\u8be5\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
"},{"location":"chapter_heap/build_heap/#822","title":"8.2.2 \u00a0 \u901a\u8fc7\u904d\u5386\u5806\u5316\u5b9e\u73b0","text":"\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9e\u73b0\u4e00\u79cd\u66f4\u4e3a\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u5171\u5206\u4e3a\u4e24\u6b65\u3002
- \u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u5730\u6dfb\u52a0\u5230\u5806\u4e2d\uff0c\u6b64\u65f6\u5806\u7684\u6027\u8d28\u5c1a\u672a\u5f97\u5230\u6ee1\u8db3\u3002
- \u5012\u5e8f\u904d\u5386\u5806\uff08\u5c42\u5e8f\u904d\u5386\u7684\u5012\u5e8f\uff09\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u975e\u53f6\u8282\u70b9\u6267\u884c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u3002
\u6bcf\u5f53\u5806\u5316\u4e00\u4e2a\u8282\u70b9\u540e\uff0c\u4ee5\u8be5\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\u5c31\u5f62\u6210\u4e00\u4e2a\u5408\u6cd5\u7684\u5b50\u5806\u3002\u800c\u7531\u4e8e\u662f\u5012\u5e8f\u904d\u5386\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u6784\u5efa\u7684\u3002
\u4e4b\u6240\u4ee5\u9009\u62e9\u5012\u5e8f\u904d\u5386\uff0c\u662f\u56e0\u4e3a\u8fd9\u6837\u80fd\u591f\u4fdd\u8bc1\u5f53\u524d\u8282\u70b9\u4e4b\u4e0b\u7684\u5b50\u6811\u5df2\u7ecf\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u8fd9\u6837\u5806\u5316\u5f53\u524d\u8282\u70b9\u624d\u662f\u6709\u6548\u7684\u3002
\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u7531\u4e8e\u53f6\u8282\u70b9\u6ca1\u6709\u5b50\u8282\u70b9\uff0c\u56e0\u6b64\u5b83\u4eec\u5929\u7136\u5c31\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u65e0\u987b\u5806\u5316\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u6700\u540e\u4e00\u4e2a\u975e\u53f6\u8282\u70b9\u662f\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u6211\u4eec\u4ece\u5b83\u5f00\u59cb\u5012\u5e8f\u904d\u5386\u5e76\u6267\u884c\u5806\u5316\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef __init__(self, nums: list[int]):\n \"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n # \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n self.max_heap = nums\n # \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in range(self.parent(self.size() - 1), -1, -1):\n self.sift_down(i)\n
my_heap.java/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector<int> nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n maxHeap = nums;\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = parent(size() - 1); i >= 0; i--) {\n siftDown(i);\n }\n}\n
my_heap.cs/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<Integer> nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n maxHeap = new ArrayList<>(nums);\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = parent(size() - 1); i >= 0; i--) {\n siftDown(i);\n }\n}\n
my_heap.go/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable<int> nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n maxHeap = new List<int>(nums);\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n var size = Parent(this.Size() - 1);\n for (int i = size; i >= 0; i--) {\n SiftDown(i);\n }\n}\n
my_heap.swift/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n h := &maxHeap{data: nums}\n for i := h.parent(len(h.data) - 1); i >= 0; i-- {\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n h.siftDown(i)\n }\n return h\n}\n
my_heap.js/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n maxHeap = nums\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in stride(from: parent(i: size() - 1), through: 0, by: -1) {\n siftDown(i: i)\n }\n}\n
my_heap.ts/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n this.#maxHeap = nums === undefined ? [] : [...nums];\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (let i = this.#parent(this.size() - 1); i >= 0; i--) {\n this.#siftDown(i);\n }\n}\n
my_heap.dart/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n this.maxHeap = nums === undefined ? [] : [...nums];\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (let i = this.parent(this.size() - 1); i >= 0; i--) {\n this.siftDown(i);\n }\n}\n
my_heap.rs/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<int> nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n _maxHeap = nums;\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = _parent(size() - 1); i >= 0; i--) {\n siftDown(i);\n }\n}\n
my_heap.c/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nfn new(nums: Vec<i32>) -> Self {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n let mut heap = MaxHeap { max_heap: nums };\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in (0..=Self::parent(heap.size() - 1)).rev() {\n heap.sift_down(i);\n }\n heap\n}\n
my_heap.zig/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n // \u6240\u6709\u5143\u7d20\u5165\u5806\n MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n maxHeap->size = size;\n memcpy(maxHeap->data, nums, size * sizeof(int));\n for (int i = parent(maxHeap, size - 1); i >= 0; i--) {\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n siftDown(maxHeap, i);\n }\n return maxHeap;\n}\n
"},{"location":"chapter_heap/build_heap/#823","title":"8.2.3 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"// \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\n if (self.max_heap != null) return;\n self.max_heap = std.ArrayList(T).init(allocator);\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n try self.max_heap.?.appendSlice(nums);\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n var i: usize = parent(self.size() - 1) + 1;\n while (i > 0) : (i -= 1) {\n try self.siftDown(i - 1);\n }\n}\n\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u63a8\u7b97\u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002
- \u5047\u8bbe\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u53f6\u8282\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u9700\u8981\u5806\u5316\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\((n - 1) / 2\\) \u3002
- \u5728\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u6700\u591a\u5806\u5316\u5230\u53f6\u8282\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(\\log n\\) \u3002
\u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u4f46\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u5e76\u4e0d\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230\u4e8c\u53c9\u6811\u5e95\u5c42\u8282\u70b9\u6570\u91cf\u8fdc\u591a\u4e8e\u9876\u5c42\u8282\u70b9\u7684\u6027\u8d28\u3002
\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8fdb\u884c\u66f4\u4e3a\u51c6\u786e\u7684\u8ba1\u7b97\u3002\u4e3a\u4e86\u964d\u4f4e\u8ba1\u7b97\u96be\u5ea6\uff0c\u5047\u8bbe\u7ed9\u5b9a\u4e00\u4e2a\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \u3001\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002
\u56fe 8-5 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u8282\u70b9\u6570\u91cf
\u5982\u56fe 8-5 \u6240\u793a\uff0c\u8282\u70b9\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8be5\u8ddd\u79bb\u6b63\u662f\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u5404\u5c42\u7684\u201c\u8282\u70b9\u6570\u91cf \\(\\times\\) \u8282\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5f97\u5230\u6240\u6709\u8282\u70b9\u7684\u5806\u5316\u8fed\u4ee3\u6b21\u6570\u7684\u603b\u548c\u3002
\\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\]\u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5bf9 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230\uff1a
\\[ \\begin{aligned} T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\]\u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u7528\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97\uff1a
\\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\]\u89c2\u5bdf\u4e0a\u5f0f\uff0c\u53d1\u73b0 \\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a
\\[ \\begin{aligned} T(h) & = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline & = 2^{h+1} - h - 2 \\newline & = O(2^h) \\end{aligned} \\]\u8fdb\u4e00\u6b65\u5730\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 \u5806","text":"\u300c\u5806 heap\u300d\u662f\u4e00\u79cd\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u4e3b\u8981\u53ef\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff0c\u5982\u56fe 8-1 \u6240\u793a\u3002
- \u300c\u5927\u9876\u5806 max heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002
- \u300c\u5c0f\u9876\u5806 min heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002
\u56fe 8-1 \u00a0 \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806
\u5806\u4f5c\u4e3a\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002
- \u6700\u5e95\u5c42\u8282\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u3002
- \u6211\u4eec\u5c06\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u9876\u201d\uff0c\u5c06\u5e95\u5c42\u6700\u9760\u53f3\u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u5e95\u201d\u3002
- \u5bf9\u4e8e\u5927\u9876\u5806\uff08\u5c0f\u9876\u5806\uff09\uff0c\u5806\u9876\u5143\u7d20\uff08\u6839\u8282\u70b9\uff09\u7684\u503c\u5206\u522b\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002
\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u300c\u4f18\u5148\u961f\u5217 priority queue\u300d\uff0c\u8fd9\u662f\u4e00\u79cd\u62bd\u8c61\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u4f18\u5148\u7ea7\u6392\u5e8f\u7684\u961f\u5217\u3002
\u5b9e\u9645\u4e0a\uff0c\u5806\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u4f18\u5148\u961f\u5217\uff0c\u5927\u9876\u5806\u76f8\u5f53\u4e8e\u5143\u7d20\u6309\u4ece\u5927\u5230\u5c0f\u7684\u987a\u5e8f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u6765\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u201c\u4f18\u5148\u961f\u5217\u201d\u548c\u201c\u5806\u201d\u770b\u4f5c\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u4e66\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u79f0\u4f5c\u201c\u5806\u201d\u3002
\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u8868 8-1 \uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002
\u8868 8-1 \u00a0 \u5806\u7684\u64cd\u4f5c\u6548\u7387
\u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) pop() \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) peek() \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) size() \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) isEmpty() \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\)\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002
\u7c7b\u4f3c\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e2a
PythonC++JavaC#GoSwiftJSTSDartRustCZig heap.pyflag\u6216\u4fee\u6539Comparator\u5b9e\u73b0\u201c\u5c0f\u9876\u5806\u201d\u4e0e\u201c\u5927\u9876\u5806\u201d\u4e4b\u95f4\u7684\u8f6c\u6362\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
heap.cpp# \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n
heap.java/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue<int, vector<int>, greater<int>> minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue<int, vector<int>, less<int>> maxHeap;\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector<int> input{1, 3, 2, 5, 4};\npriority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());\n
heap.cs/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue<Integer> minHeap = new PriorityQueue<>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4));\n
heap.go/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue<int, int> minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue<int, int> maxHeap = new(Comparer<int>.Create((x, y) => y - x));\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue(); // 5\npeek = maxHeap.Dequeue(); // 4\npeek = maxHeap.Dequeue(); // 3\npeek = maxHeap.Dequeue(); // 2\npeek = maxHeap.Dequeue(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<int, int>([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n
heap.swift// Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n // Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n // \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n *h = append(*h, x.(int))\n}\n\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n // \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\n last := (*h)[len(*h)-1]\n *h = (*h)[:len(*h)-1]\n return last\n}\n\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\n return len(*h)\n}\n\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n // \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\n return (*h)[i].(int) > (*h)[j].(int)\n}\n\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\n return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n /* \u521d\u59cb\u5316\u5806 */\n // \u521d\u59cb\u5316\u5927\u9876\u5806\n maxHeap := &intHeap{}\n heap.Init(maxHeap)\n /* \u5143\u7d20\u5165\u5806 */\n // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\n heap.Push(maxHeap, 1)\n heap.Push(maxHeap, 3)\n heap.Push(maxHeap, 2)\n heap.Push(maxHeap, 4)\n heap.Push(maxHeap, 5)\n\n /* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\n top := maxHeap.Top()\n fmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n\n /* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\n heap.Pop(maxHeap) // 5\n heap.Pop(maxHeap) // 4\n heap.Pop(maxHeap) // 3\n heap.Pop(maxHeap) // 2\n heap.Pop(maxHeap) // 1\n\n /* \u83b7\u53d6\u5806\u5927\u5c0f */\n size := len(*maxHeap)\n fmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n\n /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n isEmpty := len(*maxHeap) == 0\n fmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n
heap.js// Swift \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
heap.ts// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
heap.dart// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
heap.rs// Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
heap.cuse std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nlet mut min_heap = BinaryHeap::<Reverse<i32>>::new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nlet mut max_heap = BinaryHeap::new();\n\n/* \u5143\u7d20\u5165\u5806 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap(); // 5\nlet peek = max_heap.pop().unwrap(); // 4\nlet peek = max_heap.pop().unwrap(); // 3\nlet peek = max_heap.pop().unwrap(); // 2\nlet peek = max_heap.pop().unwrap(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = max_heap.len();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = max_heap.is_empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n
heap.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
"},{"location":"chapter_heap/heap/#812","title":"8.1.2 \u00a0 \u5806\u7684\u5b9e\u73b0","text":"\n\u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u5927\u9876\u5806\u3002\u82e5\u8981\u5c06\u5176\u8f6c\u6362\u4e3a\u5c0f\u9876\u5806\uff0c\u53ea\u9700\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\uff0c\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002
"},{"location":"chapter_heap/heap/#1","title":"1. \u00a0 \u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u8bb2\u8fc7\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u7531\u4e8e\u5806\u6b63\u662f\u4e00\u79cd\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u6b64\u6211\u4eec\u5c06\u91c7\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002
\u5f53\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u8282\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u8282\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\u3002\u8282\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002
\u5982\u56fe 8-2 \u6240\u793a\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \uff0c\u7236\u8282\u70b9\u7d22\u5f15\u4e3a \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u8868\u793a\u7a7a\u8282\u70b9\u6216\u8282\u70b9\u4e0d\u5b58\u5728\u3002
\u56fe 8-2 \u00a0 \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8
\u6211\u4eec\u53ef\u4ee5\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef left(self, i: int) -> int:\n \"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\n return 2 * i + 1\n\ndef right(self, i: int) -> int:\n \"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\n return 2 * i + 2\n\ndef parent(self, i: int) -> int:\n \"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\"\"\"\n return (i - 1) // 2 # \u5411\u4e0b\u6574\u9664\n
my_heap.java/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\n return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.cs/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\n return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.go/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint Left(int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint Right(int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint Parent(int i) {\n return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.swift/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\n return 2*i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\n return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n // \u5411\u4e0b\u6574\u9664\n return (i - 1) / 2\n}\n
my_heap.js/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc left(i: Int) -> Int {\n 2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc right(i: Int) -> Int {\n 2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc parent(i: Int) -> Int {\n (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.ts/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\n#left(i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\n#right(i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\n#parent(i) {\n return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.dart/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nleft(i: number): number {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nright(i: number): number {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nparent(i: number): number {\n return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.rs/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _left(int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _right(int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint _parent(int i) {\n return (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.c/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfn left(i: usize) -> usize {\n 2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfn right(i: usize) -> usize {\n 2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfn parent(i: usize) -> usize {\n (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.zig/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\n return (i - 1) / 2;\n}\n
"},{"location":"chapter_heap/heap/#2","title":"2. \u00a0 \u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"// \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\nfn left(i: usize) usize {\n return 2 * i + 1;\n}\n\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\nfn right(i: usize) usize {\n return 2 * i + 2;\n}\n\n// \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\nfn parent(i: usize) usize {\n // return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n return @divFloor(i - 1, 2);\n}\n\u5806\u9876\u5143\u7d20\u5373\u4e3a\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u4e5f\u5c31\u662f\u5217\u8868\u7684\u9996\u4e2a\u5143\u7d20\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\n return self.max_heap[0]\n
my_heap.java/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n return maxHeap[0];\n}\n
my_heap.cs/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n return maxHeap.get(0);\n}\n
my_heap.go/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint Peek() {\n return maxHeap[0];\n}\n
my_heap.swift/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\n return h.data[0]\n}\n
my_heap.js/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -> Int {\n maxHeap[0]\n}\n
my_heap.ts/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\n return this.#maxHeap[0];\n}\n
my_heap.dart/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\n return this.maxHeap[0];\n}\n
my_heap.rs/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n return _maxHeap[0];\n}\n
my_heap.c/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfn peek(&self) -> Option<i32> {\n self.max_heap.first().copied()\n}\n
my_heap.zig/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\n return maxHeap->data[0];\n}\n
"},{"location":"chapter_heap/heap/#3","title":"3. \u00a0 \u5143\u7d20\u5165\u5806","text":"// \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\n return self.max_heap.?.items[0];\n} \n\u7ed9\u5b9a\u5143\u7d20
val\uff0c\u6211\u4eec\u9996\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u4e4b\u540e\uff0c\u7531\u4e8e val \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u574f\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5806\u5316 heapify\u300d\u3002\u8003\u8651\u4ece\u5165\u5806\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5982\u56fe 8-3 \u6240\u793a\uff0c\u6211\u4eec\u6bd4\u8f83\u63d2\u5165\u8282\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u8282\u70b9\u66f4\u5927\uff0c\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\u3002\u7136\u540e\u7ee7\u7eed\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u76f4\u81f3\u8d8a\u8fc7\u6839\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002
<1><2><3><4><5><6><7><8><9>\u56fe 8-3 \u00a0 \u5143\u7d20\u5165\u5806\u6b65\u9aa4
\u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef push(self, val: int):\n \"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n # \u6dfb\u52a0\u8282\u70b9\n self.max_heap.append(val)\n # \u4ece\u5e95\u81f3\u9876\u5806\u5316\n self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\n while True:\n # \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n p = self.parent(i)\n # \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if p < 0 or self.max_heap[i] <= self.max_heap[p]:\n break\n # \u4ea4\u6362\u4e24\u8282\u70b9\n self.swap(i, p)\n # \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p\n
my_heap.java/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap.push_back(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || maxHeap[i] <= maxHeap[p])\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(maxHeap[i], maxHeap[p]);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.cs/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap.add(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || maxHeap.get(i) <= maxHeap.get(p))\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.go/* \u5143\u7d20\u5165\u5806 */\nvoid Push(int val) {\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap.Add(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n SiftUp(Size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid SiftUp(int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = Parent(i);\n // \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n if (p < 0 || maxHeap[i] <= maxHeap[p])\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n Swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.swift/* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n // \u6dfb\u52a0\u8282\u70b9\n h.data = append(h.data, val)\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n h.siftUp(len(h.data) - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\n for true {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n p := h.parent(i)\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if p < 0 || h.data[i].(int) <= h.data[p].(int) {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n h.swap(i, p)\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p\n }\n}\n
my_heap.js/* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap.append(val)\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(i: size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\n var i = i\n while true {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n let p = parent(i: i)\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if p < 0 || maxHeap[i] <= maxHeap[p] {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(i: i, j: p)\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p\n }\n}\n
my_heap.ts/* \u5143\u7d20\u5165\u5806 */\npush(val) {\n // \u6dfb\u52a0\u8282\u70b9\n this.#maxHeap.push(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n this.#siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n const p = this.#parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || this.#maxHeap[i] <= this.#maxHeap[p]) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n this.#swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.dart/* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n // \u6dfb\u52a0\u8282\u70b9\n this.maxHeap.push(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n this.siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n const p = this.parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || this.maxHeap[i] <= this.maxHeap[p]) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n this.swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.rs/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n // \u6dfb\u52a0\u8282\u70b9\n _maxHeap.add(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = _parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || _maxHeap[i] <= _maxHeap[p]) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n _swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.c/* \u5143\u7d20\u5165\u5806 */\nfn push(&mut self, val: i32) {\n // \u6dfb\u52a0\u8282\u70b9\n self.max_heap.push(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n self.sift_up(self.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfn sift_up(&mut self, mut i: usize) {\n loop {\n // \u8282\u70b9 i \u5df2\u7ecf\u662f\u5806\u9876\u8282\u70b9\u4e86\uff0c\u7ed3\u675f\u5806\u5316\n if i == 0 {\n break;\n }\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n let p = Self::parent(i);\n // \u5f53\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if self.max_heap[i] <= self.max_heap[p] {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n self.swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.zig/* \u5143\u7d20\u5165\u5806 */\nvoid push(MaxHeap *maxHeap, int val) {\n // \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\n if (maxHeap->size == MAX_SIZE) {\n printf(\"heap is full!\");\n return;\n }\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap->data[maxHeap->size] = val;\n maxHeap->size++;\n\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(maxHeap, maxHeap->size - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = parent(maxHeap, i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || maxHeap->data[i] <= maxHeap->data[p]) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(maxHeap, i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
"},{"location":"chapter_heap/heap/#4","title":"4. \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806","text":"// \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n // \u6dfb\u52a0\u8282\u70b9\n try self.max_heap.?.append(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n try self.siftUp(self.size() - 1);\n} \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\n var i = i_;\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n var p = parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 or self.max_heap.?.items[i] <= self.max_heap.?.items[p]) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n try self.swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n\u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u76f4\u63a5\u4ece\u5217\u8868\u4e2d\u5220\u9664\u9996\u5143\u7d20\uff0c\u90a3\u4e48\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u7d22\u5f15\u90fd\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u8fd9\u5c06\u4f7f\u5f97\u540e\u7eed\u4f7f\u7528\u5806\u5316\u8fdb\u884c\u4fee\u590d\u53d8\u5f97\u56f0\u96be\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u53d8\u52a8\uff0c\u6211\u4eec\u91c7\u7528\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\u3002
- \u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff09\u3002
- \u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u7531\u4e8e\u5df2\u7ecf\u4ea4\u6362\uff0c\u56e0\u6b64\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\u3002
- \u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\u3002
\u5982\u56fe 8-4 \u6240\u793a\uff0c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u201c\u4ece\u5e95\u81f3\u9876\u5806\u5316\u201d\u76f8\u53cd\uff0c\u6211\u4eec\u5c06\u6839\u8282\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u503c\u8fdb\u884c\u6bd4\u8f83\uff0c\u5c06\u6700\u5927\u7684\u5b50\u8282\u70b9\u4e0e\u6839\u8282\u70b9\u4ea4\u6362\u3002\u7136\u540e\u5faa\u73af\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002
<1><2><3><4><5><6><7><8><9><10>\u56fe 8-4 \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806\u6b65\u9aa4
\u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef pop(self) -> int:\n \"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n # \u5224\u7a7a\u5904\u7406\n if self.is_empty():\n raise IndexError(\"\u5806\u4e3a\u7a7a\")\n # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n self.swap(0, self.size() - 1)\n # \u5220\u9664\u8282\u70b9\n val = self.max_heap.pop()\n # \u4ece\u9876\u81f3\u5e95\u5806\u5316\n self.sift_down(0)\n # \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val\n\ndef sift_down(self, i: int):\n \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n while True:\n # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n l, r, ma = self.left(i), self.right(i), i\n if l < self.size() and self.max_heap[l] > self.max_heap[ma]:\n ma = l\n if r < self.size() and self.max_heap[r] > self.max_heap[ma]:\n ma = r\n # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i:\n break\n # \u4ea4\u6362\u4e24\u8282\u70b9\n self.swap(i, ma)\n # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n
my_heap.java/* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n // \u5224\u7a7a\u5904\u7406\n if (isEmpty()) {\n throw out_of_range(\"\u5806\u4e3a\u7a7a\");\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(maxHeap[0], maxHeap[size() - 1]);\n // \u5220\u9664\u8282\u70b9\n maxHeap.pop_back();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(0);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = left(i), r = right(i), ma = i;\n if (l < size() && maxHeap[l] > maxHeap[ma])\n ma = l;\n if (r < size() && maxHeap[r] > maxHeap[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i)\n break;\n swap(maxHeap[i], maxHeap[ma]);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.cs/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n // \u5224\u7a7a\u5904\u7406\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(0, size() - 1);\n // \u5220\u9664\u8282\u70b9\n int val = maxHeap.remove(size() - 1);\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = left(i), r = right(i), ma = i;\n if (l < size() && maxHeap.get(l) > maxHeap.get(ma))\n ma = l;\n if (r < size() && maxHeap.get(r) > maxHeap.get(ma))\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i)\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.go/* \u5143\u7d20\u51fa\u5806 */\nint Pop() {\n // \u5224\u7a7a\u5904\u7406\n if (IsEmpty())\n throw new IndexOutOfRangeException();\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n Swap(0, Size() - 1);\n // \u5220\u9664\u8282\u70b9\n int val = maxHeap.Last();\n maxHeap.RemoveAt(Size() - 1);\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n SiftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = Left(i), r = Right(i), ma = i;\n if (l < Size() && maxHeap[l] > maxHeap[ma])\n ma = l;\n if (r < Size() && maxHeap[r] > maxHeap[ma])\n ma = r;\n // \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n if (ma == i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n Swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.swift/* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n // \u5224\u7a7a\u5904\u7406\n if h.isEmpty() {\n fmt.Println(\"error\")\n return nil\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n h.swap(0, h.size()-1)\n // \u5220\u9664\u8282\u70b9\n val := h.data[len(h.data)-1]\n h.data = h.data[:len(h.data)-1]\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n h.siftDown(0)\n\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\n for true {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n l, r, max := h.left(i), h.right(i), i\n if l < h.size() && h.data[l].(int) > h.data[max].(int) {\n max = l\n }\n if r < h.size() && h.data[r].(int) > h.data[max].(int) {\n max = r\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if max == i {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n h.swap(i, max)\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = max\n }\n}\n
my_heap.js/* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -> Int {\n // \u5224\u7a7a\u5904\u7406\n if isEmpty() {\n fatalError(\"\u5806\u4e3a\u7a7a\")\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(i: 0, j: size() - 1)\n // \u5220\u9664\u8282\u70b9\n let val = maxHeap.remove(at: size() - 1)\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(i: 0)\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\n var i = i\n while true {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = left(i: i)\n let r = right(i: i)\n var ma = i\n if l < size(), maxHeap[l] > maxHeap[ma] {\n ma = l\n }\n if r < size(), maxHeap[r] > maxHeap[ma] {\n ma = r\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(i: i, j: ma)\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n }\n}\n
my_heap.ts/* \u5143\u7d20\u51fa\u5806 */\npop() {\n // \u5224\u7a7a\u5904\u7406\n if (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n this.#swap(0, this.size() - 1);\n // \u5220\u9664\u8282\u70b9\n const val = this.#maxHeap.pop();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n this.#siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n const l = this.#left(i),\n r = this.#right(i);\n let ma = i;\n if (l < this.size() && this.#maxHeap[l] > this.#maxHeap[ma]) ma = l;\n if (r < this.size() && this.#maxHeap[r] > this.#maxHeap[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma === i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n this.#swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.dart/* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n // \u5224\u7a7a\u5904\u7406\n if (this.isEmpty()) throw new RangeError('Heap is empty.');\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n this.swap(0, this.size() - 1);\n // \u5220\u9664\u8282\u70b9\n const val = this.maxHeap.pop();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n this.siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n const l = this.left(i),\n r = this.right(i);\n let ma = i;\n if (l < this.size() && this.maxHeap[l] > this.maxHeap[ma]) ma = l;\n if (r < this.size() && this.maxHeap[r] > this.maxHeap[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma === i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n this.swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.rs/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n // \u5224\u7a7a\u5904\u7406\n if (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n _swap(0, size() - 1);\n // \u5220\u9664\u8282\u70b9\n int val = _maxHeap.removeLast();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = _left(i);\n int r = _right(i);\n int ma = i;\n if (l < size() && _maxHeap[l] > _maxHeap[ma]) ma = l;\n if (r < size() && _maxHeap[r] > _maxHeap[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n _swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.c/* \u5143\u7d20\u51fa\u5806 */\nfn pop(&mut self) -> i32 {\n // \u5224\u7a7a\u5904\u7406\n if self.is_empty() {\n panic!(\"index out of bounds\");\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n self.swap(0, self.size() - 1);\n // \u5220\u9664\u8282\u70b9\n let val = self.max_heap.remove(self.size() - 1);\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n self.sift_down(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(&mut self, mut i: usize) {\n loop {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n if l < self.size() && self.max_heap[l] > self.max_heap[ma] {\n ma = l;\n }\n if r < self.size() && self.max_heap[r] > self.max_heap[ma] {\n ma = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n self.swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.zig/* \u5143\u7d20\u51fa\u5806 */\nint pop(MaxHeap *maxHeap) {\n // \u5224\u7a7a\u5904\u7406\n if (isEmpty(maxHeap)) {\n printf(\"heap is empty!\");\n return INT_MAX;\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(maxHeap, 0, size(maxHeap) - 1);\n // \u5220\u9664\u8282\u70b9\n int val = maxHeap->data[maxHeap->size - 1];\n maxHeap->size--;\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(maxHeap, 0);\n\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n int l = left(maxHeap, i);\n int r = right(maxHeap, i);\n int max = i;\n if (l < size(maxHeap) && maxHeap->data[l] > maxHeap->data[max]) {\n max = l;\n }\n if (r < size(maxHeap) && maxHeap->data[r] > maxHeap->data[max]) {\n max = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (max == i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(maxHeap, i, max);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = max;\n }\n}\n
"},{"location":"chapter_heap/heap/#813","title":"8.1.3 \u00a0 \u5806\u5e38\u89c1\u5e94\u7528","text":"// \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n // \u5224\u65ad\u5904\u7406\n if (self.isEmpty()) unreachable;\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n try self.swap(0, self.size() - 1);\n // \u5220\u9664\u8282\u70b9\n var val = self.max_heap.?.pop();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n try self.siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n} \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\n var i = i_;\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n var l = left(i);\n var r = right(i);\n var ma = i;\n if (l < self.size() and self.max_heap.?.items[l] > self.max_heap.?.items[ma]) ma = l;\n if (r < self.size() and self.max_heap.?.items[r] > self.max_heap.?.items[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n try self.swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n- \u4f18\u5148\u961f\u5217\uff1a\u5806\u901a\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \uff0c\u800c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002
- \u5806\u6392\u5e8f\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5b83\u4eec\u5efa\u7acb\u4e00\u4e2a\u5806\uff0c\u7136\u540e\u4e0d\u65ad\u5730\u6267\u884c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u6570\u636e\u3002\u7136\u800c\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u4e00\u79cd\u66f4\u4f18\u96c5\u7684\u65b9\u5f0f\u5b9e\u73b0\u5806\u6392\u5e8f\uff0c\u8be6\u89c1\u201c\u5806\u6392\u5e8f\u201d\u7ae0\u8282\u3002
- \u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1a\u8fd9\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7b97\u6cd5\u95ee\u9898\uff0c\u540c\u65f6\u4e5f\u662f\u4e00\u79cd\u5178\u578b\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u62e9\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u9500\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002
- \u5806\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002
- \u4f18\u5148\u961f\u5217\u7684\u5b9a\u4e49\u662f\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u6765\u5b9e\u73b0\u3002
- \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\)\u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u548c\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002
- \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002
- \u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u7ef4\u62a4\u5806\u7684\u6027\u8d28\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u7528\u5230\u3002
- \u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
- Top-K \u662f\u4e00\u4e2a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u6570\u636e\u7ed3\u6784\u9ad8\u6548\u89e3\u51b3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002
\u6570\u636e\u7ed3\u6784\u7684\u201c\u5806\u201d\u4e0e\u5185\u5b58\u7ba1\u7406\u7684\u201c\u5806\u201d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\u5417\uff1f
\u4e24\u8005\u4e0d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u5806\u3002\u8ba1\u7b97\u673a\u7cfb\u7edf\u5185\u5b58\u4e2d\u7684\u5806\u662f\u52a8\u6001\u5185\u5b58\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5e8f\u5728\u8fd0\u884c\u65f6\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u5b58\u50a8\u6570\u636e\u3002\u7a0b\u5e8f\u53ef\u4ee5\u8bf7\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u5185\u5b58\uff0c\u7528\u4e8e\u5b58\u50a8\u5982\u5bf9\u8c61\u548c\u6570\u7ec4\u7b49\u590d\u6742\u7ed3\u6784\u3002\u5f53\u8fd9\u4e9b\u6570\u636e\u4e0d\u518d\u9700\u8981\u65f6\uff0c\u7a0b\u5e8f\u9700\u8981\u91ca\u653e\u8fd9\u4e9b\u5185\u5b58\uff0c\u4ee5\u9632\u6b62\u5185\u5b58\u6cc4\u6f0f\u3002\u76f8\u8f83\u4e8e\u6808\u5185\u5b58\uff0c\u5806\u5185\u5b58\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8c28\u614e\uff0c\u4f7f\u7528\u4e0d\u5f53\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5185\u5b58\u6cc4\u6f0f\u548c\u91ce\u6307\u9488\u7b49\u95ee\u9898\u3002
"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-K \u95ee\u9898","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u65e0\u5e8f\u6570\u7ec4
nums\uff0c\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u524d \\(k\\) \u5927\u7684\u5143\u7d20\u3002\u5bf9\u4e8e\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u5148\u4ecb\u7ecd\u4e24\u79cd\u601d\u8def\u6bd4\u8f83\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7ecd\u6548\u7387\u66f4\u9ad8\u7684\u5806\u89e3\u6cd5\u3002
"},{"location":"chapter_heap/top_k/#831","title":"8.3.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u904d\u5386\u9009\u62e9","text":"\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u56fe 8-6 \u6240\u793a\u7684 \\(k\\) \u8f6e\u904d\u5386\uff0c\u5206\u522b\u5728\u6bcf\u8f6e\u4e2d\u63d0\u53d6\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(\\dots\\)\u3001\\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\) \u3002
\u6b64\u65b9\u6cd5\u53ea\u9002\u7528\u4e8e \\(k \\ll n\\) \u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u5f53 \\(k\\) \u4e0e \\(n\\) \u6bd4\u8f83\u63a5\u8fd1\u65f6\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411\u4e8e \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u65f6\u3002
\u56fe 8-6 \u00a0 \u904d\u5386\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20
Tip
\u5f53 \\(k = n\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5b8c\u6574\u7684\u6709\u5e8f\u5e8f\u5217\uff0c\u6b64\u65f6\u7b49\u4ef7\u4e8e\u201c\u9009\u62e9\u6392\u5e8f\u201d\u7b97\u6cd5\u3002
"},{"location":"chapter_heap/top_k/#832","title":"8.3.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"\u5982\u56fe 8-7 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u5bf9\u6570\u7ec4
nums\u8fdb\u884c\u6392\u5e8f\uff0c\u518d\u8fd4\u56de\u6700\u53f3\u8fb9\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u663e\u7136\uff0c\u8be5\u65b9\u6cd5\u201c\u8d85\u989d\u201d\u5b8c\u6210\u4efb\u52a1\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u53ea\u9700\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002
\u56fe 8-7 \u00a0 \u6392\u5e8f\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20
"},{"location":"chapter_heap/top_k/#833","title":"8.3.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806","text":"\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u5806\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3 Top-K \u95ee\u9898\uff0c\u6d41\u7a0b\u5982\u56fe 8-8 \u6240\u793a\u3002
- \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\u6700\u5c0f\u3002
- \u5148\u5c06\u6570\u7ec4\u7684\u524d \\(k\\) \u4e2a\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u3002
- \u4ece\u7b2c \\(k + 1\\) \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\uff0c\u5e76\u5c06\u5f53\u524d\u5143\u7d20\u5165\u5806\u3002
- \u904d\u5386\u5b8c\u6210\u540e\uff0c\u5806\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002
\u56fe 8-8 \u00a0 \u57fa\u4e8e\u5806\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20
\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig top_k.py
top_k.cppdef top_k_heap(nums: list[int], k: int) -> list[int]:\n \"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\n # \u521d\u59cb\u5316\u5c0f\u9876\u5806\n heap = []\n # \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for i in range(k):\n heapq.heappush(heap, nums[i])\n # \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for i in range(k, len(nums)):\n # \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if nums[i] > heap[0]:\n heapq.heappop(heap)\n heapq.heappush(heap, nums[i])\n return heap\n
top_k.java/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue<int, vector<int>, greater<int>> topKHeap(vector<int> &nums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n priority_queue<int, vector<int>, greater<int>> heap;\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (int i = 0; i < k; i++) {\n heap.push(nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < nums.size(); i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > heap.top()) {\n heap.pop();\n heap.push(nums[i]);\n }\n }\n return heap;\n}\n
top_k.cs/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue<Integer> topKHeap(int[] nums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n Queue<Integer> heap = new PriorityQueue<Integer>();\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (int i = 0; i < k; i++) {\n heap.offer(nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < nums.length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > heap.peek()) {\n heap.poll();\n heap.offer(nums[i]);\n }\n }\n return heap;\n}\n
top_k.go/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nPriorityQueue<int, int> TopKHeap(int[] nums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n PriorityQueue<int, int> heap = new();\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (int i = 0; i < k; i++) {\n heap.Enqueue(nums[i], nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < nums.Length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > heap.Peek()) {\n heap.Dequeue();\n heap.Enqueue(nums[i], nums[i]);\n }\n }\n return heap;\n}\n
top_k.swift/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n h := &minHeap{}\n heap.Init(h)\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for i := 0; i < k; i++ {\n heap.Push(h, nums[i])\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for i := k; i < len(nums); i++ {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if nums[i] > h.Top().(int) {\n heap.Pop(h)\n heap.Push(h, nums[i])\n }\n }\n return h\n}\n
top_k.js/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -> [Int] {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5e76\u5c06\u524d k \u4e2a\u5143\u7d20\u5efa\u5806\n var heap = Heap(nums.prefix(k))\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for i in stride(from: k, to: nums.count, by: 1) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if nums[i] > heap.min()! {\n _ = heap.removeMin()\n heap.insert(nums[i])\n }\n }\n return heap.unordered\n}\n
top_k.ts/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap, val) {\n // \u5143\u7d20\u53d6\u53cd\n maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return maxHeap.getMaxHeap().map((num) => -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n const maxHeap = new MaxHeap([]);\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (let i = 0; i < k; i++) {\n pushMinHeap(maxHeap, nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (let i = k; i < nums.length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > peekMinHeap(maxHeap)) {\n popMinHeap(maxHeap);\n pushMinHeap(maxHeap, nums[i]);\n }\n }\n // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n return getMinHeap(maxHeap);\n}\n
top_k.dart/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n // \u5143\u7d20\u53d6\u53cd\n maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n // \u5143\u7d20\u53d6\u53cd\n return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n // \u5143\u7d20\u53d6\u53cd\n return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n // \u5143\u7d20\u53d6\u53cd\n return maxHeap.getMaxHeap().map((num: number) => -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n const maxHeap = new MaxHeap([]);\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (let i = 0; i < k; i++) {\n pushMinHeap(maxHeap, nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (let i = k; i < nums.length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > peekMinHeap(maxHeap)) {\n popMinHeap(maxHeap);\n pushMinHeap(maxHeap, nums[i]);\n }\n }\n // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n return getMinHeap(maxHeap);\n}\n
top_k.rs/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nMinHeap topKHeap(List<int> nums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\uff0c\u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n MinHeap heap = MinHeap(nums.sublist(0, k));\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < nums.length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > heap.peek()) {\n heap.pop();\n heap.push(nums[i]);\n }\n }\n return heap;\n}\n
top_k.c/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfn top_k_heap(nums: Vec<i32>, k: usize) -> BinaryHeap<Reverse<i32>> {\n // BinaryHeap \u662f\u5927\u9876\u5806\uff0c\u4f7f\u7528 Reverse \u5c06\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u5b9e\u73b0\u5c0f\u9876\u5806\n let mut heap = BinaryHeap::<Reverse<i32>>::new();\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for &num in nums.iter().take(k) {\n heap.push(Reverse(num));\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for &num in nums.iter().skip(k) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if num > heap.peek().unwrap().0 {\n heap.pop();\n heap.push(Reverse(num));\n }\n }\n heap\n}\n
top_k.zig/* \u5143\u7d20\u5165\u5806 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n // \u5143\u7d20\u53d6\u53cd\n push(maxHeap, -val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nint popMinHeap(MaxHeap *maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return -pop(maxHeap);\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return -peek(maxHeap);\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n int *res = (int *)malloc(maxHeap->size * sizeof(int));\n for (int i = 0; i < maxHeap->size; i++) {\n res[i] = -maxHeap->data[i];\n }\n return res;\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n int *res = (int *)malloc(maxHeap->size * sizeof(int));\n for (int i = 0; i < maxHeap->size; i++) {\n res[i] = -maxHeap->data[i];\n }\n return res;\n}\n\n// \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\u7684\u51fd\u6570\nint *topKHeap(int *nums, int sizeNums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n int *empty = (int *)malloc(0);\n MaxHeap *maxHeap = newMaxHeap(empty, 0);\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (int i = 0; i < k; i++) {\n pushMinHeap(maxHeap, nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < sizeNums; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > peekMinHeap(maxHeap)) {\n popMinHeap(maxHeap);\n pushMinHeap(maxHeap, nums[i]);\n }\n }\n int *res = getMinHeap(maxHeap);\n // \u91ca\u653e\u5185\u5b58\n delMaxHeap(maxHeap);\n return res;\n}\n[class]{}-[func]{topKHeap}\n\u603b\u5171\u6267\u884c\u4e86 \\(n\\) \u8f6e\u5165\u5806\u548c\u51fa\u5806\uff0c\u5806\u7684\u6700\u5927\u957f\u5ea6\u4e3a \\(k\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002\u8be5\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u5f53 \\(k\\) \u8f83\u5c0f\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \uff1b\u5f53 \\(k\\) \u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(O(n \\log n)\\) \u3002
\u53e6\u5916\uff0c\u8be5\u65b9\u6cd5\u9002\u7528\u4e8e\u52a8\u6001\u6570\u636e\u6d41\u7684\u4f7f\u7528\u573a\u666f\u3002\u5728\u4e0d\u65ad\u52a0\u5165\u6570\u636e\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6301\u7eed\u7ef4\u62a4\u5806\u5185\u7684\u5143\u7d20\uff0c\u4ece\u800c\u5b9e\u73b0\u6700\u5927 \\(k\\) \u4e2a\u5143\u7d20\u7684\u52a8\u6001\u66f4\u65b0\u3002
"},{"location":"chapter_introduction/","title":"\u7b2c 1 \u7ae0 \u00a0 \u521d\u8bc6\u7b97\u6cd5","text":"Abstract
\u4e00\u4f4d\u5c11\u5973\u7fe9\u7fe9\u8d77\u821e\uff0c\u4e0e\u6570\u636e\u4ea4\u7ec7\u5728\u4e00\u8d77\uff0c\u88d9\u6446\u4e0a\u98d8\u626c\u7740\u7b97\u6cd5\u7684\u65cb\u5f8b\u3002
\u5979\u9080\u8bf7\u4f60\u5171\u821e\uff0c\u8bf7\u7d27\u8ddf\u5979\u7684\u6b65\u4f10\uff0c\u8e0f\u5165\u5145\u6ee1\u903b\u8f91\u4e0e\u7f8e\u611f\u7684\u7b97\u6cd5\u4e16\u754c\u3002
"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728
- 1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48
- 1.3 \u00a0 \u5c0f\u7ed3
\u5f53\u6211\u4eec\u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\u65f6\uff0c\u5f88\u81ea\u7136\u5730\u4f1a\u60f3\u5230\u6570\u5b66\u3002\u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7b97\u6cd5\u5e76\u4e0d\u6d89\u53ca\u590d\u6742\u6570\u5b66\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8d56\u57fa\u672c\u903b\u8f91\uff0c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002
\u5728\u6b63\u5f0f\u63a2\u8ba8\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u4e2a\u6709\u8da3\u7684\u4e8b\u5b9e\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u5e76\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\u6211\u5c06\u4e3e\u51e0\u4e2a\u5177\u4f53\u7684\u4f8b\u5b50\u6765\u8bc1\u5b9e\u8fd9\u4e00\u70b9\u3002
\u4f8b\u4e00\uff1a\u67e5\u9605\u5b57\u5178\u3002\u5728\u5b57\u5178\u91cc\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u5bf9\u5e94\u4e00\u4e2a\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u67e5\u627e\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u4f1a\u6309\u7167\u56fe 1-1 \u6240\u793a\u7684\u65b9\u5f0f\u5b9e\u73b0\u3002
- \u7ffb\u5f00\u5b57\u5178\u7ea6\u4e00\u534a\u7684\u9875\u6570\uff0c\u67e5\u770b\u8be5\u9875\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff0c\u5047\u8bbe\u9996\u5b57\u6bcd\u4e3a \\(m\\) \u3002
- \u7531\u4e8e\u5728\u62fc\u97f3\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u4e8e \\(m\\) \u4e4b\u540e\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u7f29\u5c0f\u5230\u540e\u534a\u90e8\u5206\u3002
- \u4e0d\u65ad\u91cd\u590d\u6b65\u9aa4
1.\u548c \u6b65\u9aa42.\uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u4e3a\u6b62\u3002
\u56fe 1-1 \u00a0 \u67e5\u5b57\u5178\u6b65\u9aa4
\u67e5\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u5fc5\u5907\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u8457\u540d\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u7b97\u6cd5\u3002\u4ece\u6570\u636e\u7ed3\u6784\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b57\u5178\u89c6\u4e3a\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u201c\u6570\u7ec4\u201d\uff1b\u4ece\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u201c\u4e8c\u5206\u67e5\u627e\u201d\u3002
\u4f8b\u4e8c\uff1a\u6574\u7406\u6251\u514b\u3002\u6211\u4eec\u5728\u6253\u724c\u65f6\uff0c\u6bcf\u5c40\u90fd\u9700\u8981\u6574\u7406\u624b\u4e2d\u7684\u6251\u514b\u724c\uff0c\u4f7f\u5176\u4ece\u5c0f\u5230\u5927\u6392\u5217\uff0c\u5b9e\u73b0\u6d41\u7a0b\u5982\u56fe 1-2 \u6240\u793a\u3002
- \u5c06\u6251\u514b\u724c\u5212\u5206\u4e3a\u201c\u6709\u5e8f\u201d\u548c\u201c\u65e0\u5e8f\u201d\u4e24\u90e8\u5206\uff0c\u5e76\u5047\u8bbe\u521d\u59cb\u72b6\u6001\u4e0b\u6700\u5de6 1 \u5f20\u6251\u514b\u724c\u5df2\u7ecf\u6709\u5e8f\u3002
- \u5728\u65e0\u5e8f\u90e8\u5206\u62bd\u51fa\u4e00\u5f20\u6251\u514b\u724c\uff0c\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\uff1b\u5b8c\u6210\u540e\u6700\u5de6 2 \u5f20\u6251\u514b\u5df2\u7ecf\u6709\u5e8f\u3002
- \u4e0d\u65ad\u5faa\u73af\u6b65\u9aa4
2.\uff0c\u6bcf\u4e00\u8f6e\u5c06\u4e00\u5f20\u6251\u514b\u724c\u4ece\u65e0\u5e8f\u90e8\u5206\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\uff0c\u76f4\u81f3\u6240\u6709\u6251\u514b\u724c\u90fd\u6709\u5e8f\u3002
\u56fe 1-2 \u00a0 \u6251\u514b\u6392\u5e8f\u6b65\u9aa4
\u4e0a\u8ff0\u6574\u7406\u6251\u514b\u724c\u7684\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u63d2\u5165\u6392\u5e8f\u201d\u7b97\u6cd5\uff0c\u5b83\u5728\u5904\u7406\u5c0f\u578b\u6570\u636e\u96c6\u65f6\u975e\u5e38\u9ad8\u6548\u3002\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u7684\u6392\u5e8f\u5e93\u51fd\u6570\u4e2d\u90fd\u6709\u63d2\u5165\u6392\u5e8f\u7684\u8eab\u5f71\u3002
\u4f8b\u4e09\uff1a\u8d27\u5e01\u627e\u96f6\u3002\u5047\u8bbe\u6211\u4eec\u5728\u8d85\u5e02\u8d2d\u4e70\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7ed9\u4e86\u6536\u94f6\u5458 \\(100\\) \u5143\uff0c\u5219\u6536\u94f6\u5458\u9700\u8981\u627e\u6211\u4eec \\(31\\) \u5143\u3002\u4ed6\u4f1a\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u5982\u56fe 1-3 \u6240\u793a\u7684\u601d\u8003\u3002
- \u53ef\u9009\u9879\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8d27\u5e01\uff0c\u5305\u62ec \\(1\\) \u5143\u3001\\(5\\) \u5143\u3001\\(10\\) \u5143\u3001\\(20\\) \u5143\u3002
- \u4ece\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u4f59 \\(31 - 20 = 11\\) \u5143\u3002
- \u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u4f59 \\(11 - 10 = 1\\) \u5143\u3002
- \u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u4f59 \\(1 - 1 = 0\\) \u5143\u3002
- \u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u4e3a \\(20 + 10 + 1 = 31\\) \u5143\u3002
\u56fe 1-3 \u00a0 \u8d27\u5e01\u627e\u96f6\u8fc7\u7a0b
\u5728\u4ee5\u4e0a\u6b65\u9aa4\u4e2d\uff0c\u6211\u4eec\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\uff08\u5c3d\u53ef\u80fd\u7528\u5927\u9762\u989d\u7684\u8d27\u5e01\uff09\uff0c\u6700\u7ec8\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u8d2a\u5fc3\u201d\u7b97\u6cd5\u3002
\u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u4f7f\u5f97\u6211\u4eec\u80fd\u591f\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u540c\u65f6\u7f16\u5199\u4ee3\u7801\u8c03\u7528 CPU \u548c GPU \u6267\u884c\u7b97\u6cd5\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u8f6c\u79fb\u5230\u8ba1\u7b97\u673a\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u51b3\u5404\u79cd\u590d\u6742\u95ee\u9898\u3002
Tip
\u5982\u679c\u4f60\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u548c\u4e8c\u5206\u67e5\u627e\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u8bf7\u7ee7\u7eed\u5f80\u4e0b\u9605\u8bfb\uff0c\u672c\u4e66\u5c06\u5f15\u5bfc\u4f60\u8fc8\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002
"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 \u5c0f\u7ed3","text":"- \u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u65e0\u5904\u4e0d\u5728\uff0c\u5e76\u4e0d\u662f\u9065\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8bc6\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u51b3\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u95ee\u9898\u3002
- \u67e5\u9605\u5b57\u5178\u7684\u539f\u7406\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u4f53\u73b0\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002
- \u6574\u7406\u6251\u514b\u7684\u8fc7\u7a0b\u4e0e\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u975e\u5e38\u7c7b\u4f3c\u3002\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u9002\u5408\u6392\u5e8f\u5c0f\u578b\u6570\u636e\u96c6\u3002
- \u8d27\u5e01\u627e\u96f6\u7684\u6b65\u9aa4\u672c\u8d28\u4e0a\u662f\u8d2a\u5fc3\u7b97\u6cd5\uff0c\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\u3002
- \u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u800c\u6570\u636e\u7ed3\u6784\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002
- \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7d27\u5bc6\u76f8\u8fde\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002
- \u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728\uff0c\u79ef\u6728\u4ee3\u8868\u6570\u636e\uff0c\u79ef\u6728\u7684\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u7b49\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6b65\u9aa4\u5219\u5bf9\u5e94\u7b97\u6cd5\u3002
\u300c\u7b97\u6cd5 algorithm\u300d\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002
- \u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002
- \u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u591f\u5728\u6709\u9650\u6b65\u9aa4\u3001\u65f6\u95f4\u548c\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002
- \u5404\u6b65\u9aa4\u90fd\u6709\u786e\u5b9a\u7684\u542b\u4e49\uff0c\u5728\u76f8\u540c\u7684\u8f93\u5165\u548c\u8fd0\u884c\u6761\u4ef6\u4e0b\uff0c\u8f93\u51fa\u59cb\u7ec8\u76f8\u540c\u3002
\u300c\u6570\u636e\u7ed3\u6784 data structure\u300d\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\uff0c\u5177\u6709\u4ee5\u4e0b\u8bbe\u8ba1\u76ee\u6807\u3002
- \u7a7a\u95f4\u5360\u7528\u5c3d\u91cf\u5c11\uff0c\u4ee5\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002
- \u6570\u636e\u64cd\u4f5c\u5c3d\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u76d6\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002
- \u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002
\u6570\u636e\u7ed3\u6784\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\u3002\u5982\u679c\u60f3\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u63d0\u5347\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u534f\u3002\u4e0b\u9762\u4e3e\u4e24\u4e2a\u4f8b\u5b50\u3002
- \u94fe\u8868\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5728\u6570\u636e\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u8bbf\u95ee\u901f\u5ea6\u3002
- \u56fe\u76f8\u8f83\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u4e30\u5bcc\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u5927\u7684\u5185\u5b58\u7a7a\u95f4\u3002
\u5982\u56fe 1-4 \u6240\u793a\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u9ad8\u5ea6\u76f8\u5173\u3001\u7d27\u5bc6\u7ed3\u5408\uff0c\u5177\u4f53\u8868\u73b0\u5728\u4ee5\u4e0b\u4e09\u4e2a\u65b9\u9762\u3002
- \u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u64cd\u4f5c\u6570\u636e\u7684\u65b9\u6cd5\u3002
- \u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u672c\u8eab\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u7ed3\u5408\u7b97\u6cd5\u624d\u80fd\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002
- \u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u4f46\u6267\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\uff0c\u9009\u62e9\u5408\u9002\u7684\u6570\u636e\u7ed3\u6784\u662f\u5173\u952e\u3002
\u56fe 1-4 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb
\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u72b9\u5982\u56fe 1-5 \u6240\u793a\u7684\u62fc\u88c5\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u5305\u542b\u8bb8\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u8fd8\u9644\u6709\u8be6\u7ec6\u7684\u7ec4\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7ec4\u88c5\u51fa\u7cbe\u7f8e\u7684\u79ef\u6728\u6a21\u578b\u3002
\u56fe 1-5 \u00a0 \u62fc\u88c5\u79ef\u6728
\u4e24\u8005\u7684\u8be6\u7ec6\u5bf9\u5e94\u5173\u7cfb\u5982\u8868 1-1 \u6240\u793a\u3002
\u8868 1-1 \u00a0 \u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728
\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 \u62fc\u88c5\u79ef\u6728 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u662f\u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u4e66\u5f97\u4ee5\u63d0\u4f9b\u57fa\u4e8e\u591a\u79cd\u7f16\u7a0b\u8bed\u8a00\u7684\u5b9e\u73b0\u3002
\u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0
\u5728\u5b9e\u9645\u8ba8\u8bba\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u201c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u201d\u7b80\u79f0\u4e3a\u201c\u7b97\u6cd5\u201d\u3002\u6bd4\u5982\u4f17\u6240\u5468\u77e5\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u67e5\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u65b9\u9762\u7684\u77e5\u8bc6\u3002
"},{"location":"chapter_preface/","title":"\u7b2c 0 \u7ae0 \u00a0 \u524d\u8a00","text":"Abstract
\u7b97\u6cd5\u72b9\u5982\u7f8e\u5999\u7684\u4ea4\u54cd\u4e50\uff0c\u6bcf\u4e00\u884c\u4ee3\u7801\u90fd\u50cf\u97f5\u5f8b\u822c\u6d41\u6dcc\u3002
\u613f\u8fd9\u672c\u4e66\u5728\u4f60\u7684\u8111\u6d77\u4e2d\u8f7b\u8f7b\u54cd\u8d77\uff0c\u7559\u4e0b\u72ec\u7279\u800c\u6df1\u523b\u7684\u65cb\u5f8b\u3002
"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 0.1 \u00a0 \u5173\u4e8e\u672c\u4e66
- 0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66
- 0.3 \u00a0 \u5c0f\u7ed3
\u672c\u9879\u76ee\u65e8\u5728\u521b\u5efa\u4e00\u672c\u5f00\u6e90\u3001\u514d\u8d39\u3001\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u6559\u7a0b\u3002
- \u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u3002
- \u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Python\u3001C++\u3001Java\u3001C#\u3001Go\u3001Swift\u3001JavaScript\u3001TypeScript\u3001Dart\u3001Rust\u3001C\u3001Zig \u7b49\u8bed\u8a00\u3002
- \u9f13\u52b1\u8bfb\u8005\u5728\u7ae0\u8282\u8ba8\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u901a\u5e38\u53ef\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\u3002
\u82e5\u4f60\u662f\u7b97\u6cd5\u521d\u5b66\u8005\uff0c\u4ece\u672a\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u4e00\u4e9b\u5237\u9898\u7ecf\u9a8c\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8ba4\u8bc6\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u672c\u4e66\u6b63\u662f\u4e3a\u4f60\u91cf\u8eab\u5b9a\u5236\u7684\uff01
\u5982\u679c\u4f60\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u5237\u9898\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u52a9\u4f60\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002
\u82e5\u4f60\u662f\u7b97\u6cd5\u5927\u795e\uff0c\u6211\u4eec\u671f\u5f85\u6536\u5230\u4f60\u7684\u5b9d\u8d35\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002
\u524d\u7f6e\u6761\u4ef6
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"},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2 \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"\u672c\u4e66\u7684\u4e3b\u8981\u5185\u5bb9\u5982\u56fe 0-1 \u6240\u793a\u3002
- \u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u4e0e\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002
- \u6570\u636e\u7ed3\u6784\uff1a\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u6570\u636e\u7ed3\u6784\u7684\u5206\u7c7b\u65b9\u6cd5\u3002\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002
- \u7b97\u6cd5\uff1a\u641c\u7d22\u3001\u6392\u5e8f\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u6548\u7387\u3001\u5e94\u7528\u573a\u666f\u3001\u89e3\u9898\u6b65\u9aa4\u548c\u793a\u4f8b\u95ee\u9898\u7b49\u3002
\u56fe 0-1 \u00a0 \u672c\u4e66\u4e3b\u8981\u5185\u5bb9
"},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3 \u00a0 \u81f4\u8c22","text":"\u5728\u672c\u4e66\u7684\u521b\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a
- \u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u4e2d\u4f60\u9f13\u52b1\u6211\u201c\u5feb\u884c\u52a8\u8d77\u6765\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\uff1b
- \u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u4e3a\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8bb8\u591a\u5b9d\u8d35\u5efa\u8bae\uff0c\u4f7f\u5f97\u672c\u4e66\u66f4\u9002\u5408\u65b0\u624b\u9605\u8bfb\uff1b
- \u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e00\u4e2a\u5bcc\u6709\u521b\u610f\u7684\u540d\u5b57\uff0c\u5524\u8d77\u5927\u5bb6\u5199\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801 \"Hello World!\" \u7684\u7f8e\u597d\u56de\u5fc6\uff1b
- \u611f\u8c22\u6821\u94e8\u5728\u77e5\u8bc6\u4ea7\u6743\u65b9\u9762\u63d0\u4f9b\u7684\u4e13\u4e1a\u5e2e\u52a9\uff0c\u8fd9\u5bf9\u672c\u5f00\u6e90\u4e66\u7684\u53d1\u5c55\u8d77\u5230\u4e86\u91cd\u8981\u7684\u4f5c\u7528\uff1b
- \u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c logo \uff0c\u5e76\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\uff1b
- \u611f\u8c22 @squidfunk \u63d0\u4f9b\u7684\u5199\u4f5c\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u4ed6\u5f00\u53d1\u7684\u5f00\u6e90\u6587\u6863\u4e3b\u9898 Material-for-MkDocs \u3002
\u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u5173\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u8fd9\u4e9b\u4f5c\u54c1\u4e3a\u672c\u4e66\u63d0\u4f9b\u4e86\u4f18\u79c0\u7684\u8303\u672c\uff0c\u786e\u4fdd\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u51c6\u786e\u6027\u4e0e\u54c1\u8d28\u3002\u5728\u6b64\u611f\u8c22\u6240\u6709\u8001\u5e08\u548c\u524d\u8f88\u4eec\u7684\u6770\u51fa\u8d21\u732e\uff01
\u672c\u4e66\u5021\u5bfc\u624b\u8111\u5e76\u7528\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u4e00\u70b9\u4e0a\u6df1\u53d7\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u7684\u542f\u53d1\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8bfb\u8005\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u4f18\u79c0\u7684\u8457\u4f5c\u3002
\u8877\u5fc3\u611f\u8c22\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u4eec\u4e00\u76f4\u4ee5\u6765\u7684\u652f\u6301\u4e0e\u9f13\u52b1\uff0c\u8ba9\u6211\u6709\u673a\u4f1a\u505a\u8fd9\u4ef6\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002
"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"Tip
\u4e3a\u4e86\u83b7\u5f97\u6700\u4f73\u7684\u9605\u8bfb\u4f53\u9a8c\uff0c\u5efa\u8bae\u4f60\u901a\u8bfb\u672c\u8282\u5185\u5bb9\u3002
"},{"location":"chapter_preface/suggestions/#021","title":"0.2.1 \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"- \u6807\u9898\u540e\u6807\u6ce8
*\u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u56f0\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002 - \u91cd\u8981\u4e13\u6709\u540d\u8bcd\u53ca\u5176\u82f1\u6587\u7ffb\u8bd1\u4f1a\u7528
\u300c \u300d\u62ec\u53f7\u6807\u6ce8\uff0c\u4f8b\u5982\u300c\u6570\u7ec4 array\u300d\u3002\u5efa\u8bae\u8bb0\u4f4f\u5b83\u4eec\uff0c\u4ee5\u4fbf\u9605\u8bfb\u6587\u732e\u3002 - \u4e13\u6709\u540d\u8bcd\u548c\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528
\u201c\u5f15\u53f7\u201d\u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002 - \u91cd\u8981\u540d\u8bcd\u3001\u91cd\u70b9\u5185\u5bb9\u548c\u603b\u7ed3\u6027\u8bed\u53e5\u4f1a\u88ab \u52a0\u7c97 \uff0c\u8fd9\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002
- \u5f53\u6d89\u53ca\u7f16\u7a0b\u8bed\u8a00\u4e4b\u95f4\u4e0d\u4e00\u81f4\u7684\u540d\u8bcd\u65f6\uff0c\u672c\u4e66\u5747\u4ee5 Python \u4e3a\u51c6\uff0c\u4f8b\u5982\u4f7f\u7528 \\(\\text{None}\\) \u6765\u8868\u793a\u201c\u7a7a\u201d\u3002
- \u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002
\"\"\"\u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\"\"\"\n\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
"},{"location":"chapter_preface/suggestions/#022","title":"0.2.2 \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"// \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n// \u591a\u884c\n// \u6ce8\u91ca\n\u76f8\u8f83\u4e8e\u6587\u5b57\uff0c\u89c6\u9891\u548c\u56fe\u7247\u5177\u6709\u66f4\u9ad8\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\uff0c\u66f4\u6613\u4e8e\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u5c06\u4e3b\u8981\u901a\u8fc7\u52a8\u753b\u548c\u56fe\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5219\u4f5c\u4e3a\u52a8\u753b\u548c\u56fe\u7247\u7684\u89e3\u91ca\u4e0e\u8865\u5145\u3002
\u5982\u679c\u4f60\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u5982\u56fe 0-2 \u6240\u793a\u7684\u52a8\u753b\u6216\u56fe\u89e3\uff0c\u8bf7\u4ee5\u56fe\u4e3a\u4e3b\u3001\u4ee5\u6587\u5b57\u4e3a\u8f85\uff0c\u7efc\u5408\u4e24\u8005\u6765\u7406\u89e3\u5185\u5bb9\u3002
\u56fe 0-2 \u00a0 \u52a8\u753b\u56fe\u89e3\u793a\u4f8b
"},{"location":"chapter_preface/suggestions/#023","title":"0.2.3 \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"\u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728 GitHub \u4ed3\u5e93\u3002\u5982\u56fe 0-3 \u6240\u793a\uff0c\u6e90\u4ee3\u7801\u9644\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u4e00\u952e\u8fd0\u884c\u3002
\u5982\u679c\u65f6\u95f4\u5141\u8bb8\uff0c\u5efa\u8bae\u4f60\u53c2\u7167\u4ee3\u7801\u81ea\u884c\u6572\u4e00\u904d\u3002\u5982\u679c\u5b66\u4e60\u65f6\u95f4\u6709\u9650\uff0c\u8bf7\u81f3\u5c11\u901a\u8bfb\u5e76\u8fd0\u884c\u6240\u6709\u4ee3\u7801\u3002
\u4e0e\u9605\u8bfb\u4ee3\u7801\u76f8\u6bd4\uff0c\u7f16\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u66f4\u591a\u6536\u83b7\u3002\u52a8\u624b\u5b66\uff0c\u624d\u662f\u771f\u7684\u5b66\u3002
\u56fe 0-3 \u00a0 \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b
\u8fd0\u884c\u4ee3\u7801\u7684\u524d\u7f6e\u5de5\u4f5c\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\u3002
\u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u8bf7\u53c2\u7167\u9644\u5f55\u6240\u793a\u7684\u6559\u7a0b\u8fdb\u884c\u5b89\u88c5\uff0c\u5982\u679c\u5df2\u5b89\u88c5\uff0c\u5219\u53ef\u8df3\u8fc7\u6b64\u6b65\u9aa4\u3002
\u7b2c\u4e8c\u6b65\uff1a\u514b\u9686\u6216\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u5e93\u3002\u524d\u5f80 GitHub \u4ed3\u5e93\u3002
\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u4ed3\u5e93\uff1a
git clone https://github.com/krahets/hello-algo.git\n\u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u5728\u56fe 0-4 \u6240\u793a\u7684\u4f4d\u7f6e\uff0c\u70b9\u51fb\u201cDownload ZIP\u201d\u6309\u94ae\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u7136\u540e\u5728\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002
\u56fe 0-4 \u00a0 \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801
\u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u5982\u56fe 0-5 \u6240\u793a\uff0c\u5bf9\u4e8e\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\u7684\u4ee3\u7801\u5757\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u4ed3\u5e93\u7684
codes\u6587\u4ef6\u5939\u5185\u627e\u5230\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u5c06\u5e2e\u52a9\u4f60\u8282\u7701\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u8ba9\u4f60\u80fd\u591f\u4e13\u6ce8\u4e8e\u5b66\u4e60\u5185\u5bb9\u3002\u56fe 0-5 \u00a0 \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6
"},{"location":"chapter_preface/suggestions/#024","title":"0.2.4 \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u8f7b\u6613\u8df3\u8fc7\u90a3\u4e9b\u6ca1\u5b66\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u63d0\u51fa\u4f60\u7684\u95ee\u9898\uff0c\u6211\u548c\u5c0f\u4f19\u4f34\u4eec\u5c06\u7aed\u8bda\u4e3a\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\u53ef\u5728\u4e24\u5929\u5185\u56de\u590d\u3002
\u5982\u56fe 0-6 \u6240\u793a\uff0c\u6bcf\u4e2a\u7ae0\u8282\u7684\u5e95\u90e8\u90fd\u914d\u6709\u8bc4\u8bba\u533a\u3002\u5e0c\u671b\u4f60\u80fd\u591a\u5173\u6ce8\u8bc4\u8bba\u533a\u7684\u5185\u5bb9\u3002\u4e00\u65b9\u9762\uff0c\u4f60\u53ef\u4ee5\u4e86\u89e3\u5927\u5bb6\u9047\u5230\u7684\u95ee\u9898\uff0c\u4ece\u800c\u67e5\u6f0f\u8865\u7f3a\uff0c\u6fc0\u53d1\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u671f\u5f85\u4f60\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u4f19\u4f34\u7684\u95ee\u9898\uff0c\u5206\u4eab\u4f60\u7684\u89c1\u89e3\uff0c\u5e2e\u52a9\u4ed6\u4eba\u8fdb\u6b65\u3002
\u56fe 0-6 \u00a0 \u8bc4\u8bba\u533a\u793a\u4f8b
"},{"location":"chapter_preface/suggestions/#025","title":"0.2.5 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"\u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5212\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\u3002
- \u9636\u6bb5\u4e00\uff1a\u7b97\u6cd5\u5165\u95e8\u3002\u6211\u4eec\u9700\u8981\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u548c\u7528\u6cd5\uff0c\u5b66\u4e60\u4e0d\u540c\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u7684\u5185\u5bb9\u3002
- \u9636\u6bb5\u4e8c\uff1a\u5237\u7b97\u6cd5\u9898\u3002\u5efa\u8bae\u4ece\u70ed\u95e8\u9898\u76ee\u5f00\u5237\uff0c\u5982\u201c\u5251\u6307 Offer\u201d\u548c\u201cLeetCode Hot 100\u201d\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521d\u6b21\u5237\u9898\u65f6\uff0c\u201c\u77e5\u8bc6\u9057\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u4e2a\u6311\u6218\uff0c\u4f46\u8bf7\u653e\u5fc3\uff0c\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u201c\u827e\u5bbe\u6d69\u65af\u9057\u5fd8\u66f2\u7ebf\u201d\u6765\u590d\u4e60\u9898\u76ee\uff0c\u901a\u5e38\u5728\u8fdb\u884c 3\uff5e5 \u8f6e\u7684\u91cd\u590d\u540e\uff0c\u5c31\u80fd\u5c06\u5176\u7262\u8bb0\u5728\u5fc3\u3002
- \u9636\u6bb5\u4e09\uff1a\u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u548c\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65ad\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u91c7\u7528\u8fdb\u9636\u5237\u9898\u7b56\u7565\uff0c\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u7684\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u627e\u5230\u3002
\u5982\u56fe 0-7 \u6240\u793a\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u6db5\u76d6\u201c\u9636\u6bb5\u4e00\u201d\uff0c\u65e8\u5728\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u5f00\u9636\u6bb5\u4e8c\u548c\u9636\u6bb5\u4e09\u7684\u5b66\u4e60\u3002
\u56fe 0-7 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf
"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 \u5c0f\u7ed3","text":"- \u672c\u4e66\u7684\u4e3b\u8981\u53d7\u4f17\u662f\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5982\u679c\u5df2\u6709\u4e00\u5b9a\u57fa\u7840\uff0c\u672c\u4e66\u80fd\u5e2e\u52a9\u4f60\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u4e66\u4e2d\u6e90\u4ee3\u7801\u4e5f\u53ef\u4f5c\u4e3a\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u4f7f\u7528\u3002
- \u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5305\u62ec\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002
- \u5bf9\u4e8e\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u81f3\u5173\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002
- \u4e66\u4e2d\u7684\u52a8\u753b\u548c\u56fe\u89e3\u901a\u5e38\u7528\u4e8e\u4ecb\u7ecd\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u3002\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5e94\u7ed9\u4e88\u8fd9\u4e9b\u5185\u5bb9\u66f4\u591a\u5173\u6ce8\u3002
- \u5b9e\u8df5\u4e43\u5b66\u4e60\u7f16\u7a0b\u4e4b\u6700\u4f73\u9014\u5f84\u3002\u5f3a\u70c8\u5efa\u8bae\u8fd0\u884c\u6e90\u4ee3\u7801\u5e76\u4eb2\u81ea\u6572\u4ee3\u7801\u3002
- \u672c\u4e66\u7f51\u9875\u7248\u7684\u6bcf\u4e2a\u7ae0\u8282\u90fd\u8bbe\u6709\u8ba8\u8bba\u533a\uff0c\u6b22\u8fce\u968f\u65f6\u5206\u4eab\u4f60\u7684\u7591\u60d1\u4e0e\u89c1\u89e3\u3002
[1] Thomas H. Cormen, et al. Introduction to Algorithms (3rd Edition).
[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1st Edition).
[3] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08C \u8bed\u8a00\u7248\uff09.
[4] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.
[5] \u9a6c\u514b \u827e\u4f26 \u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.
[6] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.
[7] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.
[8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6th Edition).
[9] Aston Zhang, et al. Dive into Deep Learning.
"},{"location":"chapter_searching/","title":"\u7b2c 10 \u7ae0 \u00a0 \u641c\u7d22","text":"Abstract
\u641c\u7d22\u662f\u4e00\u573a\u672a\u77e5\u7684\u5192\u9669\uff0c\u6211\u4eec\u6216\u8bb8\u9700\u8981\u8d70\u904d\u795e\u79d8\u7a7a\u95f4\u7684\u6bcf\u4e2a\u89d2\u843d\uff0c\u53c8\u6216\u8bb8\u53ef\u4ee5\u5feb\u901f\u9501\u5b9a\u76ee\u6807\u3002
\u5728\u8fd9\u573a\u5bfb\u89c5\u4e4b\u65c5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u63a2\u7d22\u90fd\u53ef\u80fd\u5f97\u5230\u4e00\u4e2a\u672a\u66fe\u6599\u60f3\u7684\u7b54\u6848\u3002
"},{"location":"chapter_searching/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 10.1 \u00a0 \u4e8c\u5206\u67e5\u627e
- 10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9
- 10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c
- 10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565
- 10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5
- 10.6 \u00a0 \u5c0f\u7ed3
\u300c\u4e8c\u5206\u67e5\u627e binary search\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u9ad8\u6548\u641c\u7d22\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u8303\u56f4\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u4e3a\u6b62\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4
nums\uff0c\u5143\u7d20\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u4e14\u4e0d\u91cd\u590d\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20target\u5728\u8be5\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 10-1 \u6240\u793a\u3002\u56fe 10-1 \u00a0 \u4e8c\u5206\u67e5\u627e\u793a\u4f8b\u6570\u636e
\u5982\u56fe 10-2 \u6240\u793a\uff0c\u6211\u4eec\u5148\u521d\u59cb\u5316\u6307\u9488 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u7d22\u533a\u95f4 \\([0, n - 1]\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u4e2d\u62ec\u53f7\u8868\u793a\u95ed\u533a\u95f4\uff0c\u5176\u5305\u542b\u8fb9\u754c\u503c\u672c\u8eab\u3002
\u63a5\u4e0b\u6765\uff0c\u5faa\u73af\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002
- \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\: \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002
- \u5224\u65ad
nums[m]\u548ctarget\u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e09\u79cd\u60c5\u51b5\u3002- \u5f53
nums[m] < target\u65f6\uff0c\u8bf4\u660etarget\u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(i = m + 1\\) \u3002 - \u5f53
nums[m] > target\u65f6\uff0c\u8bf4\u660etarget\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(j = m - 1\\) \u3002 - \u5f53
nums[m] = target\u65f6\uff0c\u8bf4\u660e\u627e\u5230target\uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \u3002
- \u5f53
\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20\uff0c\u641c\u7d22\u533a\u95f4\u6700\u7ec8\u4f1a\u7f29\u5c0f\u4e3a\u7a7a\u3002\u6b64\u65f6\u8fd4\u56de \\(-1\\) \u3002
<1><2><3><4><5><6><7>\u56fe 10-2 \u00a0 \u4e8c\u5206\u67e5\u627e\u6d41\u7a0b
\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e \\(i\\) \u548c \\(j\\) \u90fd\u662f
int\u7c7b\u578b\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u4f1a\u8d85\u51faint\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u4e3a\u4e86\u907f\u514d\u5927\u6570\u8d8a\u754c\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u6765\u8ba1\u7b97\u4e2d\u70b9\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search.py
binary_search.cppdef binary_search(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\"\"\"\n # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n i, j = 0, len(nums) - 1\n # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while i <= j:\n # \u7406\u8bba\u4e0a Python \u7684\u6570\u5b57\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\uff0c\u65e0\u987b\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\n m = (i + j) // 2 # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target:\n i = m + 1 # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n elif nums[m] > target:\n j = m - 1 # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n else:\n return m # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return -1 # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
binary_search.java/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector<int> &nums, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = nums.size() - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.cs/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = nums.length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.go/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint BinarySearch(int[] nums, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = nums.Length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.swift/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n i, j := 0, len(nums)-1\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n for i <= j {\n m := i + (j-i)/2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1\n } else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1\n}\n
binary_search.js/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -> Int {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n var i = 0\n var j = nums.count - 1\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while i <= j {\n let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1\n } else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1\n}\n
binary_search.ts/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n let i = 0,\n j = nums.length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n const m = parseInt(i + (j - i) / 2);\n if (nums[m] < target)\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target)\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else return m; // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.dart/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n let i = 0,\n j = nums.length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n const m = Math.floor(i + (j - i) / 2);\n if (nums[m] < target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n } else if (nums[m] > target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n }\n return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
binary_search.rs/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(List<int> nums, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = nums.length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n } else if (nums[m] > target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.c/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfn binary_search(nums: &[i32], target: i32) -> i32 {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n let mut i = 0;\n let mut j = nums.len() as i32 - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while i <= j {\n let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m as usize] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n } else if nums[m as usize] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n } \n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.zig/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int *nums, int len, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = len - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n// \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n var i: usize = 0;\n var j: usize = nums.items.len - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n var m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums.items[m] < target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n } else if (nums.items[m] > target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return @intCast(m);\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(\\log n)\\) \uff1a\u5728\u4e8c\u5206\u5faa\u73af\u4e2d\uff0c\u533a\u95f4\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\uff0c\u5faa\u73af\u6b21\u6570\u4e3a \\(\\log_2 n\\) \u3002
\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002
"},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1 \u00a0 \u533a\u95f4\u8868\u793a\u65b9\u6cd5","text":"\u9664\u4e86\u4e0a\u8ff0\u53cc\u95ed\u533a\u95f4\u5916\uff0c\u5e38\u89c1\u7684\u533a\u95f4\u8868\u793a\u8fd8\u6709\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u533a\u95f4\uff0c\u5b9a\u4e49\u4e3a \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8be5\u8868\u793a\u4e0b\uff0c\u533a\u95f4 \\([i, j)\\) \u5728 \\(i = j\\) \u65f6\u4e3a\u7a7a\u3002
\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u8be5\u8868\u793a\u5b9e\u73b0\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search.py
binary_search.cppdef binary_search_lcro(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\"\"\"\n # \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n i, j = 0, len(nums)\n # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while i < j:\n m = (i + j) // 2 # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target:\n i = m + 1 # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n elif nums[m] > target:\n j = m # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n else:\n return m # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return -1 # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
binary_search.java/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(vector<int> &nums, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = nums.size();\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.cs/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = nums.length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.go/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint BinarySearchLCRO(int[] nums, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = nums.Length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.swift/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n i, j := 0, len(nums)\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n for i < j {\n m := i + (j-i)/2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1\n } else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1\n}\n
binary_search.js/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -> Int {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n var i = 0\n var j = nums.count\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while i < j {\n let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1\n } else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1\n}\n
binary_search.ts/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums, target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n let i = 0,\n j = nums.length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n const m = parseInt(i + (j - i) / 2);\n if (nums[m] < target)\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target)\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n else return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.dart/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n let i = 0,\n j = nums.length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n const m = Math.floor(i + (j - i) / 2);\n if (nums[m] < target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n } else if (nums[m] > target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n }\n return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
binary_search.rs/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(List<int> nums, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = nums.length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n } else if (nums[m] > target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.c/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfn binary_search_lcro(nums: &[i32], target: i32) -> i32 {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n let mut i = 0;\n let mut j = nums.len() as i32;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while i < j {\n let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m as usize] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n } else if nums[m as usize] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m - 1;\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n } \n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.zig/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = len;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n// \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n var i: usize = 0;\n var j: usize = nums.items.len;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n var m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums.items[m] < target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n } else if (nums.items[m] > target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return @intCast(m);\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n\u5982\u56fe 10-3 \u6240\u793a\uff0c\u5728\u4e24\u79cd\u533a\u95f4\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u5faa\u73af\u6761\u4ef6\u548c\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002
\u7531\u4e8e\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e2d\u7684\u5de6\u53f3\u8fb9\u754c\u90fd\u88ab\u5b9a\u4e49\u4e3a\u95ed\u533a\u95f4\uff0c\u56e0\u6b64\u901a\u8fc7\u6307\u9488 \\(i\\) \u548c\u6307\u9488 \\(j\\) \u7f29\u5c0f\u533a\u95f4\u7684\u64cd\u4f5c\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\uff0c\u56e0\u6b64\u4e00\u822c\u5efa\u8bae\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002
\u56fe 10-3 \u00a0 \u4e24\u79cd\u533a\u95f4\u5b9a\u4e49
"},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"\u4e8c\u5206\u67e5\u627e\u5728\u65f6\u95f4\u548c\u7a7a\u95f4\u65b9\u9762\u90fd\u6709\u8f83\u597d\u7684\u6027\u80fd\u3002
- \u4e8c\u5206\u67e5\u627e\u7684\u65f6\u95f4\u6548\u7387\u9ad8\u3002\u5728\u5927\u6570\u636e\u91cf\u4e0b\uff0c\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002
- \u4e8c\u5206\u67e5\u627e\u65e0\u987b\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u8f83\u4e8e\u9700\u8981\u501f\u52a9\u989d\u5916\u7a7a\u95f4\u7684\u641c\u7d22\u7b97\u6cd5\uff08\u4f8b\u5982\u54c8\u5e0c\u67e5\u627e\uff09\uff0c\u4e8c\u5206\u67e5\u627e\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002
\u7136\u800c\uff0c\u4e8c\u5206\u67e5\u627e\u5e76\u975e\u9002\u7528\u4e8e\u6240\u6709\u60c5\u51b5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002
- \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u82e5\u8f93\u5165\u6570\u636e\u65e0\u5e8f\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u507f\u5931\u3002\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u9ad8\u3002\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4fdd\u6301\u6570\u7ec4\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002
- \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u4e8c\u5206\u67e5\u627e\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u6267\u884c\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u7684\u6548\u7387\u8f83\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5e94\u7528\u5728\u94fe\u8868\u6216\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
- \u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u6027\u80fd\u66f4\u4f73\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5f53\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4
nums\uff0c\u5176\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a\u5143\u7d20target\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u56de\u5fc6\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u7684\u65b9\u6cd5\uff0c\u641c\u7d22\u5b8c\u6210\u540e \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a
target\uff0c\u56e0\u6b64\u67e5\u627e\u63d2\u5165\u70b9\u672c\u8d28\u4e0a\u662f\u5728\u67e5\u627e\u6700\u5de6\u4e00\u4e2atarget\u7684\u7d22\u5f15\u3002\u8003\u8651\u901a\u8fc7\u67e5\u627e\u63d2\u5165\u70b9\u7684\u51fd\u6570\u5b9e\u73b0\u67e5\u627e\u5de6\u8fb9\u754c\u3002\u8bf7\u6ce8\u610f\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u4e0d\u5305\u542b
target\uff0c\u8fd9\u79cd\u60c5\u51b5\u53ef\u80fd\u5bfc\u81f4\u4ee5\u4e0b\u4e24\u79cd\u7ed3\u679c\u3002- \u63d2\u5165\u70b9\u7684\u7d22\u5f15 \\(i\\) \u8d8a\u754c\u3002
- \u5143\u7d20
nums[i]\u4e0etarget\u4e0d\u76f8\u7b49\u3002
\u5f53\u9047\u5230\u4ee5\u4e0a\u4e24\u79cd\u60c5\u51b5\u65f6\uff0c\u76f4\u63a5\u8fd4\u56de \\(-1\\) \u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_edge.py
binary_search_edge.cppdef binary_search_left_edge(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target\"\"\"\n # \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n i = binary_search_insertion(nums, target)\n # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if i == len(nums) or nums[i] != target:\n return -1\n # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i\n
binary_search_edge.java/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(vector<int> &nums, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == nums.size() || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.cs/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int[] nums, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binary_search_insertion.binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == nums.length || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.go/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint BinarySearchLeftEdge(int[] nums, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == nums.Length || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.swift/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n i := binarySearchInsertion(nums, target)\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if i == len(nums) || nums[i] != target {\n return -1\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i\n}\n
binary_search_edge.js/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -> Int {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n let i = binarySearchInsertion(nums: nums, target: target)\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if i == nums.count || nums[i] != target {\n return -1\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i\n}\n
binary_search_edge.ts/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums, target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n const i = binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i === nums.length || nums[i] !== target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.dart/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums: Array<number>, target: number): number {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n const i = binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i === nums.length || nums[i] !== target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.rs/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(List<int> nums, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == nums.length || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.c/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfn binary_search_left_edge(nums: &[i32], target: i32) -> i32 {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n let i = binary_search_insertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if i == nums.len() as i32 || nums[i as usize] != target {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n i\n}\n
binary_search_edge.zig/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binarySearchInsertion(nums, numSize, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == numSize || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
"},{"location":"chapter_searching/binary_search_edge/#1032","title":"10.3.2 \u00a0 \u67e5\u627e\u53f3\u8fb9\u754c","text":"[class]{}-[func]{binarySearchLeftEdge}\n\u90a3\u4e48\u5982\u4f55\u67e5\u627e\u6700\u53f3\u4e00\u4e2a
target\u5462\uff1f\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\u662f\u4fee\u6539\u4ee3\u7801\uff0c\u66ff\u6362\u5728nums[m] == target\u60c5\u51b5\u4e0b\u7684\u6307\u9488\u6536\u7f29\u64cd\u4f5c\u3002\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002\u4e0b\u9762\u6211\u4eec\u4ecb\u7ecd\u4e24\u79cd\u66f4\u52a0\u53d6\u5de7\u7684\u65b9\u6cd5\u3002
"},{"location":"chapter_searching/binary_search_edge/#1","title":"1. \u00a0 \u590d\u7528\u67e5\u627e\u5de6\u8fb9\u754c","text":"\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u67e5\u627e\u6700\u5de6\u5143\u7d20\u7684\u51fd\u6570\u6765\u67e5\u627e\u6700\u53f3\u5143\u7d20\uff0c\u5177\u4f53\u65b9\u6cd5\u4e3a\uff1a\u5c06\u67e5\u627e\u6700\u53f3\u4e00\u4e2a
target\u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2atarget + 1\u3002\u5982\u56fe 10-7 \u6240\u793a\uff0c\u67e5\u627e\u5b8c\u6210\u540e\uff0c\u6307\u9488 \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a
target + 1\uff08\u5982\u679c\u5b58\u5728\uff09\uff0c\u800c \\(j\\) \u6307\u5411\u6700\u53f3\u4e00\u4e2atarget\uff0c\u56e0\u6b64\u8fd4\u56de \\(j\\) \u5373\u53ef\u3002\u56fe 10-7 \u00a0 \u5c06\u67e5\u627e\u53f3\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5de6\u8fb9\u754c
\u8bf7\u6ce8\u610f\uff0c\u8fd4\u56de\u7684\u63d2\u5165\u70b9\u662f \\(i\\) \uff0c\u56e0\u6b64\u9700\u8981\u5c06\u5176\u51cf \\(1\\) \uff0c\u4ece\u800c\u83b7\u5f97 \\(j\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_edge.py
binary_search_edge.cppdef binary_search_right_edge(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target\"\"\"\n # \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n i = binary_search_insertion(nums, target + 1)\n # j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n j = i - 1\n # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if j == -1 or nums[j] != target:\n return -1\n # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j\n
binary_search_edge.java/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(vector<int> &nums, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.cs/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int[] nums, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.go/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint BinarySearchRightEdge(int[] nums, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.swift/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums []int, target int) int {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n i := binarySearchInsertion(nums, target+1)\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n j := i - 1\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if j == -1 || nums[j] != target {\n return -1\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j\n}\n
binary_search_edge.js/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -> Int {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n let i = binarySearchInsertion(nums: nums, target: target + 1)\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n let j = i - 1\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if j == -1 || nums[j] != target {\n return -1\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j\n}\n
binary_search_edge.ts/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums, target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n const i = binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n const j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j === -1 || nums[j] !== target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.dart/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums: Array<number>, target: number): number {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n const i = binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n const j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j === -1 || nums[j] !== target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.rs/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(List<int> nums, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.c/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfn binary_search_right_edge(nums: &[i32], target: i32) -> i32 {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n let i = binary_search_insertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n let j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if j == -1 || nums[j as usize] != target {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n j\n}\n
binary_search_edge.zig/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binarySearchInsertion(nums, numSize, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
"},{"location":"chapter_searching/binary_search_edge/#2","title":"2. \u00a0 \u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20","text":"[class]{}-[func]{binarySearchRightEdge}\n\u6211\u4eec\u77e5\u9053\uff0c\u5f53\u6570\u7ec4\u4e0d\u5305\u542b
target\u65f6\uff0c\u6700\u7ec8 \\(i\\) \u548c \\(j\\) \u4f1a\u5206\u522b\u6307\u5411\u9996\u4e2a\u5927\u4e8e\u3001\u5c0f\u4e8etarget\u7684\u5143\u7d20\u3002\u56e0\u6b64\uff0c\u5982\u56fe 10-8 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728\u7684\u5143\u7d20\uff0c\u7528\u4e8e\u67e5\u627e\u5de6\u53f3\u8fb9\u754c\u3002
- \u67e5\u627e\u6700\u5de6\u4e00\u4e2a
target\uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627etarget - 0.5\uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(i\\) \u3002 - \u67e5\u627e\u6700\u53f3\u4e00\u4e2a
target\uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627etarget + 0.5\uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(j\\) \u3002
\u56fe 10-8 \u00a0 \u5c06\u67e5\u627e\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20
\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u4ee5\u4e0b\u4e24\u70b9\u503c\u5f97\u6ce8\u610f\u3002
- \u7ed9\u5b9a\u6570\u7ec4\u4e0d\u5305\u542b\u5c0f\u6570\uff0c\u8fd9\u610f\u5473\u7740\u6211\u4eec\u65e0\u987b\u5173\u5fc3\u5982\u4f55\u5904\u7406\u76f8\u7b49\u7684\u60c5\u51b5\u3002
- \u56e0\u4e3a\u8be5\u65b9\u6cd5\u5f15\u5165\u4e86\u5c0f\u6570\uff0c\u6240\u4ee5\u9700\u8981\u5c06\u51fd\u6570\u4e2d\u7684\u53d8\u91cf
target\u6539\u4e3a\u6d6e\u70b9\u6570\u7c7b\u578b\u3002
\u4e8c\u5206\u67e5\u627e\u4e0d\u4ec5\u53ef\u7528\u4e8e\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff0c\u8fd8\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u53d8\u79cd\u95ee\u9898\uff0c\u6bd4\u5982\u641c\u7d22\u76ee\u6807\u5143\u7d20\u7684\u63d2\u5165\u4f4d\u7f6e\u3002
"},{"location":"chapter_searching/binary_search_insertion/#1021","title":"10.2.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4
nums\u548c\u4e00\u4e2a\u5143\u7d20target\uff0c\u6570\u7ec4\u4e0d\u5b58\u5728\u91cd\u590d\u5143\u7d20\u3002\u73b0\u5c06target\u63d2\u5165\u6570\u7ec4nums\u4e2d\uff0c\u5e76\u4fdd\u6301\u5176\u6709\u5e8f\u6027\u3002\u82e5\u6570\u7ec4\u4e2d\u5df2\u5b58\u5728\u5143\u7d20target\uff0c\u5219\u63d2\u5165\u5230\u5176\u5de6\u65b9\u3002\u8bf7\u8fd4\u56de\u63d2\u5165\u540etarget\u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u56fe 10-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u793a\u4f8b\u6570\u636e
\u5982\u679c\u60f3\u590d\u7528\u4e0a\u4e00\u8282\u7684\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\uff0c\u5219\u9700\u8981\u56de\u7b54\u4ee5\u4e0b\u4e24\u4e2a\u95ee\u9898\u3002
\u95ee\u9898\u4e00\uff1a\u5f53\u6570\u7ec4\u4e2d\u5305\u542b
target\u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u662f\u5426\u662f\u8be5\u5143\u7d20\u7684\u7d22\u5f15\uff1f\u9898\u76ee\u8981\u6c42\u5c06
target\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u5de6\u8fb9\uff0c\u8fd9\u610f\u5473\u7740\u65b0\u63d2\u5165\u7684target\u66ff\u6362\u4e86\u539f\u6765target\u7684\u4f4d\u7f6e\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u6570\u7ec4\u5305\u542btarget\u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u5c31\u662f\u8be5target\u7684\u7d22\u5f15\u3002\u95ee\u9898\u4e8c\uff1a\u5f53\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728
target\u65f6\uff0c\u63d2\u5165\u70b9\u662f\u54ea\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1f\u8fdb\u4e00\u6b65\u601d\u8003\u4e8c\u5206\u67e5\u627e\u8fc7\u7a0b\uff1a\u5f53
nums[m] < target\u65f6 \\(i\\) \u79fb\u52a8\uff0c\u8fd9\u610f\u5473\u7740\u6307\u9488 \\(i\\) \u5728\u5411\u5927\u4e8e\u7b49\u4e8etarget\u7684\u5143\u7d20\u9760\u8fd1\u3002\u540c\u7406\uff0c\u6307\u9488 \\(j\\) \u59cb\u7ec8\u5728\u5411\u5c0f\u4e8e\u7b49\u4e8etarget\u7684\u5143\u7d20\u9760\u8fd1\u3002\u56e0\u6b64\u4e8c\u5206\u7ed3\u675f\u65f6\u4e00\u5b9a\u6709\uff1a\\(i\\) \u6307\u5411\u9996\u4e2a\u5927\u4e8e
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_insertion.pytarget\u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8etarget\u7684\u5143\u7d20\u3002\u6613\u5f97\u5f53\u6570\u7ec4\u4e0d\u5305\u542btarget\u65f6\uff0c\u63d2\u5165\u7d22\u5f15\u4e3a \\(i\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
binary_search_insertion.cppdef binary_search_insertion_simple(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n i, j = 0, len(nums) - 1 # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j:\n m = (i + j) // 2 # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target:\n i = m + 1 # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n elif nums[m] > target:\n j = m - 1 # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n else:\n return m # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n
binary_search_insertion.java/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(vector<int> &nums, int target) {\n int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.cs/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int[] nums, int target) {\n int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.go/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.swift/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n i, j := 0, len(nums)-1\n for i <= j {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n m := i + (j-i)/2\n if nums[m] < target {\n // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1\n } else if nums[m] > target {\n // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n } else {\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n return m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n}\n
binary_search_insertion.js/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -> Int {\n var i = 0, j = nums.count - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j {\n let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target {\n i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if nums[m] > target {\n j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n}\n
binary_search_insertion.ts/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(nums, target) {\n let i = 0,\n j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.dart/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(\n nums: Array<number>,\n target: number\n): number {\n let i = 0,\n j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.rs/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(List<int> nums, int target) {\n int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.c/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &[i32], target: i32) -> i32 {\n let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j {\n let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m as usize] < target {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if nums[m as usize] > target {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n i\n}\n
binary_search_insertion.zig/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
"},{"location":"chapter_searching/binary_search_insertion/#1022","title":"10.2.2 \u00a0 \u5b58\u5728\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"[class]{}-[func]{binarySearchInsertionSimple}\nQuestion
\u5728\u4e0a\u4e00\u9898\u7684\u57fa\u7840\u4e0a\uff0c\u89c4\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u5176\u4f59\u4e0d\u53d8\u3002
\u5047\u8bbe\u6570\u7ec4\u4e2d\u5b58\u5728\u591a\u4e2a
target\uff0c\u5219\u666e\u901a\u4e8c\u5206\u67e5\u627e\u53ea\u80fd\u8fd4\u56de\u5176\u4e2d\u4e00\u4e2atarget\u7684\u7d22\u5f15\uff0c\u800c\u65e0\u6cd5\u786e\u5b9a\u8be5\u5143\u7d20\u7684\u5de6\u8fb9\u548c\u53f3\u8fb9\u8fd8\u6709\u591a\u5c11target\u3002\u9898\u76ee\u8981\u6c42\u5c06\u76ee\u6807\u5143\u7d20\u63d2\u5165\u5230\u6700\u5de6\u8fb9\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a
target\u7684\u7d22\u5f15\u3002\u521d\u6b65\u8003\u8651\u901a\u8fc7\u56fe 10-5 \u6240\u793a\u7684\u6b65\u9aa4\u5b9e\u73b0\u3002- \u6267\u884c\u4e8c\u5206\u67e5\u627e\uff0c\u5f97\u5230\u4efb\u610f\u4e00\u4e2a
target\u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\(k\\) \u3002 - \u4ece\u7d22\u5f15 \\(k\\) \u5f00\u59cb\uff0c\u5411\u5de6\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u5f53\u627e\u5230\u6700\u5de6\u8fb9\u7684
target\u65f6\u8fd4\u56de\u3002
\u56fe 10-5 \u00a0 \u7ebf\u6027\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9
\u6b64\u65b9\u6cd5\u867d\u7136\u53ef\u7528\uff0c\u4f46\u5176\u5305\u542b\u7ebf\u6027\u67e5\u627e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5f53\u6570\u7ec4\u4e2d\u5b58\u5728\u5f88\u591a\u91cd\u590d\u7684
target\u65f6\uff0c\u8be5\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\u3002\u73b0\u8003\u8651\u62d3\u5c55\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\u3002\u5982\u56fe 10-6 \u6240\u793a\uff0c\u6574\u4f53\u6d41\u7a0b\u4fdd\u6301\u4e0d\u53d8\uff0c\u6bcf\u8f6e\u5148\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65ad
target\u548cnums[m]\u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u51e0\u79cd\u60c5\u51b5\u3002- \u5f53
nums[m] < target\u6216nums[m] > target\u65f6\uff0c\u8bf4\u660e\u8fd8\u6ca1\u6709\u627e\u5230target\uff0c\u56e0\u6b64\u91c7\u7528\u666e\u901a\u4e8c\u5206\u67e5\u627e\u7684\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5411target\u9760\u8fd1\u3002 - \u5f53
nums[m] == target\u65f6\uff0c\u8bf4\u660e\u5c0f\u4e8etarget\u7684\u5143\u7d20\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u91c7\u7528 \\(j = m - 1\\) \u6765\u7f29\u5c0f\u533a\u95f4\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(j\\) \u5411\u5c0f\u4e8etarget\u7684\u5143\u7d20\u9760\u8fd1\u3002
\u5faa\u73af\u5b8c\u6210\u540e\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u8fb9\u7684
<1><2><3><4><5><6><7><8>target\uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8etarget\u7684\u5143\u7d20\uff0c\u56e0\u6b64\u7d22\u5f15 \\(i\\) \u5c31\u662f\u63d2\u5165\u70b9\u3002\u56fe 10-6 \u00a0 \u4e8c\u5206\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9\u7684\u6b65\u9aa4
\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5224\u65ad\u5206\u652f
nums[m] > target\u548cnums[m] == target\u7684\u64cd\u4f5c\u76f8\u540c\uff0c\u56e0\u6b64\u4e24\u8005\u53ef\u4ee5\u5408\u5e76\u3002\u5373\u4fbf\u5982\u6b64\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5c06\u5224\u65ad\u6761\u4ef6\u4fdd\u6301\u5c55\u5f00\uff0c\u56e0\u4e3a\u5176\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3001\u53ef\u8bfb\u6027\u66f4\u597d\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_insertion.py
binary_search_insertion.cppdef binary_search_insertion(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n i, j = 0, len(nums) - 1 # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j:\n m = (i + j) // 2 # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target:\n i = m + 1 # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n elif nums[m] > target:\n j = m - 1 # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n else:\n j = m - 1 # \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n # \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n
binary_search_insertion.java/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(vector<int> &nums, int target) {\n int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.cs/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int[] nums, int target) {\n int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.go/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertion(int[] nums, int target) {\n int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.swift/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums []int, target int) int {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n i, j := 0, len(nums)-1\n for i <= j {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n m := i + (j-i)/2\n if nums[m] < target {\n // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1\n } else if nums[m] > target {\n // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n } else {\n // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n}\n
binary_search_insertion.js/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums: [Int], target: Int) -> Int {\n var i = 0, j = nums.count - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j {\n let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target {\n i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if nums[m] > target {\n j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n}\n
binary_search_insertion.ts/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums, target) {\n let i = 0,\n j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.dart/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums: Array<number>, target: number): number {\n let i = 0,\n j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.rs/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(List<int> nums, int target) {\n int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.c/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &[i32], target: i32) -> i32 {\n let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j {\n let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m as usize] < target {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if nums[m as usize] > target {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n i\n}\n
binary_search_insertion.zig/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n[class]{}-[func]{binarySearchInsertion}\nTip
\u672c\u8282\u7684\u4ee3\u7801\u90fd\u662f\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5199\u6cd5\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5199\u6cd5\u3002
\u603b\u7684\u6765\u770b\uff0c\u4e8c\u5206\u67e5\u627e\u65e0\u975e\u5c31\u662f\u7ed9\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u522b\u8bbe\u5b9a\u641c\u7d22\u76ee\u6807\uff0c\u76ee\u6807\u53ef\u80fd\u662f\u4e00\u4e2a\u5177\u4f53\u7684\u5143\u7d20\uff08\u4f8b\u5982
target\uff09\uff0c\u4e5f\u53ef\u80fd\u662f\u4e00\u4e2a\u5143\u7d20\u8303\u56f4\uff08\u4f8b\u5982\u5c0f\u4e8etarget\u7684\u5143\u7d20\uff09\u3002\u5728\u4e0d\u65ad\u7684\u5faa\u73af\u4e8c\u5206\u4e2d\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u90fd\u9010\u6e10\u903c\u8fd1\u9884\u5148\u8bbe\u5b9a\u7684\u76ee\u6807\u3002\u6700\u7ec8\uff0c\u5b83\u4eec\u6216\u662f\u6210\u529f\u627e\u5230\u7b54\u6848\uff0c\u6216\u662f\u8d8a\u8fc7\u8fb9\u754c\u540e\u505c\u6b62\u3002
"},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565","text":"\u5728\u7b97\u6cd5\u9898\u4e2d\uff0c\u6211\u4eec\u5e38\u901a\u8fc7\u5c06\u7ebf\u6027\u67e5\u627e\u66ff\u6362\u4e3a\u54c8\u5e0c\u67e5\u627e\u6765\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u7b97\u6cd5\u9898\u6765\u52a0\u6df1\u7406\u89e3\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4
"},{"location":"chapter_searching/replace_linear_by_hashing/#1041","title":"10.4.1 \u00a0 \u7ebf\u6027\u67e5\u627e\uff1a\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4","text":"nums\u548c\u4e00\u4e2a\u76ee\u6807\u5143\u7d20target\uff0c\u8bf7\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u201c\u548c\u201d\u4e3atarget\u7684\u4e24\u4e2a\u5143\u7d20\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u4e2a\u89e3\u5373\u53ef\u3002\u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\u3002\u5982\u56fe 10-9 \u6240\u793a\uff0c\u6211\u4eec\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a
target\uff0c\u82e5\u662f\uff0c\u5219\u8fd4\u56de\u5b83\u4eec\u7684\u7d22\u5f15\u3002\u56fe 10-9 \u00a0 \u7ebf\u6027\u67e5\u627e\u6c42\u89e3\u4e24\u6570\u4e4b\u548c
\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig two_sum.py
two_sum.cppdef two_sum_brute_force(nums: list[int], target: int) -> list[int]:\n \"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\"\"\"\n # \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for i in range(len(nums) - 1):\n for j in range(i + 1, len(nums)):\n if nums[i] + nums[j] == target:\n return [i, j]\n return []\n
two_sum.java/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector<int> twoSumBruteForce(vector<int> &nums, int target) {\n int size = nums.size();\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (int i = 0; i < size - 1; i++) {\n for (int j = i + 1; j < size; j++) {\n if (nums[i] + nums[j] == target)\n return {i, j};\n }\n }\n return {};\n}\n
two_sum.cs/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\n int size = nums.length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (int i = 0; i < size - 1; i++) {\n for (int j = i + 1; j < size; j++) {\n if (nums[i] + nums[j] == target)\n return new int[] { i, j };\n }\n }\n return new int[0];\n}\n
two_sum.go/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] TwoSumBruteForce(int[] nums, int target) {\n int size = nums.Length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (int i = 0; i < size - 1; i++) {\n for (int j = i + 1; j < size; j++) {\n if (nums[i] + nums[j] == target)\n return [i, j];\n }\n }\n return [];\n}\n
two_sum.swift/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\n size := len(nums)\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for i := 0; i < size-1; i++ {\n for j := i + 1; i < size; j++ {\n if nums[i]+nums[j] == target {\n return []int{i, j}\n }\n }\n }\n return nil\n}\n
two_sum.js/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -> [Int] {\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for i in nums.indices.dropLast() {\n for j in nums.indices.dropFirst(i + 1) {\n if nums[i] + nums[j] == target {\n return [i, j]\n }\n }\n }\n return [0]\n}\n
two_sum.ts/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\n const n = nums.length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (let i = 0; i < n; i++) {\n for (let j = i + 1; j < n; j++) {\n if (nums[i] + nums[j] === target) {\n return [i, j];\n }\n }\n }\n return [];\n}\n
two_sum.dart/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n const n = nums.length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (let i = 0; i < n; i++) {\n for (let j = i + 1; j < n; j++) {\n if (nums[i] + nums[j] === target) {\n return [i, j];\n }\n }\n }\n return [];\n}\n
two_sum.rs/* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u679a\u4e3e */\nList<int> twoSumBruteForce(List<int> nums, int target) {\n int size = nums.length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (var i = 0; i < size - 1; i++) {\n for (var j = i + 1; j < size; j++) {\n if (nums[i] + nums[j] == target) return [i, j];\n }\n }\n return [0];\n}\n
two_sum.c/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\npub fn two_sum_brute_force(nums: &Vec<i32>, target: i32) -> Option<Vec<i32>> {\n let size = nums.len();\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for i in 0..size - 1 {\n for j in i + 1..size {\n if nums[i] + nums[j] == target {\n return Some(vec![i as i32, j as i32]);\n }\n }\n }\n None\n}\n
two_sum.zig/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n for (int i = 0; i < numsSize; ++i) {\n for (int j = i + 1; j < numsSize; ++j) {\n if (nums[i] + nums[j] == target) {\n int *res = malloc(sizeof(int) * 2);\n res[0] = i, res[1] = j;\n *returnSize = 2;\n return res;\n }\n }\n }\n *returnSize = 0;\n return NULL;\n}\n// \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\n var size: usize = nums.len;\n var i: usize = 0;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n while (i < size - 1) : (i += 1) {\n var j = i + 1;\n while (j < size) : (j += 1) {\n if (nums[i] + nums[j] == target) {\n return [_]i32{@intCast(i), @intCast(j)};\n }\n }\n }\n return null;\n}\n\u6b64\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002
"},{"location":"chapter_searching/replace_linear_by_hashing/#1042","title":"10.4.2 \u00a0 \u54c8\u5e0c\u67e5\u627e\uff1a\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4","text":"\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u952e\u503c\u5bf9\u5206\u522b\u4e3a\u6570\u7ec4\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u6267\u884c\u56fe 10-10 \u6240\u793a\u7684\u6b65\u9aa4\u3002
- \u5224\u65ad\u6570\u5b57
target - nums[i]\u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u3002 - \u5c06\u952e\u503c\u5bf9
nums[i]\u548c\u7d22\u5f15i\u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\u3002
\u56fe 10-10 \u00a0 \u8f85\u52a9\u54c8\u5e0c\u8868\u6c42\u89e3\u4e24\u6570\u4e4b\u548c
\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff0c\u4ec5\u9700\u5355\u5c42\u5faa\u73af\u5373\u53ef\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig two_sum.py
two_sum.cppdef two_sum_hash_table(nums: list[int], target: int) -> list[int]:\n \"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\"\"\"\n # \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n dic = {}\n # \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for i in range(len(nums)):\n if target - nums[i] in dic:\n return [dic[target - nums[i]], i]\n dic[nums[i]] = i\n return []\n
two_sum.java/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector<int> twoSumHashTable(vector<int> &nums, int target) {\n int size = nums.size();\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n unordered_map<int, int> dic;\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (int i = 0; i < size; i++) {\n if (dic.find(target - nums[i]) != dic.end()) {\n return {dic[target - nums[i]], i};\n }\n dic.emplace(nums[i], i);\n }\n return {};\n}\n
two_sum.cs/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\n int size = nums.length;\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n Map<Integer, Integer> dic = new HashMap<>();\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (int i = 0; i < size; i++) {\n if (dic.containsKey(target - nums[i])) {\n return new int[] { dic.get(target - nums[i]), i };\n }\n dic.put(nums[i], i);\n }\n return new int[0];\n}\n
two_sum.go/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] TwoSumHashTable(int[] nums, int target) {\n int size = nums.Length;\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n Dictionary<int, int> dic = [];\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (int i = 0; i < size; i++) {\n if (dic.ContainsKey(target - nums[i])) {\n return [dic[target - nums[i]], i];\n }\n dic.Add(nums[i], i);\n }\n return [];\n}\n
two_sum.swift/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n hashTable := map[int]int{}\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for idx, val := range nums {\n if preIdx, ok := hashTable[target-val]; ok {\n return []int{preIdx, idx}\n }\n hashTable[val] = idx\n }\n return nil\n}\n
two_sum.js/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -> [Int] {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n var dic: [Int: Int] = [:]\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for i in nums.indices {\n if let j = dic[target - nums[i]] {\n return [j, i]\n }\n dic[nums[i]] = i\n }\n return [0]\n}\n
two_sum.ts/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n let m = {};\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (let i = 0; i < nums.length; i++) {\n if (m[target - nums[i]] !== undefined) {\n return [m[target - nums[i]], i];\n } else {\n m[nums[i]] = i;\n }\n }\n return [];\n}\n
two_sum.dart/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n let m: Map<number, number> = new Map();\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (let i = 0; i < nums.length; i++) {\n let index = m.get(target - nums[i]);\n if (index !== undefined) {\n return [index, i];\n } else {\n m.set(nums[i], i);\n }\n }\n return [];\n}\n
two_sum.rs/* \u65b9\u6cd5\u4e8c\uff1a \u8f85\u52a9\u54c8\u5e0c\u8868 */\nList<int> twoSumHashTable(List<int> nums, int target) {\n int size = nums.length;\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n Map<int, int> dic = HashMap();\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (var i = 0; i < size; i++) {\n if (dic.containsKey(target - nums[i])) {\n return [dic[target - nums[i]]!, i];\n }\n dic.putIfAbsent(nums[i], () => i);\n }\n return [0];\n}\n
two_sum.c/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\npub fn two_sum_hash_table(nums: &Vec<i32>, target: i32) -> Option<Vec<i32>> {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n let mut dic = HashMap::new();\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (i, num) in nums.iter().enumerate() {\n match dic.get(&(target - num)) {\n Some(v) => return Some(vec![*v as i32, i as i32]),\n None => dic.insert(num, i as i32)\n };\n }\n None\n}\n
two_sum.zig/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n int key;\n int val;\n UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u54c8\u5e0c\u8868\u67e5\u8be2 */\nHashTable *find(HashTable *h, int key) {\n HashTable *tmp;\n HASH_FIND_INT(h, &key, tmp);\n return tmp;\n}\n\n/* \u54c8\u5e0c\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(HashTable *h, int key, int val) {\n HashTable *t = find(h, key);\n if (t == NULL) {\n HashTable *tmp = malloc(sizeof(HashTable));\n tmp->key = key, tmp->val = val;\n HASH_ADD_INT(h, key, tmp);\n } else {\n t->val = val;\n }\n}\n\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n HashTable *hashtable = NULL;\n for (int i = 0; i < numsSize; i++) {\n HashTable *t = find(hashtable, target - nums[i]);\n if (t != NULL) {\n int *res = malloc(sizeof(int) * 2);\n res[0] = t->val, res[1] = i;\n *returnSize = 2;\n return res;\n }\n insert(hashtable, nums[i], i);\n }\n *returnSize = 0;\n return NULL;\n}\n// \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\n var size: usize = nums.len;\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n var dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\n defer dic.deinit();\n var i: usize = 0;\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n while (i < size) : (i += 1) {\n if (dic.contains(target - nums[i])) {\n return [_]i32{dic.get(target - nums[i]).?, @intCast(i)};\n }\n try dic.put(nums[i], @intCast(i));\n }\n return null;\n}\n\u6b64\u65b9\u6cd5\u901a\u8fc7\u54c8\u5e0c\u67e5\u627e\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u8fd0\u884c\u6548\u7387\u3002
\u7531\u4e8e\u9700\u8981\u7ef4\u62a4\u4e00\u4e2a\u989d\u5916\u7684\u54c8\u5e0c\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u8be5\u65b9\u6cd5\u7684\u6574\u4f53\u65f6\u7a7a\u6548\u7387\u66f4\u4e3a\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u9898\u7684\u6700\u4f18\u89e3\u6cd5\u3002
"},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5","text":"\u300c\u641c\u7d22\u7b97\u6cd5 searching algorithm\u300d\u7528\u4e8e\u5728\u6570\u636e\u7ed3\u6784\uff08\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u6216\u56fe\uff09\u4e2d\u641c\u7d22\u4e00\u4e2a\u6216\u4e00\u7ec4\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5143\u7d20\u3002
\u641c\u7d22\u7b97\u6cd5\u53ef\u6839\u636e\u5b9e\u73b0\u601d\u8def\u5206\u4e3a\u4ee5\u4e0b\u4e24\u7c7b\u3002
- \u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\uff0c\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u904d\u5386\u7b49\u3002
- \u5229\u7528\u6570\u636e\u7ec4\u7ec7\u7ed3\u6784\u6216\u6570\u636e\u5305\u542b\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u5b9e\u73b0\u9ad8\u6548\u5143\u7d20\u67e5\u627e\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u548c\u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u7b49\u3002
\u4e0d\u96be\u53d1\u73b0\uff0c\u8fd9\u4e9b\u77e5\u8bc6\u70b9\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u56e0\u6b64\u641c\u7d22\u7b97\u6cd5\u5bf9\u4e8e\u6211\u4eec\u6765\u8bf4\u5e76\u4e0d\u964c\u751f\u3002\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4ece\u66f4\u52a0\u7cfb\u7edf\u7684\u89c6\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5ba1\u89c6\u641c\u7d22\u7b97\u6cd5\u3002
"},{"location":"chapter_searching/searching_algorithm_revisited/#1051","title":"10.5.1 \u00a0 \u66b4\u529b\u641c\u7d22","text":"\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u7684\u6bcf\u4e2a\u5143\u7d20\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002
- \u201c\u7ebf\u6027\u641c\u7d22\u201d\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\u7b49\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u9010\u4e2a\u8bbf\u95ee\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u5230\u8fbe\u53e6\u4e00\u7aef\u4ecd\u6ca1\u6709\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002
- \u201c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u548c\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u662f\u56fe\u548c\u6811\u7684\u4e24\u79cd\u904d\u5386\u7b56\u7565\u3002\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\u9010\u5c42\u641c\u7d22\uff0c\u7531\u8fd1\u53ca\u8fdc\u5730\u8bbf\u95ee\u5404\u4e2a\u8282\u70b9\u3002\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\uff0c\u6cbf\u7740\u4e00\u6761\u8def\u5f84\u8d70\u5230\u5934\uff0c\u518d\u56de\u6eaf\u5e76\u5c1d\u8bd5\u5176\u4ed6\u8def\u5f84\uff0c\u76f4\u5230\u904d\u5386\u5b8c\u6574\u4e2a\u6570\u636e\u7ed3\u6784\u3002
\u66b4\u529b\u641c\u7d22\u7684\u4f18\u70b9\u662f\u7b80\u5355\u4e14\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u505a\u9884\u5904\u7406\u548c\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\u3002
\u7136\u800c\uff0c\u6b64\u7c7b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u5143\u7d20\u6570\u91cf\uff0c\u56e0\u6b64\u5728\u6570\u636e\u91cf\u8f83\u5927\u7684\u60c5\u51b5\u4e0b\u6027\u80fd\u8f83\u5dee\u3002
"},{"location":"chapter_searching/searching_algorithm_revisited/#1052","title":"10.5.2 \u00a0 \u81ea\u9002\u5e94\u641c\u7d22","text":"\u81ea\u9002\u5e94\u641c\u7d22\u5229\u7528\u6570\u636e\u7684\u7279\u6709\u5c5e\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u6765\u4f18\u5316\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ece\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002
- \u201c\u4e8c\u5206\u67e5\u627e\u201d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\u5b9e\u73b0\u9ad8\u6548\u67e5\u627e\uff0c\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002
- \u201c\u54c8\u5e0c\u67e5\u627e\u201d\u5229\u7528\u54c8\u5e0c\u8868\u5c06\u641c\u7d22\u6570\u636e\u548c\u76ee\u6807\u6570\u636e\u5efa\u7acb\u4e3a\u952e\u503c\u5bf9\u6620\u5c04\uff0c\u4ece\u800c\u5b9e\u73b0\u67e5\u8be2\u64cd\u4f5c\u3002
- \u201c\u6811\u67e5\u627e\u201d\u5728\u7279\u5b9a\u7684\u6811\u7ed3\u6784\uff08\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\uff09\u4e2d\uff0c\u57fa\u4e8e\u6bd4\u8f83\u8282\u70b9\u503c\u6765\u5feb\u901f\u6392\u9664\u8282\u70b9\uff0c\u4ece\u800c\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002
\u6b64\u7c7b\u7b97\u6cd5\u7684\u4f18\u70b9\u662f\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002
\u7136\u800c\uff0c\u4f7f\u7528\u8fd9\u4e9b\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u67e5\u627e\u9700\u8981\u9884\u5148\u5bf9\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\uff0c\u54c8\u5e0c\u67e5\u627e\u548c\u6811\u67e5\u627e\u90fd\u9700\u8981\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7ef4\u62a4\u8fd9\u4e9b\u6570\u636e\u7ed3\u6784\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u5f00\u9500\u3002
Tip
\u81ea\u9002\u5e94\u641c\u7d22\u7b97\u6cd5\u5e38\u88ab\u79f0\u4e3a\u67e5\u627e\u7b97\u6cd5\uff0c\u4e3b\u8981\u7528\u4e8e\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u68c0\u7d22\u76ee\u6807\u5143\u7d20\u3002
"},{"location":"chapter_searching/searching_algorithm_revisited/#1053","title":"10.5.3 \u00a0 \u641c\u7d22\u65b9\u6cd5\u9009\u53d6","text":"\u7ed9\u5b9a\u5927\u5c0f\u4e3a \\(n\\) \u7684\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u7ebf\u6027\u641c\u7d22\u3001\u4e8c\u5206\u67e5\u627e\u3001\u6811\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u7b49\u591a\u79cd\u65b9\u6cd5\u4ece\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\u3002\u5404\u4e2a\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u56fe 10-11 \u6240\u793a\u3002
\u56fe 10-11 \u00a0 \u591a\u79cd\u641c\u7d22\u7b56\u7565
\u4e0a\u8ff0\u51e0\u79cd\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u4e0e\u7279\u6027\u5982\u8868 10-1 \u6240\u793a\u3002
\u8868 10-1 \u00a0 \u67e5\u627e\u7b97\u6cd5\u6548\u7387\u5bf9\u6bd4
\u7ebf\u6027\u641c\u7d22 \u4e8c\u5206\u67e5\u627e \u6811\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u989d\u5916\u7a7a\u95f4 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u6570\u636e\u9884\u5904\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6811 \\(O(n \\log n)\\) \u5efa\u54c8\u5e0c\u8868 \\(O(n)\\) \u6570\u636e\u662f\u5426\u6709\u5e8f \u65e0\u5e8f \u6709\u5e8f \u6709\u5e8f \u65e0\u5e8f\u641c\u7d22\u7b97\u6cd5\u7684\u9009\u62e9\u8fd8\u53d6\u51b3\u4e8e\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u4e0e\u66f4\u65b0\u9891\u7387\u7b49\u3002
\u7ebf\u6027\u641c\u7d22
- \u901a\u7528\u6027\u8f83\u597d\uff0c\u65e0\u987b\u4efb\u4f55\u6570\u636e\u9884\u5904\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u4eec\u4ec5\u9700\u67e5\u8be2\u4e00\u6b21\u6570\u636e\uff0c\u90a3\u4e48\u5176\u4ed6\u4e09\u79cd\u65b9\u6cd5\u7684\u6570\u636e\u9884\u5904\u7406\u7684\u65f6\u95f4\u6bd4\u7ebf\u6027\u641c\u7d22\u7684\u65f6\u95f4\u8fd8\u8981\u66f4\u957f\u3002
- \u9002\u7528\u4e8e\u4f53\u91cf\u8f83\u5c0f\u7684\u6570\u636e\uff0c\u6b64\u60c5\u51b5\u4e0b\u65f6\u95f4\u590d\u6742\u5ea6\u5bf9\u6548\u7387\u5f71\u54cd\u8f83\u5c0f\u3002
- \u9002\u7528\u4e8e\u6570\u636e\u66f4\u65b0\u9891\u7387\u8f83\u9ad8\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u8be5\u65b9\u6cd5\u4e0d\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u4efb\u4f55\u989d\u5916\u7ef4\u62a4\u3002
\u4e8c\u5206\u67e5\u627e
- \u9002\u7528\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u6548\u7387\u8868\u73b0\u7a33\u5b9a\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
- \u6570\u636e\u91cf\u4e0d\u80fd\u8fc7\u5927\uff0c\u56e0\u4e3a\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u3002
- \u4e0d\u9002\u7528\u4e8e\u9ad8\u9891\u589e\u5220\u6570\u636e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u7ef4\u62a4\u6709\u5e8f\u6570\u7ec4\u7684\u5f00\u9500\u8f83\u5927\u3002
\u54c8\u5e0c\u67e5\u627e
- \u9002\u5408\u5bf9\u67e5\u8be2\u6027\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u573a\u666f\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
- \u4e0d\u9002\u5408\u9700\u8981\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u65e0\u6cd5\u7ef4\u62a4\u6570\u636e\u7684\u6709\u5e8f\u6027\u3002
- \u5bf9\u54c8\u5e0c\u51fd\u6570\u548c\u54c8\u5e0c\u51b2\u7a81\u5904\u7406\u7b56\u7565\u7684\u4f9d\u8d56\u6027\u8f83\u9ad8\uff0c\u5177\u6709\u8f83\u5927\u7684\u6027\u80fd\u52a3\u5316\u98ce\u9669\u3002
- \u4e0d\u9002\u5408\u6570\u636e\u91cf\u8fc7\u5927\u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u9700\u8981\u989d\u5916\u7a7a\u95f4\u6765\u6700\u5927\u7a0b\u5ea6\u5730\u51cf\u5c11\u51b2\u7a81\uff0c\u4ece\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8be2\u6027\u80fd\u3002
\u6811\u67e5\u627e
- \u9002\u7528\u4e8e\u6d77\u91cf\u6570\u636e\uff0c\u56e0\u4e3a\u6811\u8282\u70b9\u5728\u5185\u5b58\u4e2d\u662f\u5206\u6563\u5b58\u50a8\u7684\u3002
- \u9002\u5408\u9700\u8981\u7ef4\u62a4\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\u3002
- \u5728\u6301\u7eed\u589e\u5220\u8282\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4ea7\u751f\u503e\u659c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
- \u82e5\u4f7f\u7528 AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u5219\u5404\u9879\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a33\u5b9a\u8fd0\u884c\uff0c\u4f46\u7ef4\u62a4\u6811\u5e73\u8861\u7684\u64cd\u4f5c\u4f1a\u589e\u52a0\u989d\u5916\u5f00\u9500\u3002
- \u4e8c\u5206\u67e5\u627e\u4f9d\u8d56\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u9010\u6b65\u7f29\u51cf\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u8fdb\u884c\u67e5\u627e\u3002\u5b83\u8981\u6c42\u8f93\u5165\u6570\u636e\u6709\u5e8f\uff0c\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
- \u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u6570\u636e\u3002\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\uff0c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u548c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u9002\u7528\u4e8e\u56fe\u548c\u6811\u3002\u6b64\u7c7b\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\uff0c\u4f46\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u8f83\u9ad8\u3002
- \u54c8\u5e0c\u67e5\u627e\u3001\u6811\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u5c5e\u4e8e\u9ad8\u6548\u641c\u7d22\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002\u6b64\u7c7b\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u3002
- \u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u548c\u66f4\u65b0\u9891\u7387\u7b49\u56e0\u7d20\u8fdb\u884c\u5177\u4f53\u5206\u6790\uff0c\u4ece\u800c\u9009\u62e9\u5408\u9002\u7684\u641c\u7d22\u65b9\u6cd5\u3002
- \u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u5c0f\u578b\u6216\u9891\u7e41\u66f4\u65b0\u7684\u6570\u636e\uff1b\u4e8c\u5206\u67e5\u627e\u9002\u7528\u4e8e\u5927\u578b\u3001\u6392\u5e8f\u7684\u6570\u636e\uff1b\u54c8\u5e0c\u67e5\u627e\u9002\u7528\u4e8e\u5bf9\u67e5\u8be2\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u4e14\u65e0\u987b\u8303\u56f4\u67e5\u8be2\u7684\u6570\u636e\uff1b\u6811\u67e5\u627e\u9002\u7528\u4e8e\u9700\u8981\u7ef4\u62a4\u987a\u5e8f\u548c\u652f\u6301\u8303\u56f4\u67e5\u8be2\u7684\u5927\u578b\u52a8\u6001\u6570\u636e\u3002
- \u7528\u54c8\u5e0c\u67e5\u627e\u66ff\u6362\u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u8fd0\u884c\u65f6\u95f4\u7684\u7b56\u7565\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002
Abstract
\u6392\u5e8f\u72b9\u5982\u4e00\u628a\u5c06\u6df7\u4e71\u53d8\u4e3a\u79e9\u5e8f\u7684\u9b54\u6cd5\u94a5\u5319\uff0c\u4f7f\u6211\u4eec\u80fd\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u7406\u89e3\u4e0e\u5904\u7406\u6570\u636e\u3002
\u65e0\u8bba\u662f\u7b80\u5355\u7684\u5347\u5e8f\uff0c\u8fd8\u662f\u590d\u6742\u7684\u5206\u7c7b\u6392\u5217\uff0c\u6392\u5e8f\u90fd\u5411\u6211\u4eec\u5c55\u793a\u4e86\u6570\u636e\u7684\u548c\u8c10\u7f8e\u611f\u3002
"},{"location":"chapter_sorting/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5
- 11.2 \u00a0 \u9009\u62e9\u6392\u5e8f
- 11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f
- 11.4 \u00a0 \u63d2\u5165\u6392\u5e8f
- 11.5 \u00a0 \u5feb\u901f\u6392\u5e8f
- 11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f
- 11.7 \u00a0 \u5806\u6392\u5e8f
- 11.8 \u00a0 \u6876\u6392\u5e8f
- 11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f
- 11.10 \u00a0 \u57fa\u6570\u6392\u5e8f
- 11.11 \u00a0 \u5c0f\u7ed3
\u300c\u5192\u6ce1\u6392\u5e8f bubble sort\u300d\u901a\u8fc7\u8fde\u7eed\u5730\u6bd4\u8f83\u4e0e\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u5b9e\u73b0\u6392\u5e8f\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u50cf\u6c14\u6ce1\u4ece\u5e95\u90e8\u5347\u5230\u9876\u90e8\u4e00\u6837\uff0c\u56e0\u6b64\u5f97\u540d\u5192\u6ce1\u6392\u5e8f\u3002
\u5982\u56fe 11-4 \u6240\u793a\uff0c\u5192\u6ce1\u8fc7\u7a0b\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6765\u6a21\u62df\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u6bd4\u8f83\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 > \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u6362\u4e8c\u8005\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6700\u5927\u7684\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u5230\u6570\u7ec4\u7684\u6700\u53f3\u7aef\u3002
<1><2><3><4><5><6><7>\u56fe 11-4 \u00a0 \u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6a21\u62df\u5192\u6ce1
"},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6b65\u9aa4\u5982\u56fe 11-5 \u6240\u793a\u3002
- \u9996\u5148\uff0c\u5bf9 \\(n\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\uff0c
- \u63a5\u4e0b\u6765\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002
- \u4ee5\u6b64\u7c7b\u63a8\uff0c\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u201c\u5192\u6ce1\u201d\u540e\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002
- \u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002
\u56fe 11-5 \u00a0 \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b
\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig bubble_sort.py
bubble_sort.cppdef bubble_sort(nums: list[int]):\n \"\"\"\u5192\u6ce1\u6392\u5e8f\"\"\"\n n = len(nums)\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in range(n - 1, 0, -1):\n # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j in range(i):\n if nums[j] > nums[j + 1]:\n # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n nums[j], nums[j + 1] = nums[j + 1], nums[j]\n
bubble_sort.java/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector<int> &nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.size() - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n swap(nums[j], nums[j + 1]);\n }\n }\n }\n}\n
bubble_sort.cs/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.go/* \u5192\u6ce1\u6392\u5e8f */\nvoid BubbleSort(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.Length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n }\n }\n }\n}\n
bubble_sort.swift/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := len(nums) - 1; i > 0; i-- {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j := 0; j < i; j++ {\n if nums[j] > nums[j+1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n nums[j], nums[j+1] = nums[j+1], nums[j]\n }\n }\n }\n}\n
bubble_sort.js/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in stride(from: 0, to: i, by: 1) {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n }\n }\n }\n}\n
bubble_sort.ts/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.dart/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.rs/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(List<int> nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.c/* \u5192\u6ce1\u6392\u5e8f */\nfn bubble_sort(nums: &mut [i32]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in (1..nums.len()).rev() {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0..i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.zig/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = size - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n int temp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = temp;\n }\n }\n }\n}\n
"},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2 \u00a0 \u6548\u7387\u4f18\u5316","text":"// \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n var i: usize = nums.len - 1;\n while (i > 0) : (i -= 1) {\n var j: usize = 0;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n while (j < i) : (j += 1) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n var tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n\u6211\u4eec\u53d1\u73b0\uff0c\u5982\u679c\u67d0\u8f6e\u201c\u5192\u6ce1\u201d\u4e2d\u6ca1\u6709\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d
flag\u6765\u76d1\u6d4b\u8fd9\u79cd\u60c5\u51b5\uff0c\u4e00\u65e6\u51fa\u73b0\u5c31\u7acb\u5373\u8fd4\u56de\u3002\u7ecf\u8fc7\u4f18\u5316\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u4f46\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u53ef\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig bubble_sort.py
bubble_sort.cppdef bubble_sort_with_flag(nums: list[int]):\n \"\"\"\u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\"\"\"\n n = len(nums)\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in range(n - 1, 0, -1):\n flag = False # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j in range(i):\n if nums[j] > nums[j + 1]:\n # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n nums[j], nums[j + 1] = nums[j + 1], nums[j]\n flag = True # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n if not flag:\n break # \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n
bubble_sort.java/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector<int> &nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.size() - 1; i > 0; i--) {\n bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n swap(nums[j], nums[j + 1]);\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag)\n break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.cs/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n boolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag)\n break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.go/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid BubbleSortWithFlag(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.Length - 1; i > 0; i--) {\n bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.swift/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := len(nums) - 1; i > 0; i-- {\n flag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j := 0; j < i; j++ {\n if nums[j] > nums[j+1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n nums[j], nums[j+1] = nums[j+1], nums[j]\n flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if flag == false { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n break\n }\n }\n}\n
bubble_sort.js/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n var flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n for j in stride(from: 0, to: i, by: 1) {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if !flag { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n break\n }\n }\n}\n
bubble_sort.ts/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.dart/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.rs/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(List<int> nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.c/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nfn bubble_sort_with_flag(nums: &mut [i32]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in (1..nums.len()).rev() {\n let mut flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0..i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if !flag {break}; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.zig/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = size - 1; i > 0; i--) {\n bool flag = false;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n int temp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = temp;\n flag = true;\n }\n }\n if (!flag)\n break;\n }\n}\n
"},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"// \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n var i: usize = nums.len - 1;\n while (i > 0) : (i -= 1) {\n var flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n var j: usize = 0;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n while (j < i) : (j += 1) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n var tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true;\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5404\u8f6e\u201c\u5192\u6ce1\u201d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4f9d\u6b21\u4e3a \\(n - 1\\)\u3001\\(n - 2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u603b\u548c\u4e3a \\((n - 1) n / 2\\) \u3002\u5728\u5f15\u5165
flag\u4f18\u5316\u540e\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(n)\\) \u3002 - \u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u6362\u3002
\u524d\u8ff0\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e\u201c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u51e0\u79cd\u201c\u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u7ebf\u6027\u9636\u3002
\u300c\u6876\u6392\u5e8f bucket sort\u300d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u987a\u5e8f\u7684\u6876\uff0c\u6bcf\u4e2a\u6876\u5bf9\u5e94\u4e00\u4e2a\u6570\u636e\u8303\u56f4\uff0c\u5c06\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff1b\u7136\u540e\uff0c\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u5206\u522b\u6267\u884c\u6392\u5e8f\uff1b\u6700\u7ec8\u6309\u7167\u6876\u7684\u987a\u5e8f\u5c06\u6240\u6709\u6570\u636e\u5408\u5e76\u3002
"},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u8003\u8651\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u5176\u5143\u7d20\u662f\u8303\u56f4 \\([0, 1)\\) \u5185\u7684\u6d6e\u70b9\u6570\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-13 \u6240\u793a\u3002
- \u521d\u59cb\u5316 \\(k\\) \u4e2a\u6876\uff0c\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\u3002
- \u5bf9\u6bcf\u4e2a\u6876\u5206\u522b\u6267\u884c\u6392\u5e8f\uff08\u8fd9\u91cc\u91c7\u7528\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff09\u3002
- \u6309\u7167\u6876\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u5408\u5e76\u7ed3\u679c\u3002
\u56fe 11-13 \u00a0 \u6876\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b
\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig bucket_sort.py
bucket_sort.cppdef bucket_sort(nums: list[float]):\n \"\"\"\u6876\u6392\u5e8f\"\"\"\n # \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n k = len(nums) // 2\n buckets = [[] for _ in range(k)]\n # 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for num in nums:\n # \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n i = int(num * k)\n # \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].append(num)\n # 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for bucket in buckets:\n # \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.sort()\n # 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n i = 0\n for bucket in buckets:\n for num in bucket:\n nums[i] = num\n i += 1\n
bucket_sort.java/* \u6876\u6392\u5e8f */\nvoid bucketSort(vector<float> &nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = nums.size() / 2;\n vector<vector<float>> buckets(k);\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (float num : nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int i = num * k;\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n buckets[i].push_back(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (vector<float> &bucket : buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n sort(bucket.begin(), bucket.end());\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n int i = 0;\n for (vector<float> &bucket : buckets) {\n for (float num : bucket) {\n nums[i++] = num;\n }\n }\n}\n
bucket_sort.cs/* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = nums.length / 2;\n List<List<Float>> buckets = new ArrayList<>();\n for (int i = 0; i < k; i++) {\n buckets.add(new ArrayList<>());\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (float num : nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int i = (int) (num * k);\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets.get(i).add(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (List<Float> bucket : buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n Collections.sort(bucket);\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n int i = 0;\n for (List<Float> bucket : buckets) {\n for (float num : bucket) {\n nums[i++] = num;\n }\n }\n}\n
bucket_sort.go/* \u6876\u6392\u5e8f */\nvoid BucketSort(float[] nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = nums.Length / 2;\n List<List<float>> buckets = [];\n for (int i = 0; i < k; i++) {\n buckets.Add([]);\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n foreach (float num in nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int i = (int)(num * k);\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].Add(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n foreach (List<float> bucket in buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.Sort();\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n int j = 0;\n foreach (List<float> bucket in buckets) {\n foreach (float num in bucket) {\n nums[j++] = num;\n }\n }\n}\n
bucket_sort.swift/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n k := len(nums) / 2\n buckets := make([][]float64, k)\n for i := 0; i < k; i++ {\n buckets[i] = make([]float64, 0)\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for _, num := range nums {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n i := int(num * float64(k))\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i] = append(buckets[i], num)\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for i := 0; i < k; i++ {\n // \u4f7f\u7528\u5185\u7f6e\u5207\u7247\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n sort.Float64s(buckets[i])\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n i := 0\n for _, bucket := range buckets {\n for _, num := range bucket {\n nums[i] = num\n i++\n }\n }\n}\n
bucket_sort.js/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n let k = nums.count / 2\n var buckets = (0 ..< k).map { _ in [Double]() }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for num in nums {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n let i = Int(num * Double(k))\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].append(num)\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for i in buckets.indices {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n buckets[i].sort()\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n var i = nums.startIndex\n for bucket in buckets {\n for num in bucket {\n nums[i] = num\n nums.formIndex(after: &i)\n }\n }\n}\n
bucket_sort.ts/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n const k = nums.length / 2;\n const buckets = [];\n for (let i = 0; i < k; i++) {\n buckets.push([]);\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (const num of nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n const i = Math.floor(num * k);\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].push(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (const bucket of buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.sort((a, b) => a - b);\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n let i = 0;\n for (const bucket of buckets) {\n for (const num of bucket) {\n nums[i++] = num;\n }\n }\n}\n
bucket_sort.dart/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n const k = nums.length / 2;\n const buckets: number[][] = [];\n for (let i = 0; i < k; i++) {\n buckets.push([]);\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (const num of nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n const i = Math.floor(num * k);\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].push(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (const bucket of buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.sort((a, b) => a - b);\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n let i = 0;\n for (const bucket of buckets) {\n for (const num of bucket) {\n nums[i++] = num;\n }\n }\n}\n
bucket_sort.rs/* \u6876\u6392\u5e8f */\nvoid bucketSort(List<double> nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = nums.length ~/ 2;\n List<List<double>> buckets = List.generate(k, (index) => []);\n\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (double _num in nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 _num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int i = (_num * k).toInt();\n // \u5c06 _num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n buckets[i].add(_num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (List<double> bucket in buckets) {\n bucket.sort();\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n int i = 0;\n for (List<double> bucket in buckets) {\n for (double _num in bucket) {\n nums[i++] = _num;\n }\n }\n}\n
bucket_sort.c/* \u6876\u6392\u5e8f */\nfn bucket_sort(nums: &mut [f64]) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n let k = nums.len() / 2;\n let mut buckets = vec![vec![]; k];\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for &mut num in &mut *nums {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n let i = (num * k as f64) as usize;\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].push(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for bucket in &mut buckets {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n let mut i = 0;\n for bucket in &mut buckets {\n for &mut num in bucket {\n nums[i] = num;\n i += 1;\n }\n }\n}\n
bucket_sort.zig/* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int size) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = size / 2;\n float **buckets = calloc(k, sizeof(float *));\n for (int i = 0; i < k; i++) {\n // \u6bcf\u4e2a\u6876\u6700\u591a\u53ef\u4ee5\u5206\u914d k \u4e2a\u5143\u7d20\n buckets[i] = calloc(ARRAY_SIZE, sizeof(float));\n }\n\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (int i = 0; i < size; i++) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int bucket_idx = nums[i] * k;\n int j = 0;\n // \u5982\u679c\u6876\u4e2d\u6709\u6570\u636e\u4e14\u6570\u636e\u5c0f\u4e8e\u5f53\u524d\u503c nums[i], \u8981\u5c06\u5176\u653e\u5230\u5f53\u524d\u6876\u7684\u540e\u9762\uff0c\u76f8\u5f53\u4e8e cpp \u4e2d\u7684 push_back\n while (buckets[bucket_idx][j] > 0 && buckets[bucket_idx][j] < nums[i]) {\n j++;\n }\n float temp = nums[i];\n while (j < ARRAY_SIZE && buckets[bucket_idx][j] > 0) {\n swap(&temp, &buckets[bucket_idx][j]);\n j++;\n }\n buckets[bucket_idx][j] = temp;\n }\n\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (int i = 0; i < k; i++) {\n qsort(buckets[i], ARRAY_SIZE, sizeof(float), compare_float);\n }\n\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n for (int i = 0, j = 0; j < k; j++) {\n for (int l = 0; l < ARRAY_SIZE; l++) {\n if (buckets[j][l] > 0) {\n nums[i++] = buckets[j][l];\n }\n }\n }\n\n // \u91ca\u653e\u4e0a\u8ff0\u5206\u914d\u7684\u5185\u5b58\n for (int i = 0; i < k; i++) {\n free(buckets[i]);\n }\n free(buckets);\n}\n
"},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"[class]{}-[func]{bucketSort}\n\u6876\u6392\u5e8f\u9002\u7528\u4e8e\u5904\u7406\u4f53\u91cf\u5f88\u5927\u7684\u6570\u636e\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u5305\u542b 100 \u4e07\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u7a7a\u95f4\u9650\u5236\uff0c\u7cfb\u7edf\u5185\u5b58\u65e0\u6cd5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u3002\u6b64\u65f6\uff0c\u53ef\u4ee5\u5c06\u6570\u636e\u5206\u6210 1000 \u4e2a\u6876\uff0c\u7136\u540e\u5206\u522b\u5bf9\u6bcf\u4e2a\u6876\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u7ed3\u679c\u5408\u5e76\u3002
- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\) \uff1a\u5047\u8bbe\u5143\u7d20\u5728\u5404\u4e2a\u6876\u5185\u5e73\u5747\u5206\u5e03\uff0c\u90a3\u4e48\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \u3002\u5047\u8bbe\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u5219\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u6bd4\u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002\u5408\u5e76\u7ed3\u679c\u65f6\u9700\u8981\u904d\u5386\u6240\u6709\u6876\u548c\u5143\u7d20\uff0c\u82b1\u8d39 \\(O(n + k)\\) \u65f6\u95f4\u3002
- \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u6570\u636e\u88ab\u5206\u914d\u5230\u4e00\u4e2a\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8be5\u6876\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9700\u8981\u501f\u52a9 \\(k\\) \u4e2a\u6876\u548c\u603b\u5171 \\(n\\) \u4e2a\u5143\u7d20\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u6876\u6392\u5e8f\u662f\u5426\u7a33\u5b9a\u53d6\u51b3\u4e8e\u6392\u5e8f\u6876\u5185\u5143\u7d20\u7684\u7b97\u6cd5\u662f\u5426\u7a33\u5b9a\u3002
\u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7406\u8bba\u4e0a\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \uff0c\u5173\u952e\u5728\u4e8e\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u56e0\u4e3a\u5b9e\u9645\u6570\u636e\u5f80\u5f80\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5c06\u6dd8\u5b9d\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 10 \u4e2a\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u4ef7\u683c\u5206\u5e03\u4e0d\u5747\uff0c\u4f4e\u4e8e 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u4e8e 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c06\u4ef7\u683c\u533a\u95f4\u5e73\u5747\u5212\u5206\u4e3a 10 \u4e2a\uff0c\u5404\u4e2a\u6876\u4e2d\u7684\u5546\u54c1\u6570\u91cf\u5dee\u8ddd\u4f1a\u975e\u5e38\u5927\u3002
\u4e3a\u5b9e\u73b0\u5e73\u5747\u5206\u914d\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u8bbe\u5b9a\u4e00\u6761\u5927\u81f4\u7684\u5206\u754c\u7ebf\uff0c\u5c06\u6570\u636e\u7c97\u7565\u5730\u5206\u5230 3 \u4e2a\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u6bd5\u540e\uff0c\u518d\u5c06\u5546\u54c1\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6570\u91cf\u5927\u81f4\u76f8\u7b49\u3002
\u5982\u56fe 11-14 \u6240\u793a\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u521b\u5efa\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u76ee\u6807\u662f\u8ba9\u53f6\u8282\u70b9\u7684\u503c\u5c3d\u53ef\u80fd\u5e73\u5747\u3002\u5f53\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f6e\u5c06\u6570\u636e\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u5177\u4f53\u5212\u5206\u65b9\u5f0f\u53ef\u6839\u636e\u6570\u636e\u7279\u70b9\u7075\u6d3b\u9009\u62e9\u3002
\u56fe 11-14 \u00a0 \u9012\u5f52\u5212\u5206\u6876
\u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u6570\u636e\u6982\u7387\u5206\u5e03\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7edf\u8ba1\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u6570\u636e\u7279\u70b9\u91c7\u7528\u67d0\u79cd\u6982\u7387\u6a21\u578b\u8fdb\u884c\u8fd1\u4f3c\u3002
\u5982\u56fe 11-15 \u6240\u793a\uff0c\u6211\u4eec\u5047\u8bbe\u5546\u54c1\u4ef7\u683c\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8bbe\u5b9a\u4ef7\u683c\u533a\u95f4\uff0c\u4ece\u800c\u5c06\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\u3002
\u56fe 11-15 \u00a0 \u6839\u636e\u6982\u7387\u5206\u5e03\u5212\u5206\u6876
"},{"location":"chapter_sorting/counting_sort/","title":"11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"\u300c\u8ba1\u6570\u6392\u5e8f counting sort\u300d\u901a\u8fc7\u7edf\u8ba1\u5143\u7d20\u6570\u91cf\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u901a\u5e38\u5e94\u7528\u4e8e\u6574\u6570\u6570\u7ec4\u3002
"},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1 \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"\u5148\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4
nums\uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8d1f\u6574\u6570\u201d\uff0c\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-16 \u6240\u793a\u3002- \u904d\u5386\u6570\u7ec4\uff0c\u627e\u51fa\u5176\u4e2d\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u7136\u540e\u521b\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4
counter\u3002 - \u501f\u52a9
counter\u7edf\u8ba1nums\u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2dcounter[num]\u5bf9\u5e94\u6570\u5b57num\u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386nums\uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3anum\uff09\uff0c\u6bcf\u8f6e\u5c06counter[num]\u589e\u52a0 \\(1\\) \u5373\u53ef\u3002 - \u7531\u4e8e
counter\u7684\u5404\u4e2a\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386counter\uff0c\u6839\u636e\u5404\u6570\u5b57\u51fa\u73b0\u6b21\u6570\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165nums\u5373\u53ef\u3002
\u56fe 11-16 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b
\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig counting_sort.py
counting_sort.cppdef counting_sort_naive(nums: list[int]):\n \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n # \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n m = 0\n for num in nums:\n m = max(m, num)\n # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n counter = [0] * (m + 1)\n for num in nums:\n counter[num] += 1\n # 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n i = 0\n for num in range(m + 1):\n for _ in range(counter[num]):\n nums[i] = num\n i += 1\n
counting_sort.java/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector<int> &nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int num : nums) {\n m = max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n vector<int> counter(m + 1, 0);\n for (int num : nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int num = 0; num < m + 1; num++) {\n for (int j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.cs/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int num : nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int[] counter = new int[m + 1];\n for (int num : nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int num = 0; num < m + 1; num++) {\n for (int j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.go/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid CountingSortNaive(int[] nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n foreach (int num in nums) {\n m = Math.Max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int[] counter = new int[m + 1];\n foreach (int num in nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int num = 0; num < m + 1; num++) {\n for (int j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.swift/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n m := 0\n for _, num := range nums {\n if num > m {\n m = num\n }\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n counter := make([]int, m+1)\n for _, num := range nums {\n counter[num]++\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n for i, num := 0, 0; num < m+1; num++ {\n for j := 0; j < counter[num]; j++ {\n nums[i] = num\n i++\n }\n }\n}\n
counting_sort.js/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = nums.max()!\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n var counter = Array(repeating: 0, count: m + 1)\n for num in nums {\n counter[num] += 1\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n var i = 0\n for num in stride(from: 0, to: m + 1, by: 1) {\n for _ in stride(from: 0, to: counter[num], by: 1) {\n nums[i] = num\n i += 1\n }\n }\n}\n
counting_sort.ts/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = 0;\n for (const num of nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n const counter = new Array(m + 1).fill(0);\n for (const num of nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n let i = 0;\n for (let num = 0; num < m + 1; num++) {\n for (let j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.dart/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums: number[]): void {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = 0;\n for (const num of nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n const counter: number[] = new Array<number>(m + 1).fill(0);\n for (const num of nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n let i = 0;\n for (let num = 0; num < m + 1; num++) {\n for (let j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.rs/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(List<int> nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int _num in nums) {\n m = max(m, _num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n List<int> counter = List.filled(m + 1, 0);\n for (int _num in nums) {\n counter[_num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int _num = 0; _num < m + 1; _num++) {\n for (int j = 0; j < counter[_num]; j++, i++) {\n nums[i] = _num;\n }\n }\n}\n
counting_sort.c/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfn counting_sort_naive(nums: &mut [i32]) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = *nums.into_iter().max().unwrap();\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n let mut counter = vec![0; m as usize + 1];\n for &num in &*nums {\n counter[num as usize] += 1;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n let mut i = 0;\n for num in 0..m + 1 {\n for _ in 0..counter[num as usize] {\n nums[i] = num;\n i += 1;\n }\n }\n}\n
counting_sort.zig/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int nums[], int size) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int i = 0; i < size; i++) {\n if (nums[i] > m) {\n m = nums[i];\n }\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int *counter = calloc(m, sizeof(int));\n for (int i = 0; i < size; i++) {\n counter[nums[i]]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int num = 0; num < m + 1; num++) {\n for (int j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n // 4. \u91ca\u653e\u5185\u5b58\n free(counter);\n}\n[class]{}-[func]{countingSortNaive}\n\u8ba1\u6570\u6392\u5e8f\u4e0e\u6876\u6392\u5e8f\u7684\u8054\u7cfb
\u4ece\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u4e2d\u7684\u8ba1\u6570\u6570\u7ec4
"},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2 \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"counter\u7684\u6bcf\u4e2a\u7d22\u5f15\u89c6\u4e3a\u4e00\u4e2a\u6876\uff0c\u5c06\u7edf\u8ba1\u6570\u91cf\u7684\u8fc7\u7a0b\u770b\u4f5c\u5c06\u5404\u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u672c\u8d28\u4e0a\uff0c\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u6570\u636e\u4e0b\u7684\u4e00\u4e2a\u7279\u4f8b\u3002\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4
3.\u5c31\u5931\u6548\u4e86\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002\u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97
\\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\]counter\u7684\u201c\u524d\u7f00\u548c\u201d\u3002\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15i\u5904\u7684\u524d\u7f00\u548cprefix[i]\u7b49\u4e8e\u6570\u7ec4\u524di\u4e2a\u5143\u7d20\u4e4b\u548c\uff1a\u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u7684\u610f\u4e49\uff0c
prefix[num] - 1\u4ee3\u8868\u5143\u7d20num\u5728\u7ed3\u679c\u6570\u7ec4res\u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u975e\u5e38\u5173\u952e\uff0c\u56e0\u4e3a\u5b83\u544a\u8bc9\u6211\u4eec\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4nums\u7684\u6bcf\u4e2a\u5143\u7d20num\uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002- \u5c06
num\u586b\u5165\u6570\u7ec4res\u7684\u7d22\u5f15prefix[num] - 1\u5904\u3002 - \u4ee4\u524d\u7f00\u548c
prefix[num]\u51cf\u5c0f \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6enum\u7684\u7d22\u5f15\u3002
\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6570\u7ec4
<1><2><3><4><5><6><7><8>res\u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528res\u8986\u76d6\u539f\u6570\u7ec4nums\u5373\u53ef\u3002\u56fe 11-17 \u5c55\u793a\u4e86\u5b8c\u6574\u7684\u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b\u3002\u56fe 11-17 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6b65\u9aa4
\u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig counting_sort.py
counting_sort.cppdef counting_sort(nums: list[int]):\n \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n # \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n m = max(nums)\n # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n counter = [0] * (m + 1)\n for num in nums:\n counter[num] += 1\n # 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n # \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for i in range(m):\n counter[i + 1] += counter[i]\n # 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n # \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n n = len(nums)\n res = [0] * n\n for i in range(n - 1, -1, -1):\n num = nums[i]\n res[counter[num] - 1] = num # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num] -= 1 # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n # \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in range(n):\n nums[i] = res[i]\n
counting_sort.java/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector<int> &nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int num : nums) {\n m = max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n vector<int> counter(m + 1, 0);\n for (int num : nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int n = nums.size();\n vector<int> res(n);\n for (int i = n - 1; i >= 0; i--) {\n int num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n nums = res;\n}\n
counting_sort.cs/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int num : nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int[] counter = new int[m + 1];\n for (int num : nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int n = nums.length;\n int[] res = new int[n];\n for (int i = n - 1; i >= 0; i--) {\n int num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n
counting_sort.go/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid CountingSort(int[] nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n foreach (int num in nums) {\n m = Math.Max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int[] counter = new int[m + 1];\n foreach (int num in nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int n = nums.Length;\n int[] res = new int[n];\n for (int i = n - 1; i >= 0; i--) {\n int num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n
counting_sort.swift/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n m := 0\n for _, num := range nums {\n if num > m {\n m = num\n }\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n counter := make([]int, m+1)\n for _, num := range nums {\n counter[num]++\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for i := 0; i < m; i++ {\n counter[i+1] += counter[i]\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n n := len(nums)\n res := make([]int, n)\n for i := n - 1; i >= 0; i-- {\n num := nums[i]\n // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n res[counter[num]-1] = num\n // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n counter[num]--\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n copy(nums, res)\n}\n
counting_sort.js/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = nums.max()!\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n var counter = Array(repeating: 0, count: m + 1)\n for num in nums {\n counter[num] += 1\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for i in stride(from: 0, to: m, by: 1) {\n counter[i + 1] += counter[i]\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n var res = Array(repeating: 0, count: nums.count)\n for i in stride(from: nums.count - 1, through: 0, by: -1) {\n let num = nums[i]\n res[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in stride(from: 0, to: nums.count, by: 1) {\n nums[i] = res[i]\n }\n}\n
counting_sort.ts/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = 0;\n for (const num of nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n const counter = new Array(m + 1).fill(0);\n for (const num of nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (let i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n const n = nums.length;\n const res = new Array(n);\n for (let i = n - 1; i >= 0; i--) {\n const num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for (let i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n
counting_sort.dart/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = 0;\n for (const num of nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n const counter: number[] = new Array<number>(m + 1).fill(0);\n for (const num of nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (let i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n const n = nums.length;\n const res: number[] = new Array<number>(n);\n for (let i = n - 1; i >= 0; i--) {\n const num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for (let i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n
counting_sort.rs/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(List<int> nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int _num in nums) {\n m = max(m, _num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n List<int> counter = List.filled(m + 1, 0);\n for (int _num in nums) {\n counter[_num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[_num]-1 \u662f _num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int n = nums.length;\n List<int> res = List.filled(n, 0);\n for (int i = n - 1; i >= 0; i--) {\n int _num = nums[i];\n res[counter[_num] - 1] = _num; // \u5c06 _num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[_num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e _num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n nums.setAll(0, res);\n}\n
counting_sort.c/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfn counting_sort(nums: &mut [i32]) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = *nums.into_iter().max().unwrap();\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n let mut counter = vec![0; m as usize + 1];\n for &num in &*nums {\n counter[num as usize] += 1;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for i in 0..m as usize {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n let n = nums.len();\n let mut res = vec![0; n];\n for i in (0..n).rev() {\n let num = nums[i];\n res[counter[num as usize] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num as usize] -= 1; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in 0..n {\n nums[i] = res[i];\n }\n}\n
counting_sort.zig/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int i = 0; i < size; i++) {\n if (nums[i] > m) {\n m = nums[i];\n }\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int *counter = calloc(m, sizeof(int));\n for (int i = 0; i < size; i++) {\n counter[nums[i]]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int *res = malloc(sizeof(int) * size);\n for (int i = size - 1; i >= 0; i--) {\n int num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n memcpy(nums, res, size * sizeof(int));\n // 5. \u91ca\u653e\u5185\u5b58\n free(counter);\n}\n
"},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"[class]{}-[func]{countingSort}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386
nums\u548c\u904d\u5386counter\uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u4e8e \\(O(n)\\) \u3002 - \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u501f\u52a9\u4e86\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \u7684\u6570\u7ec4
res\u548ccounter\u3002 - \u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5411
res\u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386nums\u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u7a33\u5b9a\u6392\u5e8f\u3002\u5b9e\u9645\u4e0a\uff0c\u6b63\u5e8f\u904d\u5386nums\u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u662f\u975e\u7a33\u5b9a\u7684\u3002
\u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u4ec5\u901a\u8fc7\u7edf\u8ba1\u6570\u91cf\u5c31\u53ef\u4ee5\u5b9e\u73b0\u9ad8\u6548\u7684\u6392\u5e8f\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\u76f8\u5bf9\u8f83\u4e3a\u4e25\u683c\u3002
\u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u5c06\u5176\u7528\u4e8e\u5176\u4ed6\u7c7b\u578b\u7684\u6570\u636e\uff0c\u9700\u8981\u786e\u4fdd\u8fd9\u4e9b\u6570\u636e\u53ef\u4ee5\u8f6c\u6362\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u5728\u8f6c\u6362\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u3002
\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u4f1a\u5360\u7528\u8fc7\u591a\u7a7a\u95f4\u3002\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002
"},{"location":"chapter_sorting/heap_sort/","title":"11.7 \u00a0 \u5806\u6392\u5e8f","text":"Tip
\u9605\u8bfb\u672c\u8282\u524d\uff0c\u8bf7\u786e\u4fdd\u5df2\u5b66\u5b8c\u201c\u5806\u201c\u7ae0\u8282\u3002
\u300c\u5806\u6392\u5e8f heap sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5806\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\u7684\u9ad8\u6548\u6392\u5e8f\u7b97\u6cd5\u3002\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5df2\u7ecf\u5b66\u8fc7\u7684\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\u5806\u6392\u5e8f\u3002
- \u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5c0f\u9876\u5806\uff0c\u6b64\u65f6\u6700\u5c0f\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002
- \u4e0d\u65ad\u6267\u884c\u51fa\u5806\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8bb0\u5f55\u51fa\u5806\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002
\u4ee5\u4e0a\u65b9\u6cd5\u867d\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u989d\u5916\u6570\u7ec4\u6765\u4fdd\u5b58\u5f39\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f83\u6d6a\u8d39\u7a7a\u95f4\u3002\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u4e00\u79cd\u66f4\u52a0\u4f18\u96c5\u7684\u5b9e\u73b0\u65b9\u5f0f\u3002
"},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5806\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-12 \u6240\u793a\u3002
- \u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5927\u9876\u5806\u3002\u5b8c\u6210\u540e\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002
- \u5c06\u5806\u9876\u5143\u7d20\uff08\u7b2c\u4e00\u4e2a\u5143\u7d20\uff09\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u6700\u540e\u4e00\u4e2a\u5143\u7d20\uff09\u4ea4\u6362\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u5806\u7684\u957f\u5ea6\u51cf \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u52a0 \\(1\\) \u3002
- \u4ece\u5806\u9876\u5143\u7d20\u5f00\u59cb\uff0c\u4ece\u9876\u5230\u5e95\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff08Sift Down\uff09\u3002\u5b8c\u6210\u5806\u5316\u540e\uff0c\u5806\u7684\u6027\u8d28\u5f97\u5230\u4fee\u590d\u3002
- \u5faa\u73af\u6267\u884c\u7b2c
2.\u6b65\u548c\u7b2c3.\u6b65\u3002\u5faa\u73af \\(n - 1\\) \u8f6e\u540e\uff0c\u5373\u53ef\u5b8c\u6210\u6570\u7ec4\u6392\u5e8f\u3002
Tip
\u5b9e\u9645\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c
<1><2><3><4><5><6><7><8><9><10><11><12>2.\u6b65\u548c\u7b2c3.\u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u4e2a\u5f39\u51fa\u5143\u7d20\u7684\u6b65\u9aa4\u3002\u56fe 11-12 \u00a0 \u5806\u6392\u5e8f\u6b65\u9aa4
\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e86\u4e0e\u201c\u5806\u201d\u7ae0\u8282\u76f8\u540c\u7684\u4ece\u9876\u81f3\u5e95\u5806\u5316
PythonC++JavaC#GoSwiftJSTSDartRustCZig heap_sort.pysift_down()\u51fd\u6570\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u5806\u7684\u957f\u5ea6\u4f1a\u968f\u7740\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u51cf\u5c0f\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7ed9sift_down()\u51fd\u6570\u6dfb\u52a0\u4e00\u4e2a\u957f\u5ea6\u53c2\u6570 \\(n\\) \uff0c\u7528\u4e8e\u6307\u5b9a\u5806\u7684\u5f53\u524d\u6709\u6548\u957f\u5ea6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
heap_sort.cppdef sift_down(nums: list[int], n: int, i: int):\n \"\"\"\u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n while True:\n # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n l = 2 * i + 1\n r = 2 * i + 2\n ma = i\n if l < n and nums[l] > nums[ma]:\n ma = l\n if r < n and nums[r] > nums[ma]:\n ma = r\n # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i:\n break\n # \u4ea4\u6362\u4e24\u8282\u70b9\n nums[i], nums[ma] = nums[ma], nums[i]\n # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n\ndef heap_sort(nums: list[int]):\n \"\"\"\u5806\u6392\u5e8f\"\"\"\n # \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in range(len(nums) // 2 - 1, -1, -1):\n sift_down(nums, len(nums), i)\n # \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for i in range(len(nums) - 1, 0, -1):\n # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n nums[0], nums[i] = nums[i], nums[0]\n # \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n sift_down(nums, i, 0)\n
heap_sort.java/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(vector<int> &nums, int n, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma])\n ma = l;\n if (r < n && nums[r] > nums[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(nums[i], nums[ma]);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(vector<int> &nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = nums.size() / 2 - 1; i >= 0; --i) {\n siftDown(nums, nums.size(), i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = nums.size() - 1; i > 0; --i) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(nums[0], nums[i]);\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.cs/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma])\n ma = l;\n if (r < n && nums[r] > nums[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i)\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n int temp = nums[i];\n nums[i] = nums[ma];\n nums[ma] = temp;\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = nums.length / 2 - 1; i >= 0; i--) {\n siftDown(nums, nums.length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = nums.length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n int tmp = nums[0];\n nums[0] = nums[i];\n nums[i] = tmp;\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.go/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int[] nums, int n, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma])\n ma = l;\n if (r < n && nums[r] > nums[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i)\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n (nums[ma], nums[i]) = (nums[i], nums[ma]);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid HeapSort(int[] nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = nums.Length / 2 - 1; i >= 0; i--) {\n SiftDown(nums, nums.Length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = nums.Length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n (nums[i], nums[0]) = (nums[0], nums[i]);\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n SiftDown(nums, i, 0);\n }\n}\n
heap_sort.swift/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\n for true {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n l := 2*i + 1\n r := 2*i + 2\n ma := i\n if l < n && (*nums)[l] > (*nums)[ma] {\n ma = l\n }\n if r < n && (*nums)[r] > (*nums)[ma] {\n ma = r\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i := len(*nums)/2 - 1; i >= 0; i-- {\n siftDown(nums, len(*nums), i)\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for i := len(*nums) - 1; i > 0; i-- {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0)\n }\n}\n
heap_sort.js/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n var i = i\n while true {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = 2 * i + 1\n let r = 2 * i + 2\n var ma = i\n if l < n, nums[l] > nums[ma] {\n ma = l\n }\n if r < n, nums[r] > nums[ma] {\n ma = r\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n nums.swapAt(i, ma)\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n siftDown(nums: &nums, n: nums.count, i: i)\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n nums.swapAt(0, i)\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums: &nums, n: i, i: 0)\n }\n}\n
heap_sort.ts/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums, n, i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = 2 * i + 1;\n let r = 2 * i + 2;\n let ma = i;\n if (l < n && nums[l] > nums[ma]) {\n ma = l;\n }\n if (r < n && nums[r] > nums[ma]) {\n ma = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma === i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n [nums[i], nums[ma]] = [nums[ma], nums[i]];\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (let i = Math.floor(nums.length / 2) - 1; i >= 0; i--) {\n siftDown(nums, nums.length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (let i = nums.length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n [nums[0], nums[i]] = [nums[i], nums[0]];\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.dart/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = 2 * i + 1;\n let r = 2 * i + 2;\n let ma = i;\n if (l < n && nums[l] > nums[ma]) {\n ma = l;\n }\n if (r < n && nums[r] > nums[ma]) {\n ma = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma === i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n [nums[i], nums[ma]] = [nums[ma], nums[i]];\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (let i = Math.floor(nums.length / 2) - 1; i >= 0; i--) {\n siftDown(nums, nums.length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (let i = nums.length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n [nums[0], nums[i]] = [nums[i], nums[0]];\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.rs/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(List<int> nums, int n, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma]) ma = l;\n if (r < n && nums[r] > nums[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n int temp = nums[i];\n nums[i] = nums[ma];\n nums[ma] = temp;\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(List<int> nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = nums.length ~/ 2 - 1; i >= 0; i--) {\n siftDown(nums, nums.length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = nums.length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n int tmp = nums[0];\n nums[0] = nums[i];\n nums[i] = tmp;\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.c/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(nums: &mut [i32], n: usize, mut i: usize) {\n loop {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = 2 * i + 1;\n let r = 2 * i + 2;\n let mut ma = i;\n if l < n && nums[l] > nums[ma] {\n ma = l;\n }\n if r < n && nums[r] > nums[ma] {\n ma = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n let temp = nums[i];\n nums[i] = nums[ma];\n nums[ma] = temp;\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfn heap_sort(nums: &mut [i32]) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in (0..=nums.len() / 2 - 1).rev() {\n sift_down(nums, nums.len(), i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for i in (1..=nums.len() - 1).rev() {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n let tmp = nums[0];\n nums[0] = nums[i];\n nums[i] = tmp;\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n sift_down(nums, i, 0);\n }\n}\n
heap_sort.zig/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int nums[], int n, int i) {\n while (1) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma])\n ma = l;\n if (r < n && nums[r] > nums[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n int temp = nums[i];\n nums[i] = nums[ma];\n nums[ma] = temp;\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = n / 2 - 1; i >= 0; --i) {\n siftDown(nums, n, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = n - 1; i > 0; --i) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n int tmp = nums[0];\n nums[0] = nums[i];\n nums[i] = tmp;\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
"},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"[class]{}-[func]{siftDown}\n\n[class]{}-[func]{heapSort}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5efa\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5171\u5faa\u73af \\(n - 1\\) \u8f6e\u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\u3002\u5143\u7d20\u4ea4\u6362\u548c\u5806\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\u3002
- \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5806\u9876\u5143\u7d20\u548c\u5806\u5e95\u5143\u7d20\u65f6\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002
\u300c\u63d2\u5165\u6392\u5e8f insertion sort\u300d\u662f\u4e00\u79cd\u7b80\u5355\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u4e0e\u624b\u52a8\u6574\u7406\u4e00\u526f\u724c\u7684\u8fc7\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002
\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u5728\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u4e00\u4e2a\u57fa\u51c6\u5143\u7d20\uff0c\u5c06\u8be5\u5143\u7d20\u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f83\u5927\u5c0f\uff0c\u5e76\u5c06\u8be5\u5143\u7d20\u63d2\u5165\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u3002
\u56fe 11-6 \u5c55\u793a\u4e86\u6570\u7ec4\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u6d41\u7a0b\u3002\u8bbe\u57fa\u51c6\u5143\u7d20\u4e3a
base\uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230base\u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u5c06base\u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002\u56fe 11-6 \u00a0 \u5355\u6b21\u63d2\u5165\u64cd\u4f5c
"},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u63d2\u5165\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-7 \u6240\u793a\u3002
- \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6570\u7ec4\u7684\u7b2c 1 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
- \u9009\u53d6\u6570\u7ec4\u7684\u7b2c 2 \u4e2a\u5143\u7d20\u4f5c\u4e3a
base\uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002 - \u9009\u53d6\u7b2c 3 \u4e2a\u5143\u7d20\u4f5c\u4e3a
base\uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 3 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002 - \u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u6700\u540e\u4e00\u8f6e\u4e2d\uff0c\u9009\u53d6\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a
base\uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002
\u56fe 11-7 \u00a0 \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b
\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig insertion_sort.py
insertion_sort.cppdef insertion_sort(nums: list[int]):\n \"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n # \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n for i in range(1, len(nums)):\n base = nums[i]\n j = i - 1\n # \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n while j >= 0 and nums[j] > base:\n nums[j + 1] = nums[j] # \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j -= 1\n nums[j + 1] = base # \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n
insertion_sort.java/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector<int> &nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < nums.size(); i++) {\n int base = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.cs/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < nums.length; i++) {\n int base = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.go/* \u63d2\u5165\u6392\u5e8f */\nvoid InsertionSort(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < nums.Length; i++) {\n int bas = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > bas) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = bas; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.swift/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := 1; i < len(nums); i++ {\n base := nums[i]\n j := i - 1\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n for j >= 0 && nums[j] > base {\n nums[j+1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--\n }\n nums[j+1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.js/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for i in stride(from: 1, to: nums.count, by: 1) {\n let base = nums[i]\n var j = i - 1\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while j >= 0, nums[j] > base {\n nums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j -= 1\n }\n nums[j + 1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.ts/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (let i = 1; i < nums.length; i++) {\n let base = nums[i],\n j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.dart/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (let i = 1; i < nums.length; i++) {\n const base = nums[i];\n let j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.rs/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List<int> nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < nums.length; i++) {\n int base = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.c/* \u63d2\u5165\u6392\u5e8f */\nfn insertion_sort(nums: &mut [i32]) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for i in 1..nums.len() {\n let (base, mut j) = (nums[i], (i - 1) as i32);\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while j >= 0 && nums[j as usize] > base {\n nums[(j + 1) as usize] = nums[j as usize]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j -= 1;\n }\n nums[(j + 1) as usize] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.zig/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < size; i++) {\n int base = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n nums[j + 1] = nums[j];\n j--;\n }\n // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n nums[j + 1] = base;\n }\n}\n
"},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"// \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n var i: usize = 1;\n while (i < nums.len) : (i += 1) {\n var base = nums[i];\n var j: usize = i;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 1 and nums[j - 1] > base) : (j -= 1) {\n nums[j] = nums[j - 1]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n }\n nums[j] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u522b\u9700\u8981\u5faa\u73af \\(n - 1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\((n - 1) n / 2\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u6570\u636e\u65f6\uff0c\u63d2\u5165\u64cd\u4f5c\u4f1a\u63d0\u524d\u7ec8\u6b62\u3002\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u63d2\u5165\u6392\u5e8f\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f1a\u5c06\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\uff0c\u4e0d\u4f1a\u6539\u53d8\u5b83\u4eec\u7684\u987a\u5e8f\u3002
\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u800c\u6211\u4eec\u5373\u5c06\u5b66\u4e60\u7684\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u5c3d\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u9ad8\uff0c\u4f46\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u7684\u60c5\u51b5\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002
\u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u7684\u9002\u7528\u60c5\u51b5\u7684\u7ed3\u8bba\u7c7b\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u8fd9\u7c7b \\(O(n \\log n)\\) \u7684\u7b97\u6cd5\u5c5e\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6570\u503c\u6bd4\u8f83\u63a5\u8fd1\uff0c\u590d\u6742\u5ea6\u4e0d\u5360\u4e3b\u5bfc\u5730\u4f4d\uff1b\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u64cd\u4f5c\u6570\u91cf\u8d77\u5230\u51b3\u5b9a\u6027\u4f5c\u7528\u3002
\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\u91c7\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u4e3a\uff1a\u5bf9\u4e8e\u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5bf9\u4e8e\u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002
\u867d\u7136\u5192\u6ce1\u6392\u5e8f\u3001\u9009\u62e9\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u9891\u7387\u663e\u8457\u9ad8\u4e8e\u5192\u6ce1\u6392\u5e8f\u548c\u9009\u62e9\u6392\u5e8f\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002
- \u5192\u6ce1\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u4ea4\u6362\u5b9e\u73b0\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\uff0c\u5171\u6d89\u53ca 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u8d4b\u503c\u5b9e\u73b0\uff0c\u4ec5\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002
- \u9009\u62e9\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002\u5982\u679c\u7ed9\u5b9a\u4e00\u7ec4\u90e8\u5206\u6709\u5e8f\u7684\u6570\u636e\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9009\u62e9\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002
- \u9009\u62e9\u6392\u5e8f\u4e0d\u7a33\u5b9a\uff0c\u65e0\u6cd5\u5e94\u7528\u4e8e\u591a\u7ea7\u6392\u5e8f\u3002
\u300c\u5f52\u5e76\u6392\u5e8f merge sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5305\u542b\u56fe 11-10 \u6240\u793a\u7684\u201c\u5212\u5206\u201d\u548c\u201c\u5408\u5e76\u201d\u9636\u6bb5\u3002
- \u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u6362\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002
- \u5408\u5e76\u9636\u6bb5\uff1a\u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u5212\u5206\uff0c\u5f00\u59cb\u5408\u5e76\uff0c\u6301\u7eed\u5730\u5c06\u5de6\u53f3\u4e24\u4e2a\u8f83\u77ed\u7684\u6709\u5e8f\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u8f83\u957f\u7684\u6709\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u7ed3\u675f\u3002
\u56fe 11-10 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5
"},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u5982\u56fe 11-11 \u6240\u793a\uff0c\u201c\u5212\u5206\u9636\u6bb5\u201d\u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002
- \u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9
mid\uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4[left, mid]\uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4[mid + 1, right]\uff09\u3002 - \u9012\u5f52\u6267\u884c\u6b65\u9aa4
1.\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u3002
\u201c\u5408\u5e76\u9636\u6bb5\u201d\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u5408\u5e76\u9636\u6bb5\u4e2d\u7684\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002
<1><2><3><4><5><6><7><8><9><10>\u56fe 11-11 \u00a0 \u5f52\u5e76\u6392\u5e8f\u6b65\u9aa4
\u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u4e0e\u4e8c\u53c9\u6811\u540e\u5e8f\u904d\u5386\u7684\u9012\u5f52\u987a\u5e8f\u662f\u4e00\u81f4\u7684\u3002
- \u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6811\uff0c\u6700\u540e\u5904\u7406\u6839\u8282\u70b9\u3002
- \u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6570\u7ec4\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6570\u7ec4\uff0c\u6700\u540e\u5904\u7406\u5408\u5e76\u3002
\u5f52\u5e76\u6392\u5e8f\u7684\u5b9e\u73b0\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\u3002\u8bf7\u6ce8\u610f\uff0c
PythonC++JavaC#GoSwiftJSTSDartRustCZig merge_sort.pynums\u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a[left, right]\uff0c\u800ctmp\u7684\u5bf9\u5e94\u533a\u95f4\u4e3a[0, right - left]\u3002
merge_sort.cppdef merge(nums: list[int], left: int, mid: int, right: int):\n \"\"\"\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\"\"\"\n # \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n # \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n tmp = [0] * (right - left + 1)\n # \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n i, j, k = left, mid + 1, 0\n # \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid and j <= right:\n if nums[i] <= nums[j]:\n tmp[k] = nums[i]\n i += 1\n else:\n tmp[k] = nums[j]\n j += 1\n k += 1\n # \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid:\n tmp[k] = nums[i]\n i += 1\n k += 1\n while j <= right:\n tmp[k] = nums[j]\n j += 1\n k += 1\n # \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for k in range(0, len(tmp)):\n nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n \"\"\"\u5f52\u5e76\u6392\u5e8f\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if left >= right:\n return # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n # \u5212\u5206\u9636\u6bb5\n mid = (left + right) // 2 # \u8ba1\u7b97\u4e2d\u70b9\n merge_sort(nums, left, mid) # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n merge_sort(nums, mid + 1, right) # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n # \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right)\n
merge_sort.java/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(vector<int> &nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n vector<int> tmp(right - left + 1);\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j])\n tmp[k++] = nums[i++];\n else\n tmp[k++] = nums[j++];\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.size(); k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector<int> &nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right)\n return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.cs/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int[] nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n int[] tmp = new int[right - left + 1];\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j])\n tmp[k++] = nums[i++];\n else\n tmp[k++] = nums[j++];\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.length; k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right)\n return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.go/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n int[] tmp = new int[right - left + 1];\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j])\n tmp[k++] = nums[i++];\n else\n tmp[k++] = nums[j++];\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.Length; ++k) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid MergeSort(int[] nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n MergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n MergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n Merge(nums, left, mid, right);\n}\n
merge_sort.swift/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums []int, left, mid, right int) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n tmp := make([]int, right-left+1)\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n i, j, k := left, mid+1, 0\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n for i <= mid && j <= right {\n if nums[i] <= nums[j] {\n tmp[k] = nums[i]\n i++\n } else {\n tmp[k] = nums[j]\n j++\n }\n k++\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n for i <= mid {\n tmp[k] = nums[i]\n i++\n k++\n }\n for j <= right {\n tmp[k] = nums[j]\n j++\n k++\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for k := 0; k < len(tmp); k++ {\n nums[left+k] = tmp[k]\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n // \u7ec8\u6b62\u6761\u4ef6\n if left >= right {\n return\n }\n // \u5212\u5206\u9636\u6bb5\n mid := (left + right) / 2\n mergeSort(nums, left, mid)\n mergeSort(nums, mid+1, right)\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right)\n}\n
merge_sort.js/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n var tmp = Array(repeating: 0, count: right - left + 1)\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n var i = left, j = mid + 1, k = 0\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid, j <= right {\n if nums[i] <= nums[j] {\n tmp[k] = nums[i]\n i += 1\n k += 1\n } else {\n tmp[k] = nums[j]\n j += 1\n k += 1\n }\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid {\n tmp[k] = nums[i]\n i += 1\n k += 1\n }\n while j <= right {\n tmp[k] = nums[j]\n j += 1\n k += 1\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for k in tmp.indices {\n nums[left + k] = tmp[k]\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n // \u7ec8\u6b62\u6761\u4ef6\n if left >= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n return\n }\n // \u5212\u5206\u9636\u6bb5\n let mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums: &nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums: &nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums: &nums, left: left, mid: mid, right: right)\n}\n
merge_sort.ts/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums, left, mid, right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n const tmp = new Array(right - left + 1);\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n let i = left,\n j = mid + 1,\n k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j]) {\n tmp[k++] = nums[i++];\n } else {\n tmp[k++] = nums[j++];\n }\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.length; k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.dart/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n const tmp = new Array(right - left + 1);\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n let i = left,\n j = mid + 1,\n k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j]) {\n tmp[k++] = nums[i++];\n } else {\n tmp[k++] = nums[j++];\n }\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.length; k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.rs/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(List<int> nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n List<int> tmp = List.filled(right - left + 1, 0);\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j])\n tmp[k++] = nums[i++];\n else\n tmp[k++] = nums[j++];\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.length; k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(List<int> nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.c/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfn merge(nums: &mut [i32], left: usize, mid: usize, right: usize) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n let tmp_size = right - left + 1;\n let mut tmp = vec![0; tmp_size];\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n let (mut i, mut j, mut k) = (left, mid + 1, 0);\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid && j <= right {\n if nums[i] <= nums[j] {\n tmp[k] = nums[j];\n i += 1;\n } else {\n tmp[k] = nums[j];\n j += 1;\n }\n k += 1;\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid {\n tmp[k] = nums[i];\n k += 1;\n i += 1;\n }\n while j <= right {\n tmp[k] = nums[j];\n k += 1;\n j += 1;\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for k in 0..tmp_size {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfn merge_sort(nums: &mut [i32], left: usize, right: usize) {\n // \u7ec8\u6b62\u6761\u4ef6\n if left >= right { return; } // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n let mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n merge_sort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n merge_sort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.zig/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int *nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n int tmpSize = right - left + 1;\n int *tmp = (int *)malloc(tmpSize * sizeof(int));\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j]) {\n tmp[k++] = nums[i++];\n } else {\n tmp[k++] = nums[j++];\n }\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmpSize; ++k) {\n nums[left + k] = tmp[k];\n }\n // \u91ca\u653e\u5185\u5b58\n free(tmp);\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right)\n return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
"},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"// \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n // \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\n var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n defer mem_arena.deinit();\n const mem_allocator = mem_arena.allocator();\n var tmp = try mem_allocator.alloc(i32, right + 1 - left);\n std.mem.copy(i32, tmp, nums[left..right+1]);\n // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15 \n var leftStart = left - left;\n var leftEnd = mid - left;\n // \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15 \n var rightStart = mid + 1 - left;\n var rightEnd = right - left;\n // i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\n var i = leftStart;\n var j = rightStart;\n // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n var k = left;\n while (k <= right) : (k += 1) {\n // \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n if (i > leftEnd) {\n nums[k] = tmp[j];\n j += 1;\n // \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n } else if (j > rightEnd or tmp[i] <= tmp[j]) {\n nums[k] = tmp[i];\n i += 1;\n // \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n } else {\n nums[k] = tmp[j];\n j += 1;\n }\n }\n}\n\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n var mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n try mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n try mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n try merge(nums, left, mid, right);\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5212\u5206\u4ea7\u751f\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002\u5408\u5e76\u64cd\u4f5c\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u53d8\u3002
\u5bf9\u4e8e\u94fe\u8868\uff0c\u5f52\u5e76\u6392\u5e8f\u76f8\u8f83\u4e8e\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\u5177\u6709\u663e\u8457\u4f18\u52bf\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u6392\u5e8f\u4efb\u52a1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(1)\\) \u3002
- \u5212\u5206\u9636\u6bb5\uff1a\u53ef\u4ee5\u4f7f\u7528\u201c\u8fed\u4ee3\u201d\u66ff\u4ee3\u201c\u9012\u5f52\u201d\u6765\u5b9e\u73b0\u94fe\u8868\u5212\u5206\u5de5\u4f5c\uff0c\u4ece\u800c\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\u3002
- \u5408\u5e76\u9636\u6bb5\uff1a\u5728\u94fe\u8868\u4e2d\uff0c\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u4ec5\u9700\u6539\u53d8\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u5b9e\u73b0\uff0c\u56e0\u6b64\u5408\u5e76\u9636\u6bb5\uff08\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\uff09\u65e0\u987b\u521b\u5efa\u989d\u5916\u94fe\u8868\u3002
\u5177\u4f53\u5b9e\u73b0\u7ec6\u8282\u6bd4\u8f83\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u76f8\u5173\u8d44\u6599\u8fdb\u884c\u5b66\u4e60\u3002
"},{"location":"chapter_sorting/quick_sort/","title":"11.5 \u00a0 \u5feb\u901f\u6392\u5e8f","text":"\u300c\u5feb\u901f\u6392\u5e8f quick sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u8fd0\u884c\u9ad8\u6548\uff0c\u5e94\u7528\u5e7f\u6cdb\u3002
\u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5176\u76ee\u6807\u662f\uff1a\u9009\u62e9\u6570\u7ec4\u4e2d\u7684\u67d0\u4e2a\u5143\u7d20\u4f5c\u4e3a\u201c\u57fa\u51c6\u6570\u201d\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u4fa7\uff0c\u800c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u4fa7\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u54e8\u5175\u5212\u5206\u7684\u6d41\u7a0b\u5982\u56fe 11-8 \u6240\u793a\u3002
- \u9009\u53d6\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488
i\u548cj\u5206\u522b\u6307\u5411\u6570\u7ec4\u7684\u4e24\u7aef\u3002 - \u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u4f7f\u7528
i\uff08j\uff09\u5206\u522b\u5bfb\u627e\u7b2c\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u540e\u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\u3002 - \u5faa\u73af\u6267\u884c\u6b65\u9aa4
2.\uff0c\u76f4\u5230i\u548cj\u76f8\u9047\u65f6\u505c\u6b62\uff0c\u6700\u540e\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\u3002
\u56fe 11-8 \u00a0 \u54e8\u5175\u5212\u5206\u6b65\u9aa4
\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u6570\u7ec4\u3001\u57fa\u51c6\u6570\u3001\u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3\u201c\u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u51c6\u6570 \\(\\leq\\) \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u53ea\u9700\u5bf9\u8fd9\u4e24\u4e2a\u5b50\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\u3002
\u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565
\u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06\u4e00\u4e2a\u8f83\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u7b80\u5316\u4e3a\u4e24\u4e2a\u8f83\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py
quick_sort.cppdef partition(self, nums: list[int], left: int, right: int) -> int:\n \"\"\"\u54e8\u5175\u5212\u5206\"\"\"\n # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n i, j = left, right\n while i < j:\n while i < j and nums[j] >= nums[left]:\n j -= 1 # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while i < j and nums[i] <= nums[left]:\n i += 1 # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n # \u5143\u7d20\u4ea4\u6362\n nums[i], nums[j] = nums[j], nums[i]\n # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n nums[i], nums[left] = nums[left], nums[i]\n return i # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
quick_sort.java/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector<int> &nums, int i, int j) {\n int tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector<int> &nums, int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.cs/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\n int tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.go/* \u5143\u7d20\u4ea4\u6362 */\nvoid Swap(int[] nums, int i, int j) {\n (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint Partition(int[] nums, int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n Swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.swift/* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n i, j := left, right\n for i < j {\n for i < j && nums[j] >= nums[left] {\n j-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n for i < j && nums[i] <= nums[left] {\n i++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n // \u5143\u7d20\u4ea4\u6362\n nums[i], nums[j] = nums[j], nums[i]\n }\n // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n nums[i], nums[left] = nums[left], nums[i]\n return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.js/* \u5143\u7d20\u4ea4\u6362 */\nfunc swap(nums: inout [Int], i: Int, j: Int) {\n let tmp = nums[i]\n nums[i] = nums[j]\n nums[j] = tmp\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -> Int {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n var i = left\n var j = right\n while i < j {\n while i < j, nums[j] >= nums[left] {\n j -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while i < j, nums[i] <= nums[left] {\n i += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n swap(nums: &nums, i: i, j: j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums: &nums, i: i, j: left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.ts/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\n let tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let i = left,\n j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while (i < j && nums[i] <= nums[left]) {\n i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n // \u5143\u7d20\u4ea4\u6362\n this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.dart/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\n let tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let i = left,\n j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while (i < j && nums[i] <= nums[left]) {\n i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n // \u5143\u7d20\u4ea4\u6362\n this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.rs/* \u5143\u7d20\u4ea4\u6362 */\nvoid _swap(List<int> nums, int i, int j) {\n int tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint _partition(List<int> nums, int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.c/* \u54e8\u5175\u5212\u5206 */\nfn partition(nums: &mut [i32], left: usize, right: usize) -> usize {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let (mut i, mut j) = (left, right);\n while i < j {\n while i < j && nums[j] >= nums[left] {\n j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while i < j && nums[i] <= nums[left] {\n i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n nums.swap(i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.zig/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int nums[], int i, int j) {\n int tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n j--;\n }\n while (i < j && nums[i] <= nums[left]) {\n // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n i++;\n }\n // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n swap(nums, i, j);\n }\n // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n swap(nums, i, left);\n // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n return i;\n}\n
"},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"// \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\n var tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n var i = left;\n var j = right;\n while (i < j) {\n while (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n\u5feb\u901f\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-9 \u6240\u793a\u3002
- \u9996\u5148\uff0c\u5bf9\u539f\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u3002
- \u7136\u540e\uff0c\u5bf9\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5206\u522b\u9012\u5f52\u6267\u884c\u201c\u54e8\u5175\u5212\u5206\u201d\u3002
- \u6301\u7eed\u9012\u5f52\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\uff0c\u4ece\u800c\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\u3002
\u56fe 11-9 \u00a0 \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b
PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py
quick_sort.cppdef quick_sort(self, nums: list[int], left: int, right: int):\n \"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if left >= right:\n return\n # \u54e8\u5175\u5212\u5206\n pivot = self.partition(nums, left, right)\n # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n self.quick_sort(nums, left, pivot - 1)\n self.quick_sort(nums, pivot + 1, right)\n
quick_sort.java/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector<int> &nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right)\n return;\n // \u54e8\u5175\u5212\u5206\n int pivot = partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n
quick_sort.cs/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right)\n return;\n // \u54e8\u5175\u5212\u5206\n int pivot = partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n
quick_sort.go/* \u5feb\u901f\u6392\u5e8f */\nvoid QuickSort(int[] nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right)\n return;\n // \u54e8\u5175\u5212\u5206\n int pivot = Partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n QuickSort(nums, left, pivot - 1);\n QuickSort(nums, pivot + 1, right);\n}\n
quick_sort.swift/* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if left >= right {\n return\n }\n // \u54e8\u5175\u5212\u5206\n pivot := q.partition(nums, left, right)\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n q.quickSort(nums, left, pivot-1)\n q.quickSort(nums, pivot+1, right)\n}\n
quick_sort.js/* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if left >= right {\n return\n }\n // \u54e8\u5175\u5212\u5206\n let pivot = partition(nums: &nums, left: left, right: right)\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums: &nums, left: left, right: pivot - 1)\n quickSort(nums: &nums, left: pivot + 1, right: right)\n}\n
quick_sort.ts/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) return;\n // \u54e8\u5175\u5212\u5206\n const pivot = this.partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n this.quickSort(nums, left, pivot - 1);\n this.quickSort(nums, pivot + 1, right);\n}\n
quick_sort.dart/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) {\n return;\n }\n // \u54e8\u5175\u5212\u5206\n const pivot = this.partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n this.quickSort(nums, left, pivot - 1);\n this.quickSort(nums, pivot + 1, right);\n}\n
quick_sort.rs/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List<int> nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) return;\n // \u54e8\u5175\u5212\u5206\n int pivot = _partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n
quick_sort.c/* \u5feb\u901f\u6392\u5e8f */\npub fn quick_sort(left: i32, right: i32, nums: &mut [i32]) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if left >= right {\n return;\n }\n // \u54e8\u5175\u5212\u5206\n let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n Self::quick_sort(left, pivot - 1, nums);\n Self::quick_sort(pivot + 1, right, nums);\n}\n
quick_sort.zig/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n j--;\n }\n while (i < j && nums[i] <= nums[left]) {\n // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n i++;\n }\n // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n swap(nums, i, j);\n }\n // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n swap(nums, i, left);\n // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n return i;\n}\n\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u5feb\u901f\u6392\u5e8f\nvoid quickSort(int nums[], int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) {\n return;\n }\n // \u54e8\u5175\u5212\u5206\n int pivot = partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n
"},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"// \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) return;\n // \u54e8\u5175\u5212\u5206\n var pivot = partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
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\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u4f2a\u968f\u673a\u6570\u201d\u3002\u5982\u679c\u6211\u4eec\u9488\u5bf9\u4f2a\u968f\u673a\u6570\u5e8f\u5217\u6784\u5efa\u4e00\u4e2a\u7279\u5b9a\u7684\u6d4b\u8bd5\u6837\u4f8b\uff0c\u90a3\u4e48\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002
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\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py
quick_sort.cppdef median_three(self, nums: list[int], left: int, mid: int, right: int) -> int:\n \"\"\"\u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\"\"\"\n # \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n # \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if (nums[left] < nums[mid]) ^ (nums[left] < nums[right]):\n return left\n elif (nums[mid] < nums[left]) ^ (nums[mid] < nums[right]):\n return mid\n return right\n\ndef partition(self, nums: list[int], left: int, right: int) -> int:\n \"\"\"\u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\"\"\"\n # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n med = self.median_three(nums, left, (left + right) // 2, right)\n # \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n nums[left], nums[med] = nums[med], nums[left]\n # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n i, j = left, right\n while i < j:\n while i < j and nums[j] >= nums[left]:\n j -= 1 # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while i < j and nums[i] <= nums[left]:\n i += 1 # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n # \u5143\u7d20\u4ea4\u6362\n nums[i], nums[j] = nums[j], nums[i]\n # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n nums[i], nums[left] = nums[left], nums[i]\n return i # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
quick_sort.java/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector<int> &nums, int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector<int> &nums, int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = medianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.cs/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = medianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.go/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint Partition(int[] nums, int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = MedianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n Swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n Swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.swift/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\uff08!= \u5728\u8fd9\u91cc\u8d77\u5230\u5f02\u6216\u7684\u4f5c\u7528\uff09\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\n return left\n } else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\n return mid\n }\n return right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n med := q.medianThree(nums, left, (left+right)/2, right)\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n nums[left], nums[med] = nums[med], nums[left]\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n i, j := left, right\n for i < j {\n for i < j && nums[j] >= nums[left] {\n j-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n for i < j && nums[i] <= nums[left] {\n i++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n //\u5143\u7d20\u4ea4\u6362\n nums[i], nums[j] = nums[j], nums[i]\n }\n //\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n nums[i], nums[left] = nums[left], nums[i]\n return i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.js/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -> Int {\n if (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\n return left\n } else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\n return mid\n } else {\n return right\n }\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -> Int {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n let med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums: &nums, i: left, j: med)\n return partition(nums: &nums, left: left, right: right)\n}\n
quick_sort.ts/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right])) return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n let med = this.medianThree(\n nums,\n left,\n Math.floor((left + right) / 2),\n right\n );\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n this.swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let i = left,\n j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.dart/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(\n nums: number[],\n left: number,\n mid: number,\n right: number\n): number {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if (Number(nums[left] < nums[mid]) ^ Number(nums[left] < nums[right])) {\n return left;\n } else if (\n Number(nums[mid] < nums[left]) ^ Number(nums[mid] < nums[right])\n ) {\n return mid;\n } else {\n return right;\n }\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n let med = this.medianThree(\n nums,\n left,\n Math.floor((left + right) / 2),\n right\n );\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n this.swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let i = left,\n j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while (i < j && nums[i] <= nums[left]) {\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.rs/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint _medianThree(List<int> nums, int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint _partition(List<int> nums, int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n _swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.c/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfn median_three(nums: &mut [i32], left: usize, mid: usize, right: usize) -> usize {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if (nums[left] < nums[mid]) ^ (nums[left] < nums[right]) {\n return left;\n } else if (nums[mid] < nums[left]) ^ (nums[mid] < nums[right]) {\n return mid;\n } \n right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfn partition(nums: &mut [i32], left: usize, right: usize) -> usize {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n let med = Self::median_three(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n nums.swap(left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let (mut i, mut j) = (left, right);\n while i < j {\n while i < j && nums[j] >= nums[left] {\n j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while i < j && nums[i] <= nums[left] {\n i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n nums.swap(i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.zig/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */ \nint partitionMedian(int nums[], int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = medianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
"},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5 \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) != (nums[left] < nums[right])) {\n return left;\n } else if ((nums[mid] < nums[left]) != (nums[mid] < nums[right])) {\n return mid;\n } else {\n return right;\n }\n}\n\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n var med = medianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n var i = left;\n var j = right;\n while (i < j) {\n while (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u5360\u7528\u7a7a\u95f4\u8f83\u591a\u3002\u4ee5\u5b8c\u5168\u6709\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u8bbe\u9012\u5f52\u4e2d\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(m\\) \uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u4ea7\u751f\u957f\u5ea6\u4e3a \\(0\\) \u7684\u5de6\u5b50\u6570\u7ec4\u548c\u957f\u5ea6\u4e3a \\(m - 1\\) \u7684\u53f3\u5b50\u6570\u7ec4\uff0c\u8fd9\u610f\u5473\u7740\u6bcf\u4e00\u5c42\u9012\u5f52\u8c03\u7528\u51cf\u5c11\u7684\u95ee\u9898\u89c4\u6a21\u975e\u5e38\u5c0f\uff08\u53ea\u51cf\u5c11\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4f1a\u8fbe\u5230 \\(n - 1\\) \uff0c\u6b64\u65f6\u9700\u8981\u5360\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002
\u4e3a\u4e86\u9632\u6b62\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u6bd4\u8f83\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\uff0c\u4ec5\u5bf9\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u8fdb\u884c\u9012\u5f52\u3002\u7531\u4e8e\u8f83\u77ed\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(n / 2\\) \uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u80fd\u786e\u4fdd\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u4ece\u800c\u5c06\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py
quick_sort.cppdef quick_sort(self, nums: list[int], left: int, right: int):\n \"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\"\"\"\n # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while left < right:\n # \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n pivot = self.partition(nums, left, right)\n # \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if pivot - left < right - pivot:\n self.quick_sort(nums, left, pivot - 1) # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1 # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n else:\n self.quick_sort(nums, pivot + 1, right) # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1 # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n
quick_sort.java/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector<int> &nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.cs/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.go/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid QuickSort(int[] nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = Partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n QuickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n QuickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.swift/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n for left < right {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n pivot := q.partition(nums, left, right)\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if pivot-left < right-pivot {\n q.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n q.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.js/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n var left = left\n var right = right\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while left < right {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n let pivot = partition(nums: &nums, left: left, right: right)\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left) < (right - pivot) {\n quickSortTailCall(nums: &nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSortTailCall(nums: &nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.ts/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n let pivot = this.partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.dart/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n let pivot = this.partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.rs/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(List<int> nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = _partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.c/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &mut [i32]) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while left < right {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if pivot - left < right - pivot {\n Self::quick_sort(left, pivot - 1, nums); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n Self::quick_sort(pivot + 1, right, nums); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.zig/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\n// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nvoid quickSortTailCall(int nums[], int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSortTailCall(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSortTailCall(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
"},{"location":"chapter_sorting/radix_sort/","title":"11.10 \u00a0 \u57fa\u6570\u6392\u5e8f","text":"// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\n var left = left_;\n var right = right_;\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n var pivot = partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u8ba1\u6570\u6392\u5e8f\uff0c\u5b83\u9002\u7528\u4e8e\u6570\u636e\u91cf \\(n\\) \u8f83\u5927\u4f46\u6570\u636e\u8303\u56f4 \\(m\\) \u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u5bf9 \\(n = 10^6\\) \u4e2a\u5b66\u53f7\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u5b66\u53f7\u662f\u4e00\u4e2a \\(8\\) \u4f4d\u6570\u5b57\uff0c\u8fd9\u610f\u5473\u7740\u6570\u636e\u8303\u56f4 \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u5185\u5b58\u7a7a\u95f4\uff0c\u800c\u57fa\u6570\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u3002
\u300c\u57fa\u6570\u6392\u5e8f radix sort\u300d\u7684\u6838\u5fc3\u601d\u60f3\u4e0e\u8ba1\u6570\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u901a\u8fc7\u7edf\u8ba1\u4e2a\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u57fa\u6570\u6392\u5e8f\u5229\u7528\u6570\u5b57\u5404\u4f4d\u4e4b\u95f4\u7684\u9012\u8fdb\u5173\u7cfb\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e00\u4f4d\u8fdb\u884c\u6392\u5e8f\uff0c\u4ece\u800c\u5f97\u5230\u6700\u7ec8\u7684\u6392\u5e8f\u7ed3\u679c\u3002
"},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u4ee5\u5b66\u53f7\u6570\u636e\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6570\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6570\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-18 \u6240\u793a\u3002
- \u521d\u59cb\u5316\u4f4d\u6570 \\(k = 1\\) \u3002
- \u5bf9\u5b66\u53f7\u7684\u7b2c \\(k\\) \u4f4d\u6267\u884c\u201c\u8ba1\u6570\u6392\u5e8f\u201d\u3002\u5b8c\u6210\u540e\uff0c\u6570\u636e\u4f1a\u6839\u636e\u7b2c \\(k\\) \u4f4d\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u3002
- \u5c06 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u540e\u8fd4\u56de\u6b65\u9aa4
2.\u7ee7\u7eed\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u540e\u7ed3\u675f\u3002
\u56fe 11-18 \u00a0 \u57fa\u6570\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b
\u4e0b\u9762\u5256\u6790\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e00\u4e2a \\(d\\) \u8fdb\u5236\u7684\u6570\u5b57 \\(x\\) \uff0c\u8981\u83b7\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8ba1\u7b97\u516c\u5f0f\uff1a
\\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\]\u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5bf9\u6d6e\u70b9\u6570 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\: d\\) \u8868\u793a\u5bf9 \\(d\\) \u53d6\u4f59\u3002\u5bf9\u4e8e\u5b66\u53f7\u6570\u636e\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002
\u6b64\u5916\uff0c\u6211\u4eec\u9700\u8981\u5c0f\u5e45\u6539\u52a8\u8ba1\u6570\u6392\u5e8f\u4ee3\u7801\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u636e\u6570\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u8fdb\u884c\u6392\u5e8f\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig radix_sort.py
radix_sort.cppdef digit(num: int, exp: int) -> int:\n \"\"\"\u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n # \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n \"\"\"\u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n # \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n counter = [0] * 10\n n = len(nums)\n # \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for i in range(n):\n d = digit(nums[i], exp) # \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d] += 1 # \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n # \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for i in range(1, 10):\n counter[i] += counter[i - 1]\n # \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n res = [0] * n\n for i in range(n - 1, -1, -1):\n d = digit(nums[i], exp)\n j = counter[d] - 1 # \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i] # \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d] -= 1 # \u5c06 d \u7684\u6570\u91cf\u51cf 1\n # \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in range(n):\n nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n \"\"\"\u57fa\u6570\u6392\u5e8f\"\"\"\n # \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n m = max(nums)\n # \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n exp = 1\n while exp <= m:\n # \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n # k = 1 -> exp = 1\n # k = 2 -> exp = 10\n # \u5373 exp = 10^(k-1)\n counting_sort_digit(nums, exp)\n exp *= 10\n
radix_sort.java/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector<int> &nums, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n vector<int> counter(10, 0);\n int n = nums.size();\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < n; i++) {\n int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n vector<int> res(n, 0);\n for (int i = n - 1; i >= 0; i--) {\n int d = digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++)\n nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(vector<int> &nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n int m = *max_element(nums.begin(), nums.end());\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; exp <= m; exp *= 10)\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n}\n
radix_sort.cs/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n int[] counter = new int[10];\n int n = nums.length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < n; i++) {\n int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n int[] res = new int[n];\n for (int i = n - 1; i >= 0; i--) {\n int d = digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++)\n nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n int m = Integer.MIN_VALUE;\n for (int num : nums)\n if (num > m)\n m = num;\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; exp <= m; exp *= 10)\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n}\n
radix_sort.go/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint Digit(int num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid CountingSortDigit(int[] nums, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n int[] counter = new int[10];\n int n = nums.Length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < n; i++) {\n int d = Digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n int[] res = new int[n];\n for (int i = n - 1; i >= 0; i--) {\n int d = Digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid RadixSort(int[] nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n int m = int.MinValue;\n foreach (int num in nums) {\n if (num > m) m = num;\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; exp <= m; exp *= 10) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n CountingSortDigit(nums, exp);\n }\n}\n
radix_sort.swift/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n counter := make([]int, 10)\n n := len(nums)\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for i := 0; i < n; i++ {\n d := digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++ // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for i := 1; i < 10; i++ {\n counter[i] += counter[i-1]\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n res := make([]int, n)\n for i := n - 1; i >= 0; i-- {\n d := digit(nums[i], exp)\n j := counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]-- // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for i := 0; i < n; i++ {\n nums[i] = res[i]\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums []int) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n max := math.MinInt\n for _, num := range nums {\n if num > max {\n max = num\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for exp := 1; max >= exp; exp *= 10 {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp)\n }\n}\n
radix_sort.js/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -> Int {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n var counter = Array(repeating: 0, count: 10)\n let n = nums.count\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for i in nums.indices {\n let d = digit(num: nums[i], exp: exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d] += 1 // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for i in 1 ..< 10 {\n counter[i] += counter[i - 1]\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n var res = Array(repeating: 0, count: n)\n for i in stride(from: n - 1, through: 0, by: -1) {\n let d = digit(num: nums[i], exp: exp)\n let j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d] -= 1 // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in nums.indices {\n nums[i] = res[i]\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n var m = Int.min\n for num in nums {\n if num > m {\n m = num\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for exp in sequence(first: 1, next: { m >= ($0 * 10) ? $0 * 10 : nil }) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums: &nums, exp: exp)\n }\n}\n
radix_sort.ts/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n const counter = new Array(10).fill(0);\n const n = nums.length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (let i = 0; i < n; i++) {\n const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (let i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n const res = new Array(n).fill(0);\n for (let i = n - 1; i >= 0; i--) {\n const d = digit(nums[i], exp);\n const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (let i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n let m = Number.MIN_VALUE;\n for (const num of nums) {\n if (num > m) {\n m = num;\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (let exp = 1; exp <= m; exp *= 10) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n }\n}\n
radix_sort.dart/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n const counter = new Array(10).fill(0);\n const n = nums.length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (let i = 0; i < n; i++) {\n const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (let i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n const res = new Array(n).fill(0);\n for (let i = n - 1; i >= 0; i--) {\n const d = digit(nums[i], exp);\n const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (let i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n let m = Number.MIN_VALUE;\n for (const num of nums) {\n if (num > m) {\n m = num;\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (let exp = 1; exp <= m; exp *= 10) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n }\n}\n
radix_sort.rs/* \u83b7\u53d6\u5143\u7d20 _num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int _num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (_num ~/ exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List<int> nums, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n List<int> counter = List<int>.filled(10, 0);\n int n = nums.length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < n; i++) {\n int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n List<int> res = List<int>.filled(n, 0);\n for (int i = n - 1; i >= 0; i--) {\n int d = digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++) nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(List<int> nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n // dart \u4e2d int \u7684\u957f\u5ea6\u662f 64 \u4f4d\u7684\n int m = -1 << 63;\n for (int _num in nums) if (_num > m) m = _num;\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; exp <= m; exp *= 10)\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n}\n
radix_sort.c/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -> usize {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return ((num / exp) % 10) as usize;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfn counting_sort_digit(nums: &mut [i32], exp: i32) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n let mut counter = [0; 10];\n let n = nums.len();\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for i in 0..n {\n let d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for i in 1..10 {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n let mut res = vec![0; n];\n for i in (0..n).rev() {\n let d = digit(nums[i], exp);\n let j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d] -= 1; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in 0..n {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfn radix_sort(nums: &mut [i32]) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n let m = *nums.into_iter().max().unwrap();\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n let mut exp = 1;\n while exp <= m {\n counting_sort_digit(nums, exp);\n exp *= 10;\n }\n}\n
radix_sort.zig/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n int *counter = (int *)malloc((sizeof(int) * 10));\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < size; i++) {\n // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n int d = digit(nums[i], exp);\n // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n counter[d]++;\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n int *res = (int *)malloc(sizeof(int) * size);\n for (int i = size - 1; i >= 0; i--) {\n int d = digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < size; i++) {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n int max = INT32_MIN;\n for (size_t i = 0; i < size - 1; i++) {\n if (nums[i] > max) {\n max = nums[i];\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; max >= exp; exp *= 10)\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, size, exp);\n}\n// \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return @mod(@divFloor(num, exp), 10);\n}\n\n// \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n // defer mem_arena.deinit();\n const mem_allocator = mem_arena.allocator();\n var counter = try mem_allocator.alloc(usize, 10);\n @memset(counter, 0);\n var n = nums.len;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (nums) |num| {\n var d: u32 = @bitCast(digit(num, exp)); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n var i: usize = 1;\n while (i < 10) : (i += 1) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n var res = try mem_allocator.alloc(i32, n);\n i = n - 1;\n while (i >= 0) : (i -= 1) {\n var d: u32 = @bitCast(digit(nums[i], exp));\n var j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d] -= 1; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n if (i == 0) break;\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n i = 0;\n while (i < n) : (i += 1) {\n nums[i] = res[i];\n }\n}\n\n// \u57fa\u6570\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n var m: i32 = std.math.minInt(i32);\n for (nums) |num| {\n if (num > m) m = num;\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n var exp: i32 = 1;\n while (exp <= m) : (exp *= 10) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n try countingSortDigit(nums, exp); \n }\n} \n\u4e3a\u4ec0\u4e48\u4ece\u6700\u4f4e\u4f4d\u5f00\u59cb\u6392\u5e8f\uff1f
\u5728\u8fde\u7eed\u7684\u6392\u5e8f\u8f6e\u6b21\u4e2d\uff0c\u540e\u4e00\u8f6e\u6392\u5e8f\u4f1a\u8986\u76d6\u524d\u4e00\u8f6e\u6392\u5e8f\u7684\u7ed3\u679c\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u5982\u679c\u7b2c\u4e00\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a < b\\) \uff0c\u800c\u7b2c\u4e8c\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a > b\\) \uff0c\u90a3\u4e48\u7b2c\u4e8c\u8f6e\u7684\u7ed3\u679c\u5c06\u53d6\u4ee3\u7b2c\u4e00\u8f6e\u7684\u7ed3\u679c\u3002\u7531\u4e8e\u6570\u5b57\u7684\u9ad8\u4f4d\u4f18\u5148\u7ea7\u9ad8\u4e8e\u4f4e\u4f4d\uff0c\u56e0\u6b64\u5e94\u8be5\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002
"},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"\u76f8\u8f83\u4e8e\u8ba1\u6570\u6392\u5e8f\uff0c\u57fa\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u503c\u8303\u56f4\u8f83\u5927\u7684\u60c5\u51b5\uff0c\u4f46\u524d\u63d0\u662f\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6570\u4e0d\u80fd\u8fc7\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u70b9\u6570\u4e0d\u9002\u5408\u4f7f\u7528\u57fa\u6570\u6392\u5e8f\uff0c\u56e0\u4e3a\u5176\u4f4d\u6570 \\(k\\) \u8fc7\u5927\uff0c\u53ef\u80fd\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002
- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk)\\)\uff1a\u8bbe\u6570\u636e\u91cf\u4e3a \\(n\\)\u3001\u6570\u636e\u4e3a \\(d\\) \u8fdb\u5236\u3001\u6700\u5927\u4f4d\u6570\u4e3a \\(k\\) \uff0c\u5219\u5bf9\u67d0\u4e00\u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u65f6\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5bf9\u8f83\u5c0f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + d)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6570\u6392\u5e8f\u9700\u8981\u501f\u52a9\u957f\u5ea6\u4e3a \\(n\\) \u548c \\(d\\) \u7684\u6570\u7ec4
res\u548ccounter\u3002 - \u7a33\u5b9a\u6392\u5e8f\uff1a\u5f53\u8ba1\u6570\u6392\u5e8f\u7a33\u5b9a\u65f6\uff0c\u57fa\u6570\u6392\u5e8f\u4e5f\u7a33\u5b9a\uff1b\u5f53\u8ba1\u6570\u6392\u5e8f\u4e0d\u7a33\u5b9a\u65f6\uff0c\u57fa\u6570\u6392\u5e8f\u65e0\u6cd5\u4fdd\u8bc1\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\u3002
\u300c\u9009\u62e9\u6392\u5e8f selection sort\u300d\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u7b80\u5355\uff1a\u5f00\u542f\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4ece\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c06\u5176\u653e\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u672b\u5c3e\u3002
\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u9009\u62e9\u6392\u5e8f\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 11-2 \u6240\u793a\u3002
- \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u533a\u95f4\u4e3a \\([0, n-1]\\) \u3002
- \u9009\u53d6\u533a\u95f4 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(0\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 1 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
- \u9009\u53d6\u533a\u95f4 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(1\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
- \u4ee5\u6b64\u7c7b\u63a8\u3002\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u9009\u62e9\u4e0e\u4ea4\u6362\u540e\uff0c\u6570\u7ec4\u524d \\(n - 1\\) \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
- \u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002
\u56fe 11-2 \u00a0 \u9009\u62e9\u6392\u5e8f\u6b65\u9aa4
\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u7528 \\(k\\) \u6765\u8bb0\u5f55\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig selection_sort.py
selection_sort.cppdef selection_sort(nums: list[int]):\n \"\"\"\u9009\u62e9\u6392\u5e8f\"\"\"\n n = len(nums)\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for i in range(n - 1):\n # \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n k = i\n for j in range(i + 1, n):\n if nums[j] < nums[k]:\n k = j # \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n # \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n nums[i], nums[k] = nums[k], nums[i]\n
selection_sort.java/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(vector<int> &nums) {\n int n = nums.size();\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k])\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n swap(nums[i], nums[k]);\n }\n}\n
selection_sort.cs/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\n int n = nums.length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k])\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n int temp = nums[i];\n nums[i] = nums[k];\n nums[k] = temp;\n }\n}\n
selection_sort.go/* \u9009\u62e9\u6392\u5e8f */\nvoid SelectionSort(int[] nums) {\n int n = nums.Length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k])\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n (nums[k], nums[i]) = (nums[i], nums[k]);\n }\n}\n
selection_sort.swift/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums []int) {\n n := len(nums)\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for i := 0; i < n-1; i++ {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n k := i\n for j := i + 1; j < n; j++ {\n if nums[j] < nums[k] {\n // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n k = j\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n nums[i], nums[k] = nums[k], nums[i]\n\n }\n}\n
selection_sort.js/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for i in nums.indices.dropLast() {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n var k = i\n for j in nums.indices.dropFirst(i + 1) {\n if nums[j] < nums[k] {\n k = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n nums.swapAt(i, k)\n }\n}\n
selection_sort.ts/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums) {\n let n = nums.length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (let i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n let k = i;\n for (let j = i + 1; j < n; j++) {\n if (nums[j] < nums[k]) {\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n [nums[i], nums[k]] = [nums[k], nums[i]];\n }\n}\n
selection_sort.dart/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\n let n = nums.length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (let i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n let k = i;\n for (let j = i + 1; j < n; j++) {\n if (nums[j] < nums[k]) {\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n [nums[i], nums[k]] = [nums[k], nums[i]];\n }\n}\n
selection_sort.rs/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(List<int> nums) {\n int n = nums.length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k]) k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n int temp = nums[i];\n nums[i] = nums[k];\n nums[k] = temp;\n }\n}\n
selection_sort.c/* \u9009\u62e9\u6392\u5e8f */\nfn selection_sort(nums: &mut [i32]) {\n let n = nums.len();\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for i in 0..n-1 {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n let mut k = i;\n for j in i+1..n {\n if nums[j] < nums[k] {\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n nums.swap(i, k);\n }\n}\n
selection_sort.zig/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k])\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n int temp = nums[i];\n nums[i] = nums[k];\n nums[k] = temp;\n }\n}\n
"},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"[class]{}-[func]{selectionSort}\n- \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5916\u5faa\u73af\u5171 \\(n - 1\\) \u8f6e\uff0c\u7b2c\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u6700\u540e\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(2\\) \uff0c\u5373\u5404\u8f6e\u5916\u5faa\u73af\u5206\u522b\u5305\u542b \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(3\\)\u3001\\(2\\) \u8f6e\u5185\u5faa\u73af\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5982\u56fe 11-3 \u6240\u793a\uff0c\u5143\u7d20
nums[i]\u6709\u53ef\u80fd\u88ab\u4ea4\u6362\u81f3\u4e0e\u5176\u76f8\u7b49\u7684\u5143\u7d20\u7684\u53f3\u8fb9\uff0c\u5bfc\u81f4\u4e24\u8005\u7684\u76f8\u5bf9\u987a\u5e8f\u53d1\u751f\u6539\u53d8\u3002
\u56fe 11-3 \u00a0 \u9009\u62e9\u6392\u5e8f\u975e\u7a33\u5b9a\u793a\u4f8b
"},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5","text":"\u300c\u6392\u5e8f\u7b97\u6cd5 sorting algorithm\u300d\u7528\u4e8e\u5bf9\u4e00\u7ec4\u6570\u636e\u6309\u7167\u7279\u5b9a\u987a\u5e8f\u8fdb\u884c\u6392\u5217\u3002\u6392\u5e8f\u7b97\u6cd5\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u56e0\u4e3a\u6709\u5e8f\u6570\u636e\u901a\u5e38\u80fd\u591f\u88ab\u66f4\u9ad8\u6548\u5730\u67e5\u627e\u3001\u5206\u6790\u548c\u5904\u7406\u3002
\u5982\u56fe 11-1 \u6240\u793a\uff0c\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u6570\u636e\u7c7b\u578b\u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7b49\u3002\u6392\u5e8f\u7684\u5224\u65ad\u89c4\u5219\u53ef\u6839\u636e\u9700\u6c42\u8bbe\u5b9a\uff0c\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u6216\u81ea\u5b9a\u4e49\u89c4\u5219\u3002
\u56fe 11-1 \u00a0 \u6570\u636e\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219\u793a\u4f8b
"},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1 \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"\u8fd0\u884c\u6548\u7387\uff1a\u6211\u4eec\u671f\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c3d\u91cf\u4f4e\uff0c\u4e14\u603b\u4f53\u64cd\u4f5c\u6570\u91cf\u8f83\u5c11\uff08\u65f6\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\uff09\u3002\u5bf9\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u8fd0\u884c\u6548\u7387\u663e\u5f97\u5c24\u4e3a\u91cd\u8981\u3002
\u5c31\u5730\u6027\uff1a\u987e\u540d\u601d\u4e49\uff0c\u300c\u539f\u5730\u6392\u5e8f\u300d\u901a\u8fc7\u5728\u539f\u6570\u7ec4\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\uff0c\u65e0\u987b\u501f\u52a9\u989d\u5916\u7684\u8f85\u52a9\u6570\u7ec4\uff0c\u4ece\u800c\u8282\u7701\u5185\u5b58\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u6570\u636e\u642c\u8fd0\u64cd\u4f5c\u8f83\u5c11\uff0c\u8fd0\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002
\u7a33\u5b9a\u6027\uff1a\u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u4e0d\u53d1\u751f\u6539\u53d8\u3002
\u7a33\u5b9a\u6392\u5e8f\u662f\u591a\u7ea7\u6392\u5e8f\u573a\u666f\u7684\u5fc5\u8981\u6761\u4ef6\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1 \u5217\u548c\u7b2c 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u53ef\u80fd\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e27\u5931\uff1a
# \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n ('A', 19)\n ('B', 18)\n ('C', 21)\n ('D', 19)\n ('E', 23)\n\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n ('B', 18)\n ('D', 19)\n ('A', 19)\n ('C', 21)\n ('E', 23)\n\u81ea\u9002\u5e94\u6027\uff1a\u300c\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u53d7\u8f93\u5165\u6570\u636e\u7684\u5f71\u54cd\uff0c\u5373\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5e76\u4e0d\u5b8c\u5168\u76f8\u7b49\u3002
\u81ea\u9002\u5e94\u6027\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u8bc4\u4f30\u3002\u5982\u679c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u5dee\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8bf4\u660e\u6392\u5e8f\u7b97\u6cd5\u5728\u67d0\u4e9b\u6570\u636e\u4e0b\u6027\u80fd\u53ef\u80fd\u52a3\u5316\uff0c\u56e0\u6b64\u88ab\u89c6\u4e3a\u8d1f\u9762\u5c5e\u6027\uff1b\u800c\u5982\u679c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5219\u88ab\u89c6\u4e3a\u6b63\u9762\u5c5e\u6027\u3002
\u662f\u5426\u57fa\u4e8e\u6bd4\u8f83\uff1a\u300c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u300d\u4f9d\u8d56\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff08\\(<\\)\u3001\\(=\\)\u3001\\(>\\)\uff09\u6765\u5224\u65ad\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\uff0c\u4ece\u800c\u6392\u5e8f\u6574\u4e2a\u6570\u7ec4\uff0c\u7406\u8bba\u6700\u4f18\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u800c\u300c\u975e\u6bd4\u8f83\u6392\u5e8f\u300d\u4e0d\u4f7f\u7528\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5bf9\u8f83\u5dee\u3002
"},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2 \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"\u8fd0\u884c\u5feb\u3001\u539f\u5730\u3001\u7a33\u5b9a\u3001\u6b63\u5411\u81ea\u9002\u5e94\u3001\u901a\u7528\u6027\u597d\u3002\u663e\u7136\uff0c\u8fc4\u4eca\u4e3a\u6b62\u5c1a\u672a\u53d1\u73b0\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u65f6\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u7684\u6570\u636e\u7279\u70b9\u548c\u95ee\u9898\u9700\u6c42\u6765\u51b3\u5b9a\u3002
\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5171\u540c\u5b66\u4e60\u5404\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5e76\u57fa\u4e8e\u4e0a\u8ff0\u8bc4\u4ef7\u7ef4\u5ea6\u5bf9\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u4f18\u7f3a\u70b9\u8fdb\u884c\u5206\u6790\u3002
"},{"location":"chapter_sorting/summary/","title":"11.11 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_sorting/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"- \u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(n)\\) \u3002
- \u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u867d\u7136\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u7531\u4e8e\u5355\u5143\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u56e0\u6b64\u5728\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u4efb\u52a1\u4e2d\u975e\u5e38\u53d7\u6b22\u8fce\u3002
- \u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u4ee5\u964d\u4f4e\u8fd9\u79cd\u52a3\u5316\u7684\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u51cf\u5c11\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(\\log n)\\) \u3002
- \u5f52\u5e76\u6392\u5e8f\u5305\u62ec\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u5178\u578b\u5730\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5f52\u5e76\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u521b\u5efa\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002
- \u6876\u6392\u5e8f\u5305\u542b\u4e09\u4e2a\u6b65\u9aa4\uff1a\u6570\u636e\u5206\u6876\u3001\u6876\u5185\u6392\u5e8f\u548c\u5408\u5e76\u7ed3\u679c\u3002\u5b83\u540c\u6837\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9002\u7528\u4e8e\u6570\u636e\u4f53\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u3002\u6876\u6392\u5e8f\u7684\u5173\u952e\u5728\u4e8e\u5bf9\u6570\u636e\u8fdb\u884c\u5e73\u5747\u5206\u914d\u3002
- \u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5b83\u901a\u8fc7\u7edf\u8ba1\u6570\u636e\u51fa\u73b0\u7684\u6b21\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u6709\u9650\u7684\u60c5\u51b5\uff0c\u5e76\u4e14\u8981\u6c42\u6570\u636e\u80fd\u591f\u8f6c\u6362\u4e3a\u6b63\u6574\u6570\u3002
- \u57fa\u6570\u6392\u5e8f\u901a\u8fc7\u9010\u4f4d\u6392\u5e8f\u6765\u5b9e\u73b0\u6570\u636e\u6392\u5e8f\uff0c\u8981\u6c42\u6570\u636e\u80fd\u591f\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u6570\u5b57\u3002
- \u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a33\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7b49\u4f18\u70b9\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e00\u6837\uff0c\u6ca1\u6709\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\u80fd\u591f\u540c\u65f6\u6ee1\u8db3\u6240\u6709\u8fd9\u4e9b\u6761\u4ef6\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u6570\u636e\u7684\u7279\u6027\u6765\u9009\u62e9\u5408\u9002\u7684\u6392\u5e8f\u7b97\u6cd5\u3002
- \u56fe 11-19 \u5bf9\u6bd4\u4e86\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u6548\u7387\u3001\u7a33\u5b9a\u6027\u3001\u5c31\u5730\u6027\u548c\u81ea\u9002\u5e94\u6027\u7b49\u3002
\u56fe 11-19 \u00a0 \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4
"},{"location":"chapter_sorting/summary/#2-q-a","title":"2. \u00a0 Q & A","text":"\u6392\u5e8f\u7b97\u6cd5\u7a33\u5b9a\u6027\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f\u5fc5\u9700\u7684\uff1f
\u5728\u73b0\u5b9e\u4e2d\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u662f\u57fa\u4e8e\u5bf9\u8c61\u7684\u67d0\u4e2a\u5c5e\u6027\u8fdb\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b66\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u4e24\u4e2a\u5c5e\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u591a\u7ea7\u6392\u5e8f\uff1a
\u5148\u6309\u7167\u59d3\u540d\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230
(A, 180) (B, 185) (C, 170) (D, 170)\uff1b\u518d\u5bf9\u8eab\u9ad8\u8fdb\u884c\u6392\u5e8f\u3002\u7531\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e0d\u7a33\u5b9a\uff0c\u56e0\u6b64\u53ef\u80fd\u5f97\u5230(D, 170) (C, 170) (A, 180) (B, 185)\u3002\u53ef\u4ee5\u53d1\u73b0\uff0c\u5b66\u751f D \u548c C \u7684\u4f4d\u7f6e\u53d1\u751f\u4e86\u4ea4\u6362\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u574f\u4e86\uff0c\u800c\u8fd9\u662f\u6211\u4eec\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002
\u54e8\u5175\u5212\u5206\u4e2d\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u7684\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f
\u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002
\u54e8\u5175\u5212\u5206
partition()\u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362nums[left]\u548cnums[i]\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd<=\u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524dnums[left] >= nums[i]\u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728i == j\u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fdnums[j] == nums[i] > nums[left]\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4
[0, 0, 0, 0, 1]\uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a[1, 0, 0, 0, 0]\uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u6b63\u786e\u7684\u3002\u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9
nums[right]\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u5173\u4e8e\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u4e3a\u4ec0\u4e48\u9009\u77ed\u7684\u6570\u7ec4\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff1f
\u9012\u5f52\u6df1\u5ea6\u5c31\u662f\u5f53\u524d\u672a\u8fd4\u56de\u7684\u9012\u5f52\u65b9\u6cd5\u7684\u6570\u91cf\u3002\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u6211\u4eec\u5c06\u539f\u6570\u7ec4\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002\u5728\u5c3e\u9012\u5f52\u4f18\u5316\u540e\uff0c\u5411\u4e0b\u9012\u5f52\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u6700\u5927\u4e3a\u539f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a\u3002\u5047\u8bbe\u6700\u5dee\u60c5\u51b5\uff0c\u4e00\u76f4\u4e3a\u4e00\u534a\u957f\u5ea6\uff0c\u90a3\u4e48\u6700\u7ec8\u7684\u9012\u5f52\u6df1\u5ea6\u5c31\u662f \\(\\log n\\) \u3002
\u56de\u987e\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u4f1a\u8fde\u7eed\u5730\u9012\u5f52\u957f\u5ea6\u8f83\u5927\u7684\u6570\u7ec4\uff0c\u6700\u5dee\u60c5\u51b5\u4e0b\u4e3a \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002\u5c3e\u9012\u5f52\u4f18\u5316\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u51fa\u73b0\u3002
\u5f53\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u65f6\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n^2)\\) \u5417\uff1f\u8be5\u5982\u4f55\u5904\u7406\u8fd9\u79cd\u9000\u5316\u60c5\u51b5\uff1f
\u662f\u7684\u3002\u5bf9\u4e8e\u8fd9\u79cd\u60c5\u51b5\uff0c\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u54e8\u5175\u5212\u5206\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a\u5c0f\u4e8e\u3001\u7b49\u4e8e\u3001\u5927\u4e8e\u57fa\u51c6\u6570\u3002\u4ec5\u5411\u4e0b\u9012\u5f52\u5c0f\u4e8e\u548c\u5927\u4e8e\u7684\u4e24\u90e8\u5206\u3002\u5728\u8be5\u65b9\u6cd5\u4e0b\uff0c\u8f93\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u6570\u7ec4\uff0c\u4ec5\u4e00\u8f6e\u54e8\u5175\u5212\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002
\u6876\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n^2)\\) \uff1f
\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u4e2a\u6876\u4e2d\u3002\u5982\u679c\u6211\u4eec\u91c7\u7528\u4e00\u4e2a \\(O(n^2)\\) \u7b97\u6cd5\u6765\u6392\u5e8f\u8fd9\u4e9b\u5143\u7d20\uff0c\u5219\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002
"},{"location":"chapter_stack_and_queue/","title":"\u7b2c 5 \u7ae0 \u00a0 \u6808\u4e0e\u961f\u5217","text":"Abstract
\u6808\u5982\u540c\u53e0\u732b\u732b\uff0c\u800c\u961f\u5217\u5c31\u50cf\u732b\u732b\u6392\u961f\u3002
\u4e24\u8005\u5206\u522b\u4ee3\u8868\u5148\u5165\u540e\u51fa\u548c\u5148\u5165\u5148\u51fa\u7684\u903b\u8f91\u5173\u7cfb\u3002
"},{"location":"chapter_stack_and_queue/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 5.1 \u00a0 \u6808
- 5.2 \u00a0 \u961f\u5217
- 5.3 \u00a0 \u53cc\u5411\u961f\u5217
- 5.4 \u00a0 \u5c0f\u7ed3
\u5728\u961f\u5217\u4e2d\uff0c\u6211\u4eec\u4ec5\u80fd\u5220\u9664\u5934\u90e8\u5143\u7d20\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u3002\u5982\u56fe 5-7 \u6240\u793a\uff0c\u300c\u53cc\u5411\u961f\u5217 double-ended queue\u300d\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u7075\u6d3b\u6027\uff0c\u5141\u8bb8\u5728\u5934\u90e8\u548c\u5c3e\u90e8\u6267\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002
\u56fe 5-7 \u00a0 \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c
"},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1 \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"\u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-3 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u79f0\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002
\u8868 5-3 \u00a0 \u53cc\u5411\u961f\u5217\u64cd\u4f5c\u6548\u7387
\u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 pushFirst() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) pushLast() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) popFirst() \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) popLast() \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) peekFirst() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) peekLast() \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\)\u540c\u6837\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5df2\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig deque.py
deque.cppfrom collections import deque\n\n# \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217\ndeque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\ndeque.append(2) # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3) # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n\n# \u8bbf\u95ee\u5143\u7d20\nfront: int = deque[0] # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1] # \u961f\u5c3e\u5143\u7d20\n\n# \u5143\u7d20\u51fa\u961f\npop_front: int = deque.popleft() # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop() # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n# \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(deque)\n\n# \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(deque) == 0\n
deque.java/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque<int> deque;\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back(); // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back(); // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n
deque.cs/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque<Integer> deque = new LinkedList<>();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst(); // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast(); // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast(); // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n
deque_test.go/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList<int> deque = new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value; // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast(); // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n
deque.swift/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back() // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front) // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear) // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n
deque.js/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n
deque.ts/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack = deque.pop();\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
deque.dart/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast: number = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack: number = deque.pop() as number;\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
deque.rs/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u4e49\u4e3a\u53cc\u5411\u961f\u5217\nQueue<int> deque = Queue<int>();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.addLast(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.addFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.first; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.last; // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.removeFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.removeLast(); // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.length;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.isEmpty;W\n
deque.c/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nlet mut deque: VecDeque<u32> = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nif let Some(front) = deque.front() { // \u961f\u9996\u5143\u7d20\n}\nif let Some(rear) = deque.back() { // \u961f\u5c3e\u5143\u7d20\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop_front) = deque.pop_front() { // \u961f\u9996\u5143\u7d20\u51fa\u961f\n}\nif let Some(pop_rear) = deque.pop_back() { // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n
deque.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u53cc\u5411\u961f\u5217\n
"},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2 \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"\n\u53cc\u5411\u961f\u5217\u7684\u5b9e\u73b0\u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53ef\u4ee5\u9009\u62e9\u94fe\u8868\u6216\u6570\u7ec4\u4f5c\u4e3a\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
"},{"location":"chapter_stack_and_queue/deque/#1","title":"1. \u00a0 \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"\u56de\u987e\u4e0a\u4e00\u8282\u5185\u5bb9\uff0c\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\uff0c\u56e0\u4e3a\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u5934\u8282\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0\u65b0\u8282\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\u3002
\u5bf9\u4e8e\u53cc\u5411\u961f\u5217\u800c\u8a00\uff0c\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u201c\u53cc\u5411\u94fe\u8868\u201d\u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
\u5982\u56fe 5-8 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u8282\u70b9\u548c\u5c3e\u8282\u70b9\u89c6\u4e3a\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u540c\u65f6\u5b9e\u73b0\u5728\u4e24\u7aef\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u7684\u529f\u80fd\u3002
LinkedListDequepushLast()pushFirst()popLast()popFirst()\u56fe 5-8 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c
\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_deque.py
linkedlist_deque.cppclass ListNode:\n \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\n\n def __init__(self, val: int):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self.val: int = val\n self.next: ListNode | None = None # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n self.prev: ListNode | None = None # \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\nclass LinkedListDeque:\n \"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._front: ListNode | None = None # \u5934\u8282\u70b9 front\n self._rear: ListNode | None = None # \u5c3e\u8282\u70b9 rear\n self._size: int = 0 # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n return self.size() == 0\n\n def push(self, num: int, is_front: bool):\n \"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\n node = ListNode(num)\n # \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if self.is_empty():\n self._front = self._rear = node\n # \u961f\u9996\u5165\u961f\u64cd\u4f5c\n elif is_front:\n # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n self._front.prev = node\n node.next = self._front\n self._front = node # \u66f4\u65b0\u5934\u8282\u70b9\n # \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else:\n # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n self._rear.next = node\n node.prev = self._rear\n self._rear = node # \u66f4\u65b0\u5c3e\u8282\u70b9\n self._size += 1 # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n\n def push_first(self, num: int):\n \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n self.push(num, True)\n\n def push_last(self, num: int):\n \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n self.push(num, False)\n\n def pop(self, is_front: bool) -> int:\n \"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n # \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if is_front:\n val: int = self._front.val # \u6682\u5b58\u5934\u8282\u70b9\u503c\n # \u5220\u9664\u5934\u8282\u70b9\n fnext: ListNode | None = self._front.next\n if fnext != None:\n fnext.prev = None\n self._front.next = None\n self._front = fnext # \u66f4\u65b0\u5934\u8282\u70b9\n # \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else:\n val: int = self._rear.val # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n # \u5220\u9664\u5c3e\u8282\u70b9\n rprev: ListNode | None = self._rear.prev\n if rprev != None:\n rprev.next = None\n self._rear.prev = None\n self._rear = rprev # \u66f4\u65b0\u5c3e\u8282\u70b9\n self._size -= 1 # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val\n\n def pop_first(self) -> int:\n \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n return self.pop(True)\n\n def pop_last(self) -> int:\n \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n return self.pop(False)\n\n def peek_first(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n return self._front.val\n\n def peek_last(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n return self._rear.val\n\n def to_array(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n node = self._front\n res = [0] * self.size()\n for i in range(self.size()):\n res[i] = node.val\n node = node.next\n return res\n
linkedlist_deque.java/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\n int val; // \u8282\u70b9\u503c\n DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\n DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n }\n};\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n private:\n DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n LinkedListDeque() : front(nullptr), rear(nullptr) {\n }\n\n /* \u6790\u6784\u65b9\u6cd5 */\n ~LinkedListDeque() {\n // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n DoublyListNode *pre, *cur = front;\n while (cur != nullptr) {\n pre = cur;\n cur = cur->next;\n delete pre;\n }\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n void push(int num, bool isFront) {\n DoublyListNode *node = new DoublyListNode(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (isEmpty())\n front = rear = node;\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if (isFront) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n front->prev = node;\n node->next = front;\n front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n rear->next = node;\n node->prev = rear;\n rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n void pushFirst(int num) {\n push(num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n void pushLast(int num) {\n push(num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n int pop(bool isFront) {\n if (isEmpty())\n throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n int val;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (isFront) {\n val = front->val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n DoublyListNode *fNext = front->next;\n if (fNext != nullptr) {\n fNext->prev = nullptr;\n front->next = nullptr;\n delete front;\n }\n front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n } else {\n val = rear->val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n DoublyListNode *rPrev = rear->prev;\n if (rPrev != nullptr) {\n rPrev->next = nullptr;\n rear->prev = nullptr;\n delete rear;\n }\n rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n int popFirst() {\n return pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n int popLast() {\n return pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peekFirst() {\n if (isEmpty())\n throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return front->val;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n int peekLast() {\n if (isEmpty())\n throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return rear->val;\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n vector<int> toVector() {\n DoublyListNode *node = front;\n vector<int> res(size());\n for (int i = 0; i < res.size(); i++) {\n res[i] = node->val;\n node = node->next;\n }\n return res;\n }\n};\n
linkedlist_deque.cs/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n ListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n ListNode(int val) {\n this.val = val;\n prev = next = null;\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n private int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n public LinkedListDeque() {\n front = rear = null;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n public int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n private void push(int num, boolean isFront) {\n ListNode node = new ListNode(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (isEmpty())\n front = rear = node;\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if (isFront) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n front.prev = node;\n node.next = front;\n front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n rear.next = node;\n node.prev = rear;\n rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n public void pushFirst(int num) {\n push(num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n public void pushLast(int num) {\n push(num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n private int pop(boolean isFront) {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n int val;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (isFront) {\n val = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n ListNode fNext = front.next;\n if (fNext != null) {\n fNext.prev = null;\n front.next = null;\n }\n front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n } else {\n val = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n ListNode rPrev = rear.prev;\n if (rPrev != null) {\n rPrev.next = null;\n rear.prev = null;\n }\n rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n public int popFirst() {\n return pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n public int popLast() {\n return pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int peekFirst() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return front.val;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n public int peekLast() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return rear.val;\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n public int[] toArray() {\n ListNode node = front;\n int[] res = new int[size()];\n for (int i = 0; i < res.length; i++) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_deque.go/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(int val) {\n public int val = val; // \u8282\u70b9\u503c\n public ListNode? next = null; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n public ListNode? prev = null; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n ListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\n int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n public LinkedListDeque() {\n front = null;\n rear = null;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n public int Size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return Size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n void Push(int num, bool isFront) {\n ListNode node = new(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (IsEmpty()) {\n front = node;\n rear = node;\n }\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if (isFront) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n front!.prev = node;\n node.next = front;\n front = node; // \u66f4\u65b0\u5934\u8282\u70b9 \n }\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n rear!.next = node;\n node.prev = rear;\n rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n\n queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n public void PushFirst(int num) {\n Push(num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n public void PushLast(int num) {\n Push(num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n int? Pop(bool isFront) {\n if (IsEmpty())\n throw new Exception();\n int? val;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (isFront) {\n val = front?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n ListNode? fNext = front?.next;\n if (fNext != null) {\n fNext.prev = null;\n front!.next = null;\n }\n front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else {\n val = rear?.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n ListNode? rPrev = rear?.prev;\n if (rPrev != null) {\n rPrev.next = null;\n rear!.prev = null;\n }\n rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n\n queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n public int? PopFirst() {\n return Pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n public int? PopLast() {\n return Pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int? PeekFirst() {\n if (IsEmpty())\n throw new Exception();\n return front?.val;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n public int? PeekLast() {\n if (IsEmpty())\n throw new Exception();\n return rear?.val;\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n public int?[] ToArray() {\n ListNode? node = front;\n int?[] res = new int?[Size()];\n for (int i = 0; i < res.Length; i++) {\n res[i] = node?.val;\n node = node?.next;\n }\n\n return res;\n }\n}\n
linkedlist_deque.swift/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n // \u4f7f\u7528\u5185\u7f6e\u5305 list\n data *list.List\n}\n\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\n return &linkedListDeque{\n data: list.New(),\n }\n}\n\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\n s.data.PushFront(value)\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\n s.data.PushBack(value)\n}\n\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Front()\n s.data.Remove(e)\n return e.Value\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Back()\n s.data.Remove(e)\n return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Front()\n return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Back()\n return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\n return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\n return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\n return s.data\n}\n
linkedlist_deque.js/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n var val: Int // \u8282\u70b9\u503c\n var next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n weak var prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n init(val: Int) {\n self.val = val\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n private var front: ListNode? // \u5934\u8282\u70b9 front\n private var rear: ListNode? // \u5c3e\u8282\u70b9 rear\n private var queSize: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n init() {\n queSize = 0\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n func size() -> Int {\n queSize\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n size() == 0\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n private func push(num: Int, isFront: Bool) {\n let node = ListNode(val: num)\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if isEmpty() {\n front = node\n rear = node\n }\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if isFront {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n front?.prev = node\n node.next = front\n front = node // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n rear?.next = node\n node.prev = rear\n rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n func pushFirst(num: Int) {\n push(num: num, isFront: true)\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n func pushLast(num: Int) {\n push(num: num, isFront: false)\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n private func pop(isFront: Bool) -> Int {\n if isEmpty() {\n fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n }\n let val: Int\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if isFront {\n val = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n let fNext = front?.next\n if fNext != nil {\n fNext?.prev = nil\n front?.next = nil\n }\n front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else {\n val = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n let rPrev = rear?.prev\n if rPrev != nil {\n rPrev?.next = nil\n rear?.prev = nil\n }\n rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val\n }\n\n /* \u961f\u9996\u51fa\u961f */\n func popFirst() -> Int {\n pop(isFront: true)\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n func popLast() -> Int {\n pop(isFront: false)\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n func peekFirst() -> Int? {\n isEmpty() ? nil : front?.val\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n func peekLast() -> Int? {\n isEmpty() ? nil : rear?.val\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n func toArray() -> [Int] {\n var node = front\n var res = Array(repeating: 0, count: size())\n for i in res.indices {\n res[i] = node!.val\n node = node?.next\n }\n return res\n }\n}\n
linkedlist_deque.ts/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n val; // \u8282\u70b9\u503c\n\n constructor(val) {\n this.val = val;\n this.next = null;\n this.prev = null;\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n #front; // \u5934\u8282\u70b9 front\n #rear; // \u5c3e\u8282\u70b9 rear\n #queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n constructor() {\n this.#front = null;\n this.#rear = null;\n this.#queSize = 0;\n }\n\n /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n pushLast(val) {\n const node = new ListNode(val);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (this.#queSize === 0) {\n this.#front = node;\n this.#rear = node;\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n this.#rear.next = node;\n node.prev = this.#rear;\n this.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n this.#queSize++;\n }\n\n /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n pushFirst(val) {\n const node = new ListNode(val);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (this.#queSize === 0) {\n this.#front = node;\n this.#rear = node;\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n this.#front.prev = node;\n node.next = this.#front;\n this.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n this.#queSize++;\n }\n\n /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n popLast() {\n if (this.#queSize === 0) {\n return null;\n }\n const value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n let temp = this.#rear.prev;\n if (temp !== null) {\n temp.next = null;\n this.#rear.prev = null;\n }\n this.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n this.#queSize--;\n return value;\n }\n\n /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n popFirst() {\n if (this.#queSize === 0) {\n return null;\n }\n const value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n let temp = this.#front.next;\n if (temp !== null) {\n temp.prev = null;\n this.#front.next = null;\n }\n this.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n this.#queSize--;\n return value;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n peekLast() {\n return this.#queSize === 0 ? null : this.#rear.val;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peekFirst() {\n return this.#queSize === 0 ? null : this.#front.val;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n size() {\n return this.#queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.#queSize === 0;\n }\n\n /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n print() {\n const arr = [];\n let temp = this.#front;\n while (temp !== null) {\n arr.push(temp.val);\n temp = temp.next;\n }\n console.log('[' + arr.join(', ') + ']');\n }\n}\n
linkedlist_deque.dart/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n prev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n next: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n val: number; // \u8282\u70b9\u503c\n\n constructor(val: number) {\n this.val = val;\n this.next = null;\n this.prev = null;\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n private front: ListNode; // \u5934\u8282\u70b9 front\n private rear: ListNode; // \u5c3e\u8282\u70b9 rear\n private queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n constructor() {\n this.front = null;\n this.rear = null;\n this.queSize = 0;\n }\n\n /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n pushLast(val: number): void {\n const node: ListNode = new ListNode(val);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (this.queSize === 0) {\n this.front = node;\n this.rear = node;\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n this.rear.next = node;\n node.prev = this.rear;\n this.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n this.queSize++;\n }\n\n /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n pushFirst(val: number): void {\n const node: ListNode = new ListNode(val);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (this.queSize === 0) {\n this.front = node;\n this.rear = node;\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n this.front.prev = node;\n node.next = this.front;\n this.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n this.queSize++;\n }\n\n /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n popLast(): number {\n if (this.queSize === 0) {\n return null;\n }\n const value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n let temp: ListNode = this.rear.prev;\n if (temp !== null) {\n temp.next = null;\n this.rear.prev = null;\n }\n this.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n this.queSize--;\n return value;\n }\n\n /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n popFirst(): number {\n if (this.queSize === 0) {\n return null;\n }\n const value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n let temp: ListNode = this.front.next;\n if (temp !== null) {\n temp.prev = null;\n this.front.next = null;\n }\n this.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n this.queSize--;\n return value;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n peekLast(): number {\n return this.queSize === 0 ? null : this.rear.val;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peekFirst(): number {\n return this.queSize === 0 ? null : this.front.val;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n size(): number {\n return this.queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.queSize === 0;\n }\n\n /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n print(): void {\n const arr: number[] = [];\n let temp: ListNode = this.front;\n while (temp !== null) {\n arr.push(temp.val);\n temp = temp.next;\n }\n console.log('[' + arr.join(', ') + ']');\n }\n}\n
linkedlist_deque.rs/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n ListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n ListNode(this.val, {this.next, this.prev});\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\n late ListNode? _front; // \u5934\u8282\u70b9 _front\n late ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n int _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n LinkedListDeque() {\n this._front = null;\n this._rear = null;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\n int size() {\n return this._queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n void push(int _num, bool isFront) {\n final ListNode node = ListNode(_num);\n if (isEmpty()) {\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n _front = _rear = node;\n } else if (isFront) {\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n _front!.prev = node;\n node.next = _front;\n _front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n } else {\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n _rear!.next = node;\n node.prev = _rear;\n _rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n _queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n void pushFirst(int _num) {\n push(_num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n void pushLast(int _num) {\n push(_num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n int? pop(bool isFront) {\n // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\n if (isEmpty()) {\n return null;\n }\n final int val;\n if (isFront) {\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n val = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n ListNode? fNext = _front!.next;\n if (fNext != null) {\n fNext.prev = null;\n _front!.next = null;\n }\n _front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n } else {\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n val = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n ListNode? rPrev = _rear!.prev;\n if (rPrev != null) {\n rPrev.next = null;\n _rear!.prev = null;\n }\n _rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n _queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n int? popFirst() {\n return pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n int? popLast() {\n return pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int? peekFirst() {\n return _front?.val;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n int? peekLast() {\n return _rear?.val;\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n List<int> toArray() {\n ListNode? node = _front;\n final List<int> res = [];\n for (int i = 0; i < _queSize; i++) {\n res.add(node!.val);\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_deque.c/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\npub struct ListNode<T> {\n pub val: T, // \u8282\u70b9\u503c\n pub next: Option<Rc<RefCell<ListNode<T>>>>, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n pub prev: Option<Rc<RefCell<ListNode<T>>>>, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n}\n\nimpl<T> ListNode<T> {\n pub fn new(val: T) -> Rc<RefCell<ListNode<T>>> {\n Rc::new(RefCell::new(ListNode {\n val,\n next: None,\n prev: None,\n }))\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque<T> {\n front: Option<Rc<RefCell<ListNode<T>>>>, // \u5934\u8282\u70b9 front\n rear: Option<Rc<RefCell<ListNode<T>>>>, // \u5c3e\u8282\u70b9 rear \n que_size: usize, // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl<T: Copy> LinkedListDeque<T> {\n pub fn new() -> Self {\n Self {\n front: None,\n rear: None,\n que_size: 0, \n }\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n pub fn size(&self) -> usize {\n return self.que_size;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n pub fn is_empty(&self) -> bool {\n return self.size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n pub fn push(&mut self, num: T, is_front: bool) {\n let node = ListNode::new(num);\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n if is_front {\n match self.front.take() {\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n None => {\n self.rear = Some(node.clone());\n self.front = Some(node);\n }\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n Some(old_front) => {\n old_front.borrow_mut().prev = Some(node.clone());\n node.borrow_mut().next = Some(old_front);\n self.front = Some(node); // \u66f4\u65b0\u5934\u8282\u70b9\n }\n }\n } \n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else {\n match self.rear.take() {\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n None => {\n self.front = Some(node.clone());\n self.rear = Some(node);\n }\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n Some(old_rear) => {\n old_rear.borrow_mut().next = Some(node.clone());\n node.borrow_mut().prev = Some(old_rear);\n self.rear = Some(node); // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n }\n }\n self.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n pub fn push_first(&mut self, num: T) {\n self.push(num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n pub fn push_last(&mut self, num: T) {\n self.push(num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n pub fn pop(&mut self, is_front: bool) -> Option<T> {\n // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\n if self.is_empty() { \n return None \n };\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if is_front {\n self.front.take().map(|old_front| {\n match old_front.borrow_mut().next.take() {\n Some(new_front) => {\n new_front.borrow_mut().prev.take();\n self.front = Some(new_front); // \u66f4\u65b0\u5934\u8282\u70b9\n }\n None => {\n self.rear.take();\n }\n }\n self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n })\n\n } \n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else {\n self.rear.take().map(|old_rear| {\n match old_rear.borrow_mut().prev.take() {\n Some(new_rear) => {\n new_rear.borrow_mut().next.take();\n self.rear = Some(new_rear); // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n None => {\n self.front.take();\n }\n }\n self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n Rc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n })\n }\n }\n\n /* \u961f\u9996\u51fa\u961f */\n pub fn pop_first(&mut self) -> Option<T> {\n return self.pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n pub fn pop_last(&mut self) -> Option<T> {\n return self.pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n pub fn peek_first(&self) -> Option<&Rc<RefCell<ListNode<T>>>> {\n self.front.as_ref()\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n pub fn peek_last(&self) -> Option<&Rc<RefCell<ListNode<T>>>> {\n self.rear.as_ref()\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n pub fn to_array(&self, head: Option<&Rc<RefCell<ListNode<T>>>>) -> Vec<T> {\n if let Some(node) = head {\n let mut nums = self.to_array(node.borrow().next.as_ref());\n nums.insert(0, node.borrow().val);\n return nums;\n }\n return Vec::new();\n }\n}\n
linkedlist_deque.zig/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\ntypedef struct DoublyListNode {\n int val; // \u8282\u70b9\u503c\n struct DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\n struct DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n} DoublyListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nDoublyListNode *newDoublyListNode(int num) {\n DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n new->val = num;\n new->next = NULL;\n new->prev = NULL;\n return new;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(DoublyListNode *node) {\n free(node);\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n int queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n} LinkedListDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListDeque *newLinkedListDeque() {\n LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n deque->front = NULL;\n deque->rear = NULL;\n deque->queSize = 0;\n return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n // \u91ca\u653e\u6240\u6709\u8282\u70b9\n for (int i = 0; i < deque->queSize && deque->front != NULL; i++) {\n DoublyListNode *tmp = deque->front;\n deque->front = deque->front->next;\n free(tmp);\n }\n // \u91ca\u653e deque \u7ed3\u6784\u4f53\n free(deque);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListDeque *deque) {\n return deque->queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListDeque *deque) {\n return (size(deque) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n DoublyListNode *node = newDoublyListNode(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411node\n if (empty(deque)) {\n deque->front = deque->rear = node;\n }\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if (isFront) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n deque->front->prev = node;\n node->next = deque->front;\n deque->front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n deque->rear->next = node;\n node->prev = deque->rear;\n deque->rear = node;\n }\n deque->queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n push(deque, num, true);\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(LinkedListDeque *deque, int num) {\n push(deque, num, false);\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\n assert(size(deque) && deque->front);\n return deque->front->val;\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\n assert(size(deque) && deque->rear);\n return deque->rear->val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListDeque *deque, bool isFront) {\n if (empty(deque))\n return -1;\n int val;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (isFront) {\n val = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\n DoublyListNode *fNext = deque->front->next;\n if (fNext) {\n fNext->prev = NULL;\n deque->front->next = NULL;\n delDoublyListNode(deque->front);\n }\n deque->front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else {\n val = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n DoublyListNode *rPrev = deque->rear->prev;\n if (rPrev) {\n rPrev->next = NULL;\n deque->rear->prev = NULL;\n delDoublyListNode(deque->rear);\n }\n deque->rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n deque->queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(LinkedListDeque *deque) {\n return pop(deque, true);\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(LinkedListDeque *deque) {\n return pop(deque, false);\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n int *arr = malloc(sizeof(int) * deque->queSize);\n // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n int i;\n DoublyListNode *node;\n for (i = 0, node = deque->front; i < deque->queSize; i++) {\n arr[i] = node->val;\n node = node->next;\n }\n printArray(arr, deque->queSize);\n free(arr);\n}\n
"},{"location":"chapter_stack_and_queue/deque/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"// \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\n return struct {\n const Self = @This();\n\n val: T = undefined, // \u8282\u70b9\u503c\n next: ?*Self = null, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n prev: ?*Self = null, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n\n // Initialize a list node with specific value\n pub fn init(self: *Self, x: i32) void {\n self.val = x;\n self.next = null;\n self.prev = null;\n }\n };\n}\n\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\n return struct {\n const Self = @This();\n\n front: ?*ListNode(T) = null, // \u5934\u8282\u70b9 front\n rear: ?*ListNode(T) = null, // \u5c3e\u8282\u70b9 rear\n que_size: usize = 0, // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.front = null;\n self.rear = null;\n self.que_size = 0;\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.que_size;\n }\n\n // \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.size() == 0;\n }\n\n // \u5165\u961f\u64cd\u4f5c\n pub fn push(self: *Self, num: T, is_front: bool) !void {\n var node = try self.mem_allocator.create(ListNode(T));\n node.init(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (self.isEmpty()) {\n self.front = node;\n self.rear = node;\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n } else if (is_front) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n self.front.?.prev = node;\n node.next = self.front;\n self.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n self.rear.?.next = node;\n node.prev = self.rear;\n self.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n self.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n } \n\n // \u961f\u9996\u5165\u961f\n pub fn pushFirst(self: *Self, num: T) !void {\n try self.push(num, true);\n } \n\n // \u961f\u5c3e\u5165\u961f\n pub fn pushLast(self: *Self, num: T) !void {\n try self.push(num, false);\n } \n\n // \u51fa\u961f\u64cd\u4f5c\n pub fn pop(self: *Self, is_front: bool) T {\n if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n var val: T = undefined;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (is_front) {\n val = self.front.?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n var fNext = self.front.?.next;\n if (fNext != null) {\n fNext.?.prev = null;\n self.front.?.next = null;\n }\n self.front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n } else {\n val = self.rear.?.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n var rPrev = self.rear.?.prev;\n if (rPrev != null) {\n rPrev.?.next = null;\n self.rear.?.prev = null;\n }\n self.rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n } \n\n // \u961f\u9996\u51fa\u961f\n pub fn popFirst(self: *Self) T {\n return self.pop(true);\n } \n\n // \u961f\u5c3e\u51fa\u961f\n pub fn popLast(self: *Self) T {\n return self.pop(false);\n } \n\n // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n pub fn peekFirst(self: *Self) T {\n if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return self.front.?.val;\n } \n\n // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\n pub fn peekLast(self: *Self) T {\n if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return self.rear.?.val;\n }\n\n // \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\n pub fn toArray(self: *Self) ![]T {\n var node = self.front;\n var res = try self.mem_allocator.alloc(T, self.size());\n @memset(res, @as(T, 0));\n var i: usize = 0;\n while (i < res.len) : (i += 1) {\n res[i] = node.?.val;\n node = node.?.next;\n }\n return res;\n }\n };\n}\n\u5982\u56fe 5-9 \u6240\u793a\uff0c\u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002
ArrayDequepushLast()pushFirst()popLast()popFirst()\u56fe 5-9 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c
\u5728\u961f\u5217\u7684\u5b9e\u73b0\u57fa\u7840\u4e0a\uff0c\u4ec5\u9700\u589e\u52a0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u7684\u65b9\u6cd5\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array_deque.py
array_deque.cppclass ArrayDeque:\n \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n def __init__(self, capacity: int):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._nums: list[int] = [0] * capacity\n self._front: int = 0\n self._size: int = 0\n\n def capacity(self) -> int:\n \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n return len(self._nums)\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n return self._size == 0\n\n def index(self, i: int) -> int:\n \"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + self.capacity()) % self.capacity()\n\n def push_first(self, num: int):\n \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n if self._size == self.capacity():\n print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n # \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n self._front = self.index(self._front - 1)\n # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n self._nums[self._front] = num\n self._size += 1\n\n def push_last(self, num: int):\n \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n if self._size == self.capacity():\n print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n # \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n rear = self.index(self._front + self._size)\n # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n self._nums[rear] = num\n self._size += 1\n\n def pop_first(self) -> int:\n \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n num = self.peek_first()\n # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n self._front = self.index(self._front + 1)\n self._size -= 1\n return num\n\n def pop_last(self) -> int:\n \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n num = self.peek_last()\n self._size -= 1\n return num\n\n def peek_first(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n return self._nums[self._front]\n\n def peek_last(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n # \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n last = self.index(self._front + self._size - 1)\n return self._nums[last]\n\n def to_array(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n res = []\n for i in range(self._size):\n res.append(self._nums[self.index(self._front + i)])\n return res\n
array_deque.java/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n private:\n vector<int> nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n ArrayDeque(int capacity) {\n nums.resize(capacity);\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n int capacity() {\n return nums.size();\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return queSize == 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n int index(int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + capacity()) % capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n void pushFirst(int num) {\n if (queSize == capacity()) {\n cout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n front = index(front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n nums[front] = num;\n queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n void pushLast(int num) {\n if (queSize == capacity()) {\n cout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = index(front + queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n int popFirst() {\n int num = peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n front = index(front + 1);\n queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n int popLast() {\n int num = peekLast();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peekFirst() {\n if (isEmpty())\n throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return nums[front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n int peekLast() {\n if (isEmpty())\n throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n int last = index(front + queSize - 1);\n return nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n vector<int> toVector() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n vector<int> res(queSize);\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[index(j)];\n }\n return res;\n }\n};\n
array_deque.cs/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public ArrayDeque(int capacity) {\n this.nums = new int[capacity];\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n public int capacity() {\n return nums.length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n public int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return queSize == 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n private int index(int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + capacity()) % capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n public void pushFirst(int num) {\n if (queSize == capacity()) {\n System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n front = index(front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n nums[front] = num;\n queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n public void pushLast(int num) {\n if (queSize == capacity()) {\n System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = index(front + queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n public int popFirst() {\n int num = peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n front = index(front + 1);\n queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n public int popLast() {\n int num = peekLast();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int peekFirst() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return nums[front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n public int peekLast() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n int last = index(front + queSize - 1);\n return nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n public int[] toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] res = new int[queSize];\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[index(j)];\n }\n return res;\n }\n}\n
array_deque.go/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public ArrayDeque(int capacity) {\n nums = new int[capacity];\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n int Capacity() {\n return nums.Length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n public int Size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return queSize == 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n int Index(int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + Capacity()) % Capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n public void PushFirst(int num) {\n if (queSize == Capacity()) {\n Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n front = Index(front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n nums[front] = num;\n queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n public void PushLast(int num) {\n if (queSize == Capacity()) {\n Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = Index(front + queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n public int PopFirst() {\n int num = PeekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n front = Index(front + 1);\n queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n public int PopLast() {\n int num = PeekLast();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int PeekFirst() {\n if (IsEmpty()) {\n throw new InvalidOperationException();\n }\n return nums[front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n public int PeekLast() {\n if (IsEmpty()) {\n throw new InvalidOperationException();\n }\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n int last = Index(front + queSize - 1);\n return nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n public int[] ToArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] res = new int[queSize];\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[Index(j)];\n }\n return res;\n }\n}\n
array_deque.swift/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\n nums []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n front int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n queSize int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n queCapacity int // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n return &arrayDeque{\n nums: make([]int, queCapacity),\n queCapacity: queCapacity,\n front: 0,\n queSize: 0,\n }\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\n return q.queSize\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\n return q.queSize == 0\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + q.queCapacity) % q.queCapacity\n}\n\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\n if q.queSize == q.queCapacity {\n fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n q.front = q.index(q.front - 1)\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n q.nums[q.front] = num\n q.queSize++\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\n if q.queSize == q.queCapacity {\n fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n rear := q.index(q.front + q.queSize)\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n q.nums[rear] = num\n q.queSize++\n}\n\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\n num := q.peekFirst()\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n q.front = q.index(q.front + 1)\n q.queSize--\n return num\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\n num := q.peekLast()\n q.queSize--\n return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\n if q.isEmpty() {\n return nil\n }\n return q.nums[q.front]\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\n if q.isEmpty() {\n return nil\n }\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n last := q.index(q.front + q.queSize - 1)\n return q.nums[last]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n res := make([]int, q.queSize)\n for i, j := 0, q.front; i < q.queSize; i++ {\n res[i] = q.nums[q.index(j)]\n j++\n }\n return res\n}\n
array_deque.js/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private var queSize: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init(capacity: Int) {\n nums = Array(repeating: 0, count: capacity)\n front = 0\n queSize = 0\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n func capacity() -> Int {\n nums.count\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n func size() -> Int {\n queSize\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n size() == 0\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n private func index(i: Int) -> Int {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n (i + capacity()) % capacity()\n }\n\n /* \u961f\u9996\u5165\u961f */\n func pushFirst(num: Int) {\n if size() == capacity() {\n print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n front = index(i: front - 1)\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n nums[front] = num\n queSize += 1\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n func pushLast(num: Int) {\n if size() == capacity() {\n print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n let rear = index(i: front + size())\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num\n queSize += 1\n }\n\n /* \u961f\u9996\u51fa\u961f */\n func popFirst() -> Int {\n let num = peekFirst()\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n front = index(i: front + 1)\n queSize -= 1\n return num\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n func popLast() -> Int {\n let num = peekLast()\n queSize -= 1\n return num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n func peekFirst() -> Int {\n if isEmpty() {\n fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n }\n return nums[front]\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n func peekLast() -> Int {\n if isEmpty() {\n fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n }\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n let last = index(i: front + size() - 1)\n return nums[last]\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n func toArray() -> [Int] {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n var res = Array(repeating: 0, count: size())\n for (i, j) in sequence(first: (0, front), next: { $0 < self.size() - 1 ? ($0 + 1, $1 + 1) : nil }) {\n res[i] = nums[index(i: j)]\n }\n return res\n }\n}\n
array_deque.ts/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n #nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n #front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n #queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(capacity) {\n this.#nums = new Array(capacity);\n this.#front = 0;\n this.#queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n capacity() {\n return this.#nums.length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n size() {\n return this.#queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.#queSize === 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n index(i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + this.capacity()) % this.capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n pushFirst(num) {\n if (this.#queSize === this.capacity()) {\n console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n this.#front = this.index(this.#front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n this.#nums[this.#front] = num;\n this.#queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n pushLast(num) {\n if (this.#queSize === this.capacity()) {\n console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n const rear = this.index(this.#front + this.#queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n this.#nums[rear] = num;\n this.#queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n popFirst() {\n const num = this.peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n this.#front = this.index(this.#front + 1);\n this.#queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n popLast() {\n const num = this.peekLast();\n this.#queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peekFirst() {\n if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n return this.#nums[this.#front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n peekLast() {\n if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n const last = this.index(this.#front + this.#queSize - 1);\n return this.#nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const res = [];\n for (let i = 0, j = this.#front; i < this.#queSize; i++, j++) {\n res[i] = this.#nums[this.index(j)];\n }\n return res;\n }\n}\n
array_deque.dart/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(capacity: number) {\n this.nums = new Array(capacity);\n this.front = 0;\n this.queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n capacity(): number {\n return this.nums.length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n size(): number {\n return this.queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.queSize === 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n index(i: number): number {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + this.capacity()) % this.capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n pushFirst(num: number): void {\n if (this.queSize === this.capacity()) {\n console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n this.front = this.index(this.front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n this.nums[this.front] = num;\n this.queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n pushLast(num: number): void {\n if (this.queSize === this.capacity()) {\n console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n const rear: number = this.index(this.front + this.queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n this.nums[rear] = num;\n this.queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n popFirst(): number {\n const num: number = this.peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n this.front = this.index(this.front + 1);\n this.queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n popLast(): number {\n const num: number = this.peekLast();\n this.queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peekFirst(): number {\n if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n return this.nums[this.front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n peekLast(): number {\n if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n const last = this.index(this.front + this.queSize - 1);\n return this.nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n toArray(): number[] {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const res: number[] = [];\n for (let i = 0, j = this.front; i < this.queSize; i++, j++) {\n res[i] = this.nums[this.index(j)];\n }\n return res;\n }\n}\n
array_deque.rs/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n late List<int> _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n late int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n ArrayDeque(int capacity) {\n this._nums = List.filled(capacity, 0);\n this._front = this._queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n int capacity() {\n return _nums.length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return _queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _queSize == 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n int index(int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + capacity()) % capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n void pushFirst(int _num) {\n if (_queSize == capacity()) {\n throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n _front = index(_front - 1);\n // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u9996\n _nums[_front] = _num;\n _queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n void pushLast(int _num) {\n if (_queSize == capacity()) {\n throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = index(_front + _queSize);\n // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n _nums[rear] = _num;\n _queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n int popFirst() {\n int _num = peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n _front = index(_front + 1);\n _queSize--;\n return _num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n int popLast() {\n int _num = peekLast();\n _queSize--;\n return _num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peekFirst() {\n if (isEmpty()) {\n throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n }\n return _nums[_front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n int peekLast() {\n if (isEmpty()) {\n throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n }\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n int last = index(_front + _queSize - 1);\n return _nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n List<int> toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n List<int> res = List.filled(_queSize, 0);\n for (int i = 0, j = _front; i < _queSize; i++, j++) {\n res[i] = _nums[index(j)];\n }\n return res;\n }\n}\n
array_deque.c/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct ArrayDeque {\n nums: Vec<i32>, // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n front: usize, // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n que_size: usize, // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n}\n\nimpl ArrayDeque {\n /* \u6784\u9020\u65b9\u6cd5 */\n pub fn new(capacity: usize) -> Self {\n Self {\n nums: vec![0; capacity],\n front: 0,\n que_size: 0,\n }\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n pub fn capacity(&self) -> usize {\n self.nums.len()\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n pub fn size(&self) -> usize {\n self.que_size\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n pub fn is_empty(&self) -> bool {\n self.que_size == 0\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n fn index(&self, i: i32) -> usize {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n }\n\n /* \u961f\u9996\u5165\u961f */\n pub fn push_first(&mut self, num: i32) {\n if self.que_size == self.capacity() {\n println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n self.front = self.index(self.front as i32 - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n self.nums[self.front] = num;\n self.que_size += 1;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n pub fn push_last(&mut self, num: i32) {\n if self.que_size == self.capacity() {\n println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n let rear = self.index(self.front as i32 + self.que_size as i32);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n self.nums[rear] = num;\n self.que_size += 1;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n fn pop_first(&mut self) -> i32 {\n let num = self.peek_first();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n self.front = self.index(self.front as i32 + 1);\n self.que_size -= 1;\n num\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n fn pop_last(&mut self) -> i32 {\n let num = self.peek_last();\n self.que_size -= 1;\n num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n fn peek_first(&self) -> i32 {\n if self.is_empty() { panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\") };\n self.nums[self.front]\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n fn peek_last(&self) -> i32 {\n if self.is_empty() { panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\") };\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n self.nums[last]\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n fn to_array(&self) -> Vec<i32> {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n let mut res = vec![0; self.que_size];\n let mut j = self.front;\n for i in 0..self.que_size {\n res[i] = self.nums[self.index(j as i32)];\n j += 1;\n }\n res\n }\n}\n
array_deque.zig/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n int *nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayDeque *newArrayDeque(int capacity) {\n ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n // \u521d\u59cb\u5316\u6570\u7ec4\n deque->queCapacity = capacity;\n deque->nums = (int *)malloc(sizeof(int) * deque->queCapacity);\n deque->front = deque->queSize = 0;\n return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(ArrayDeque *deque) {\n free(deque->nums);\n free(deque);\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\n return deque->queCapacity;\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayDeque *deque) {\n return deque->queSize;\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayDeque *deque) {\n return deque->queSize == 0;\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(ArrayDeque *deque, int num) {\n if (deque->queSize == capacity(deque)) {\n printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\n deque->front = dequeIndex(deque, deque->front - 1);\n // \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\n deque->nums[deque->front] = num;\n deque->queSize++;\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(ArrayDeque *deque, int num) {\n if (deque->queSize == capacity(deque)) {\n printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = dequeIndex(deque, deque->front + deque->queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n deque->nums[rear] = num;\n deque->queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n assert(empty(deque) == 0);\n return deque->nums[deque->front];\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n assert(empty(deque) == 0);\n int last = dequeIndex(deque, deque->front + deque->queSize - 1);\n return deque->nums[last];\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(ArrayDeque *deque) {\n int num = peekFirst(deque);\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n deque->front = dequeIndex(deque, deque->front + 1);\n deque->queSize--;\n return num;\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(ArrayDeque *deque) {\n int num = peekLast(deque);\n deque->queSize--;\n return num;\n}\n
"},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3 \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"[class]{ArrayDeque}-[func]{}\n\u53cc\u5411\u961f\u5217\u517c\u5177\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5b9e\u73b0\u8fd9\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\u573a\u666f\uff0c\u540c\u65f6\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002
\u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1a\u7cfb\u7edf\u5c06\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c
"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 \u961f\u5217","text":"push\u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7pop\u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u7684\u9650\u5236\uff0c\u8f6f\u4ef6\u901a\u5e38\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\u3002\u5f53\u6808\u7684\u957f\u5ea6\u8d85\u8fc7 \\(50\\) \u65f6\uff0c\u8f6f\u4ef6\u9700\u8981\u5728\u6808\u5e95\uff08\u961f\u9996\uff09\u6267\u884c\u5220\u9664\u64cd\u4f5c\u3002\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\u8be5\u529f\u80fd\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u8bf7\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u9075\u5faa\u6808\u7684\u5148\u5165\u540e\u51fa\u539f\u5219\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u80fd\u591f\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u903b\u8f91\u3002\u300c\u961f\u5217 queue\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u4e86\u6392\u961f\u73b0\u8c61\uff0c\u5373\u65b0\u6765\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u5c3e\u90e8\uff0c\u800c\u4f4d\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u9010\u4e2a\u79bb\u5f00\u3002
\u5982\u56fe 5-4 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u961f\u5217\u5934\u90e8\u79f0\u4e3a\u201c\u961f\u9996\u201d\uff0c\u5c3e\u90e8\u79f0\u4e3a\u201c\u961f\u5c3e\u201d\uff0c\u5c06\u628a\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u5165\u961f\u201d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u51fa\u961f\u201d\u3002
\u56fe 5-4 \u00a0 \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219
"},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1 \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"\u961f\u5217\u7684\u5e38\u89c1\u64cd\u4f5c\u5982\u8868 5-2 \u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u79f0\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u6211\u4eec\u5728\u6b64\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002
\u8868 5-2 \u00a0 \u961f\u5217\u64cd\u4f5c\u6548\u7387
\u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) pop() \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) peek() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\)\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u73b0\u6210\u7684\u961f\u5217\u7c7b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig queue.py
queue.cppfrom collections import deque\n\n# \u521d\u59cb\u5316\u961f\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u5f53\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u63a8\u8350\nque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# \u8bbf\u95ee\u961f\u9996\u5143\u7d20\nfront: int = que[0];\n\n# \u5143\u7d20\u51fa\u961f\npop: int = que.popleft()\n\n# \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(que)\n\n# \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(que) == 0\n
queue.java/* \u521d\u59cb\u5316\u961f\u5217 */\nqueue<int> queue;\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n
queue.cs/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<Integer> queue = new LinkedList<>();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n
queue_test.go/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<int> queue = new();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count == 0;\n
queue.swift/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n
queue.js/* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n
queue.ts/* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
queue.dart/* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
queue.rs/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u961f\u5217\u7c7b Qeque \u662f\u53cc\u5411\u961f\u5217\uff0c\u4e5f\u53ef\u4f5c\u4e3a\u961f\u5217\u4f7f\u7528\nQueue<int> queue = Queue();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.first;\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.removeFirst();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.isEmpty;\n
queue.c/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Rust \u4e2d\u4f7f\u7528\u53cc\u5411\u961f\u5217\u4f5c\u4e3a\u666e\u901a\u961f\u5217\u6765\u4f7f\u7528\nlet mut deque: VecDeque<u32> = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nif let Some(front) = deque.front() {\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n
queue.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u961f\u5217\n
"},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2 \u00a0 \u961f\u5217\u5b9e\u73b0","text":"\n\u4e3a\u4e86\u5b9e\u73b0\u961f\u5217\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\u5143\u7d20\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u5143\u7d20\u3002\u94fe\u8868\u548c\u6570\u7ec4\u90fd\u7b26\u5408\u8981\u6c42\u3002
"},{"location":"chapter_stack_and_queue/queue/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"\u5982\u56fe 5-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u201c\u5934\u8282\u70b9\u201d\u548c\u201c\u5c3e\u8282\u70b9\u201d\u5206\u522b\u89c6\u4e3a\u201c\u961f\u9996\u201d\u548c\u201c\u961f\u5c3e\u201d\uff0c\u89c4\u5b9a\u961f\u5c3e\u4ec5\u53ef\u6dfb\u52a0\u8282\u70b9\uff0c\u961f\u9996\u4ec5\u53ef\u5220\u9664\u8282\u70b9\u3002
LinkedListQueuepush()pop()\u56fe 5-5 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c
\u4ee5\u4e0b\u662f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_queue.py
linkedlist_queue.cppclass LinkedListQueue:\n \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._front: ListNode | None = None # \u5934\u8282\u70b9 front\n self._rear: ListNode | None = None # \u5c3e\u8282\u70b9 rear\n self._size: int = 0\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n return not self._front\n\n def push(self, num: int):\n \"\"\"\u5165\u961f\"\"\"\n # \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n node = ListNode(num)\n # \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if self._front is None:\n self._front = node\n self._rear = node\n # \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n else:\n self._rear.next = node\n self._rear = node\n self._size += 1\n\n def pop(self) -> int:\n \"\"\"\u51fa\u961f\"\"\"\n num = self.peek()\n # \u5220\u9664\u5934\u8282\u70b9\n self._front = self._front.next\n self._size -= 1\n return num\n\n def peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n return self._front.val\n\n def to_list(self) -> list[int]:\n \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n queue = []\n temp = self._front\n while temp:\n queue.append(temp.val)\n temp = temp.next\n return queue\n
linkedlist_queue.java/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n private:\n ListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n int queSize;\n\n public:\n LinkedListQueue() {\n front = nullptr;\n rear = nullptr;\n queSize = 0;\n }\n\n ~LinkedListQueue() {\n // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n freeMemoryLinkedList(front);\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return queSize == 0;\n }\n\n /* \u5165\u961f */\n void push(int num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n ListNode *node = new ListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (front == nullptr) {\n front = node;\n rear = node;\n }\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n else {\n rear->next = node;\n rear = node;\n }\n queSize++;\n }\n\n /* \u51fa\u961f */\n int pop() {\n int num = peek();\n // \u5220\u9664\u5934\u8282\u70b9\n ListNode *tmp = front;\n front = front->next;\n // \u91ca\u653e\u5185\u5b58\n delete tmp;\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peek() {\n if (size() == 0)\n throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n return front->val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n vector<int> toVector() {\n ListNode *node = front;\n vector<int> res(size());\n for (int i = 0; i < res.size(); i++) {\n res[i] = node->val;\n node = node->next;\n }\n return res;\n }\n};\n
linkedlist_queue.cs/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n private int queSize = 0;\n\n public LinkedListQueue() {\n front = null;\n rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n public int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f */\n public void push(int num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n ListNode node = new ListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (front == null) {\n front = node;\n rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else {\n rear.next = node;\n rear = node;\n }\n queSize++;\n }\n\n /* \u51fa\u961f */\n public int pop() {\n int num = peek();\n // \u5220\u9664\u5934\u8282\u70b9\n front = front.next;\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int peek() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return front.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] toArray() {\n ListNode node = front;\n int[] res = new int[size()];\n for (int i = 0; i < res.length; i++) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_queue.go/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n ListNode? front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \n int queSize = 0;\n\n public LinkedListQueue() {\n front = null;\n rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n public int Size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return Size() == 0;\n }\n\n /* \u5165\u961f */\n public void Push(int num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n ListNode node = new(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (front == null) {\n front = node;\n rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else if (rear != null) {\n rear.next = node;\n rear = node;\n }\n queSize++;\n }\n\n /* \u51fa\u961f */\n public int Pop() {\n int num = Peek();\n // \u5220\u9664\u5934\u8282\u70b9\n front = front?.next;\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int Peek() {\n if (IsEmpty())\n throw new Exception();\n return front!.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] ToArray() {\n if (front == null)\n return [];\n\n ListNode? node = front;\n int[] res = new int[Size()];\n for (int i = 0; i < res.Length; i++) {\n res[i] = node!.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_queue.swift/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\n data *list.List\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\n return &linkedListQueue{\n data: list.New(),\n }\n}\n\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\n s.data.PushBack(value)\n}\n\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Front()\n s.data.Remove(e)\n return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Front()\n return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\n return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\n return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\n return s.data\n}\n
linkedlist_queue.js/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n private var front: ListNode? // \u5934\u8282\u70b9\n private var rear: ListNode? // \u5c3e\u8282\u70b9\n private var _size = 0\n\n init() {}\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n func size() -> Int {\n _size\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n size() == 0\n }\n\n /* \u5165\u961f */\n func push(num: Int) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n let node = ListNode(x: num)\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if front == nil {\n front = node\n rear = node\n }\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n else {\n rear?.next = node\n rear = node\n }\n _size += 1\n }\n\n /* \u51fa\u961f */\n @discardableResult\n func pop() -> Int {\n let num = peek()\n // \u5220\u9664\u5934\u8282\u70b9\n front = front?.next\n _size -= 1\n return num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n func peek() -> Int {\n if isEmpty() {\n fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n }\n return front!.val\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n func toArray() -> [Int] {\n var node = front\n var res = Array(repeating: 0, count: size())\n for i in res.indices {\n res[i] = node!.val\n node = node?.next\n }\n return res\n }\n}\n
linkedlist_queue.ts/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n #front; // \u5934\u8282\u70b9 #front\n #rear; // \u5c3e\u8282\u70b9 #rear\n #queSize = 0;\n\n constructor() {\n this.#front = null;\n this.#rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n get size() {\n return this.#queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.size === 0;\n }\n\n /* \u5165\u961f */\n push(num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n const node = new ListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (!this.#front) {\n this.#front = node;\n this.#rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else {\n this.#rear.next = node;\n this.#rear = node;\n }\n this.#queSize++;\n }\n\n /* \u51fa\u961f */\n pop() {\n const num = this.peek();\n // \u5220\u9664\u5934\u8282\u70b9\n this.#front = this.#front.next;\n this.#queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peek() {\n if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n return this.#front.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n toArray() {\n let node = this.#front;\n const res = new Array(this.size);\n for (let i = 0; i < res.length; i++) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_queue.dart/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n private front: ListNode | null; // \u5934\u8282\u70b9 front\n private rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\n private queSize: number = 0;\n\n constructor() {\n this.front = null;\n this.rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n get size(): number {\n return this.queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.size === 0;\n }\n\n /* \u5165\u961f */\n push(num: number): void {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n const node = new ListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (!this.front) {\n this.front = node;\n this.rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else {\n this.rear!.next = node;\n this.rear = node;\n }\n this.queSize++;\n }\n\n /* \u51fa\u961f */\n pop(): number {\n const num = this.peek();\n if (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n // \u5220\u9664\u5934\u8282\u70b9\n this.front = this.front.next;\n this.queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peek(): number {\n if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n return this.front!.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n toArray(): number[] {\n let node = this.front;\n const res = new Array<number>(this.size);\n for (let i = 0; i < res.length; i++) {\n res[i] = node!.val;\n node = node!.next;\n }\n return res;\n }\n}\n
linkedlist_queue.rs/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n ListNode? _front; // \u5934\u8282\u70b9 _front\n ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n int _queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n LinkedListQueue() {\n _front = null;\n _rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return _queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _queSize == 0;\n }\n\n /* \u5165\u961f */\n void push(int _num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 _num\n final node = ListNode(_num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (_front == null) {\n _front = node;\n _rear = node;\n } else {\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n _rear!.next = node;\n _rear = node;\n }\n _queSize++;\n }\n\n /* \u51fa\u961f */\n int pop() {\n final int _num = peek();\n // \u5220\u9664\u5934\u8282\u70b9\n _front = _front!.next;\n _queSize--;\n return _num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peek() {\n if (_queSize == 0) {\n throw Exception('\u961f\u5217\u4e3a\u7a7a');\n }\n return _front!.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n List<int> toArray() {\n ListNode? node = _front;\n final List<int> queue = [];\n while (node != null) {\n queue.add(node.val);\n node = node.next;\n }\n return queue;\n }\n}\n
linkedlist_queue.c/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue<T> {\n front: Option<Rc<RefCell<ListNode<T>>>>, // \u5934\u8282\u70b9 front\n rear: Option<Rc<RefCell<ListNode<T>>>>, // \u5c3e\u8282\u70b9 rear \n que_size: usize, // \u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl<T: Copy> LinkedListQueue<T> {\n pub fn new() -> Self {\n Self {\n front: None,\n rear: None,\n que_size: 0, \n }\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n pub fn size(&self) -> usize {\n return self.que_size;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n pub fn is_empty(&self) -> bool {\n return self.size() == 0;\n }\n\n /* \u5165\u961f */\n pub fn push(&mut self, num: T) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n let new_rear = ListNode::new(num);\n match self.rear.take() {\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n Some(old_rear) => {\n old_rear.borrow_mut().next = Some(new_rear.clone());\n self.rear = Some(new_rear);\n }\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n None => {\n self.front = Some(new_rear.clone());\n self.rear = Some(new_rear);\n }\n }\n self.que_size += 1;\n }\n\n /* \u51fa\u961f */\n pub fn pop(&mut self) -> Option<T> {\n self.front.take().map(|old_front| {\n match old_front.borrow_mut().next.take() {\n Some(new_front) => {\n self.front = Some(new_front);\n }\n None => {\n self.rear.take();\n }\n }\n self.que_size -= 1;\n Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n })\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n pub fn peek(&self) -> Option<&Rc<RefCell<ListNode<T>>>> {\n self.front.as_ref()\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n pub fn to_array(&self, head: Option<&Rc<RefCell<ListNode<T>>>>) -> Vec<T> {\n if let Some(node) = head {\n let mut nums = self.to_array(node.borrow().next.as_ref());\n nums.insert(0, node.borrow().val);\n return nums;\n }\n return Vec::new();\n }\n}\n
linkedlist_queue.zig/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n ListNode *front, *rear;\n int queSize;\n} LinkedListQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListQueue *newLinkedListQueue() {\n LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n queue->front = NULL;\n queue->rear = NULL;\n queue->queSize = 0;\n return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n // \u91ca\u653e\u6240\u6709\u8282\u70b9\n for (int i = 0; i < queue->queSize && queue->front != NULL; i++) {\n ListNode *tmp = queue->front;\n queue->front = queue->front->next;\n free(tmp);\n }\n // \u91ca\u653e queue \u7ed3\u6784\u4f53\n free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListQueue *queue) {\n return queue->queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListQueue *queue) {\n return (size(queue) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListQueue *queue, int num) {\n // \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\n ListNode *node = newListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (queue->front == NULL) {\n queue->front = node;\n queue->rear = node;\n }\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n else {\n queue->rear->next = node;\n queue->rear = node;\n }\n queue->queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\n assert(size(queue) && queue->front);\n return queue->front->val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListQueue *queue) {\n int num = peek(queue);\n ListNode *tmp = queue->front;\n queue->front = queue->front->next;\n free(tmp);\n queue->queSize--;\n return num;\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n int *arr = malloc(sizeof(int) * queue->queSize);\n // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n int i;\n ListNode *node;\n for (i = 0, node = queue->front; i < queue->queSize; i++) {\n arr[i] = node->val;\n node = node->next;\n }\n printArray(arr, queue->queSize);\n free(arr);\n}\n
"},{"location":"chapter_stack_and_queue/queue/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\n return struct {\n const Self = @This();\n\n front: ?*inc.ListNode(T) = null, // \u5934\u8282\u70b9 front\n rear: ?*inc.ListNode(T) = null, // \u5c3e\u8282\u70b9 rear\n que_size: usize = 0, // \u961f\u5217\u7684\u957f\u5ea6\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.front = null;\n self.rear = null;\n self.que_size = 0;\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.que_size;\n }\n\n // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.size() == 0;\n }\n\n // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n pub fn peek(self: *Self) T {\n if (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n return self.front.?.val;\n } \n\n // \u5165\u961f\n pub fn push(self: *Self, num: T) !void {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n var node = try self.mem_allocator.create(inc.ListNode(T));\n node.init(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (self.front == null) {\n self.front = node;\n self.rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else {\n self.rear.?.next = node;\n self.rear = node;\n }\n self.que_size += 1;\n } \n\n // \u51fa\u961f\n pub fn pop(self: *Self) T {\n var num = self.peek();\n // \u5220\u9664\u5934\u8282\u70b9\n self.front = self.front.?.next;\n self.que_size -= 1;\n return num;\n } \n\n // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n pub fn toArray(self: *Self) ![]T {\n var node = self.front;\n var res = try self.mem_allocator.alloc(T, self.size());\n @memset(res, @as(T, 0));\n var i: usize = 0;\n while (i < res.len) : (i += 1) {\n res[i] = node.?.val;\n node = node.?.next;\n }\n return res;\n }\n };\n}\n\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002
\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf
front\u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u4e00\u4e2a\u53d8\u91cfsize\u7528\u4e8e\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u5b9a\u4e49rear = front + size\uff0c\u8fd9\u4e2a\u516c\u5f0f\u8ba1\u7b97\u51fa\u7684rear\u6307\u5411\u961f\u5c3e\u5143\u7d20\u4e4b\u540e\u7684\u4e0b\u4e00\u4e2a\u4f4d\u7f6e\u3002\u57fa\u4e8e\u6b64\u8bbe\u8ba1\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a
[front, rear - 1]\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u6cd5\u5982\u56fe 5-6 \u6240\u793a\u3002- \u5165\u961f\u64cd\u4f5c\uff1a\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9
rear\u7d22\u5f15\u5904\uff0c\u5e76\u5c06size\u589e\u52a0 1 \u3002 - \u51fa\u961f\u64cd\u4f5c\uff1a\u53ea\u9700\u5c06
front\u589e\u52a0 1 \uff0c\u5e76\u5c06size\u51cf\u5c11 1 \u3002
\u53ef\u4ee5\u770b\u5230\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u90fd\u53ea\u9700\u8fdb\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002
ArrayQueuepush()pop()\u56fe 5-6 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c
\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u8fdb\u884c\u5165\u961f\u548c\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0c
front\u548crear\u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5f53\u5b83\u4eec\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u65f6\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u89c6\u4e3a\u9996\u5c3e\u76f8\u63a5\u7684\u201c\u73af\u5f62\u6570\u7ec4\u201d\u3002\u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u8ba9
PythonC++JavaC#GoSwiftJSTSDartRustCZig array_queue.pyfront\u6216rear\u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u76f4\u63a5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u201c\u53d6\u4f59\u64cd\u4f5c\u201d\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
array_queue.cppclass ArrayQueue:\n \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n def __init__(self, size: int):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._nums: list[int] = [0] * size # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n self._front: int = 0 # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n self._size: int = 0 # \u961f\u5217\u957f\u5ea6\n\n def capacity(self) -> int:\n \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n return len(self._nums)\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n return self._size == 0\n\n def push(self, num: int):\n \"\"\"\u5165\u961f\"\"\"\n if self._size == self.capacity():\n raise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n # \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n rear: int = (self._front + self._size) % self.capacity()\n # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n self._nums[rear] = num\n self._size += 1\n\n def pop(self) -> int:\n \"\"\"\u51fa\u961f\"\"\"\n num: int = self.peek()\n # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n self._front = (self._front + 1) % self.capacity()\n self._size -= 1\n return num\n\n def peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n return self._nums[self._front]\n\n def to_list(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n res = [0] * self.size()\n j: int = self._front\n for i in range(self.size()):\n res[i] = self._nums[(j % self.capacity())]\n j += 1\n return res\n
array_queue.java/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n private:\n int *nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u961f\u5217\u957f\u5ea6\n int queCapacity; // \u961f\u5217\u5bb9\u91cf\n\n public:\n ArrayQueue(int capacity) {\n // \u521d\u59cb\u5316\u6570\u7ec4\n nums = new int[capacity];\n queCapacity = capacity;\n front = queSize = 0;\n }\n\n ~ArrayQueue() {\n delete[] nums;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n int capacity() {\n return queCapacity;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f */\n void push(int num) {\n if (queSize == queCapacity) {\n cout << \"\u961f\u5217\u5df2\u6ee1\" << endl;\n return;\n }\n // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (front + queSize) % queCapacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u51fa\u961f */\n int pop() {\n int num = peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n front = (front + 1) % queCapacity;\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peek() {\n if (isEmpty())\n throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n return nums[front];\n }\n\n /* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n vector<int> toVector() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n vector<int> arr(queSize);\n for (int i = 0, j = front; i < queSize; i++, j++) {\n arr[i] = nums[j % queCapacity];\n }\n return arr;\n }\n};\n
array_queue.cs/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private int queSize; // \u961f\u5217\u957f\u5ea6\n\n public ArrayQueue(int capacity) {\n nums = new int[capacity];\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n public int capacity() {\n return nums.length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n public int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return queSize == 0;\n }\n\n /* \u5165\u961f */\n public void push(int num) {\n if (queSize == capacity()) {\n System.out.println(\"\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (front + queSize) % capacity();\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u51fa\u961f */\n public int pop() {\n int num = peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n front = (front + 1) % capacity();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int peek() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return nums[front];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4 */\n public int[] toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] res = new int[queSize];\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[j % capacity()];\n }\n return res;\n }\n}\n
array_queue.go/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u961f\u5217\u957f\u5ea6\n\n public ArrayQueue(int capacity) {\n nums = new int[capacity];\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n int Capacity() {\n return nums.Length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n public int Size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return queSize == 0;\n }\n\n /* \u5165\u961f */\n public void Push(int num) {\n if (queSize == Capacity()) {\n Console.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (front + queSize) % Capacity();\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u51fa\u961f */\n public int Pop() {\n int num = Peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n front = (front + 1) % Capacity();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int Peek() {\n if (IsEmpty())\n throw new Exception();\n return nums[front];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4 */\n public int[] ToArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] res = new int[queSize];\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[j % this.Capacity()];\n }\n return res;\n }\n}\n
array_queue.swift/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\n nums []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n front int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n queSize int // \u961f\u5217\u957f\u5ea6\n queCapacity int // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n return &arrayQueue{\n nums: make([]int, queCapacity),\n queCapacity: queCapacity,\n front: 0,\n queSize: 0,\n }\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\n return q.queSize\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\n return q.queSize == 0\n}\n\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n // \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\n if q.queSize == q.queCapacity {\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n rear := (q.front + q.queSize) % q.queCapacity\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n q.nums[rear] = num\n q.queSize++\n}\n\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\n num := q.peek()\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n q.front = (q.front + 1) % q.queCapacity\n q.queSize--\n return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\n if q.isEmpty() {\n return nil\n }\n return q.nums[q.front]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\n rear := (q.front + q.queSize)\n if rear >= q.queCapacity {\n rear %= q.queCapacity\n return append(q.nums[q.front:], q.nums[:rear]...)\n }\n return q.nums[q.front:rear]\n}\n
array_queue.js/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private var queSize = 0 // \u961f\u5217\u957f\u5ea6\n\n init(capacity: Int) {\n // \u521d\u59cb\u5316\u6570\u7ec4\n nums = Array(repeating: 0, count: capacity)\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n func capacity() -> Int {\n nums.count\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n func size() -> Int {\n queSize\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n queSize == 0\n }\n\n /* \u5165\u961f */\n func push(num: Int) {\n if size() == capacity() {\n print(\"\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n let rear = (front + queSize) % capacity()\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num\n queSize += 1\n }\n\n /* \u51fa\u961f */\n @discardableResult\n func pop() -> Int {\n let num = peek()\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n front = (front + 1) % capacity()\n queSize -= 1\n return num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n func peek() -> Int {\n if isEmpty() {\n fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n }\n return nums[front]\n }\n\n /* \u8fd4\u56de\u6570\u7ec4 */\n func toArray() -> [Int] {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n var res = Array(repeating: 0, count: queSize)\n for (i, j) in sequence(first: (0, front), next: { $0 < self.queSize - 1 ? ($0 + 1, $1 + 1) : nil }) {\n res[i] = nums[j % capacity()]\n }\n return res\n }\n}\n
array_queue.ts/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n #nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n #front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n #queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n constructor(capacity) {\n this.#nums = new Array(capacity);\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n get capacity() {\n return this.#nums.length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n get size() {\n return this.#queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.#queSize === 0;\n }\n\n /* \u5165\u961f */\n push(num) {\n if (this.size === this.capacity) {\n console.log('\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n const rear = (this.#front + this.size) % this.capacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n this.#nums[rear] = num;\n this.#queSize++;\n }\n\n /* \u51fa\u961f */\n pop() {\n const num = this.peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n this.#front = (this.#front + 1) % this.capacity;\n this.#queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peek() {\n if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n return this.#nums[this.#front];\n }\n\n /* \u8fd4\u56de Array */\n toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const arr = new Array(this.size);\n for (let i = 0, j = this.#front; i < this.size; i++, j++) {\n arr[i] = this.#nums[j % this.capacity];\n }\n return arr;\n }\n}\n
array_queue.dart/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private queSize: number; // \u961f\u5217\u957f\u5ea6\n\n constructor(capacity: number) {\n this.nums = new Array(capacity);\n this.front = this.queSize = 0;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n get capacity(): number {\n return this.nums.length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n get size(): number {\n return this.queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.queSize === 0;\n }\n\n /* \u5165\u961f */\n push(num: number): void {\n if (this.size === this.capacity) {\n console.log('\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n const rear = (this.front + this.queSize) % this.capacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n this.nums[rear] = num;\n this.queSize++;\n }\n\n /* \u51fa\u961f */\n pop(): number {\n const num = this.peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n this.front = (this.front + 1) % this.capacity;\n this.queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peek(): number {\n if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n return this.nums[this.front];\n }\n\n /* \u8fd4\u56de Array */\n toArray(): number[] {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const arr = new Array(this.size);\n for (let i = 0, j = this.front; i < this.size; i++, j++) {\n arr[i] = this.nums[j % this.capacity];\n }\n return arr;\n }\n}\n
array_queue.rs/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n late List<int> _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n late int _queSize; // \u961f\u5217\u957f\u5ea6\n\n ArrayQueue(int capacity) {\n _nums = List.filled(capacity, 0);\n _front = _queSize = 0;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n int capaCity() {\n return _nums.length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return _queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _queSize == 0;\n }\n\n /* \u5165\u961f */\n void push(int _num) {\n if (_queSize == capaCity()) {\n throw Exception(\"\u961f\u5217\u5df2\u6ee1\");\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (_front + _queSize) % capaCity();\n // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n _nums[rear] = _num;\n _queSize++;\n }\n\n /* \u51fa\u961f */\n int pop() {\n int _num = peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n _front = (_front + 1) % capaCity();\n _queSize--;\n return _num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peek() {\n if (isEmpty()) {\n throw Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n }\n return _nums[_front];\n }\n\n /* \u8fd4\u56de Array */\n List<int> toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n final List<int> res = List.filled(_queSize, 0);\n for (int i = 0, j = _front; i < _queSize; i++, j++) {\n res[i] = _nums[j % capaCity()];\n }\n return res;\n }\n}\n
array_queue.c/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct ArrayQueue {\n nums: Vec<i32>, // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n front: i32, // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n que_size: i32, // \u961f\u5217\u957f\u5ea6\n que_capacity: i32, // \u961f\u5217\u5bb9\u91cf\n}\n\nimpl ArrayQueue {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new(capacity: i32) -> ArrayQueue {\n ArrayQueue {\n nums: vec![0; capacity as usize],\n front: 0,\n que_size: 0,\n que_capacity: capacity,\n }\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n fn capacity(&self) -> i32 {\n self.que_capacity\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n fn size(&self) -> i32 {\n self.que_size\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n fn is_empty(&self) -> bool {\n self.que_size == 0\n }\n\n /* \u5165\u961f */\n fn push(&mut self, num: i32) {\n if self.que_size == self.capacity() {\n println!(\"\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n let rear = (self.front + self.que_size) % self.que_capacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n self.nums[rear as usize] = num;\n self.que_size += 1;\n }\n\n /* \u51fa\u961f */\n fn pop(&mut self) -> i32 {\n let num = self.peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n self.front = (self.front + 1) % self.que_capacity;\n self.que_size -= 1;\n num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n fn peek(&self) -> i32 {\n if self.is_empty() {\n panic!(\"index out of bounds\");\n }\n self.nums[self.front as usize]\n }\n\n /* \u8fd4\u56de\u6570\u7ec4 */\n fn to_vector(&self) -> Vec<i32> {\n let cap = self.que_capacity;\n let mut j = self.front;\n let mut arr = vec![0; self.que_size as usize];\n for i in 0..self.que_size {\n arr[i as usize] = self.nums[(j % cap) as usize];\n j += 1;\n }\n arr\n }\n}\n
array_queue.zig/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n int *nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayQueue *newArrayQueue(int capacity) {\n ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n // \u521d\u59cb\u5316\u6570\u7ec4\n queue->queCapacity = capacity;\n queue->nums = (int *)malloc(sizeof(int) * queue->queCapacity);\n queue->front = queue->queSize = 0;\n return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(ArrayQueue *queue) {\n free(queue->nums);\n free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\n return queue->queCapacity;\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayQueue *queue) {\n return queue->queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayQueue *queue) {\n return queue->queSize == 0;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\n assert(size(queue) != 0);\n return queue->nums[queue->front];\n}\n\n/* \u5165\u961f */\nvoid push(ArrayQueue *queue, int num) {\n if (size(queue) == capacity(queue)) {\n printf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\n return;\n }\n // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (queue->front + queue->queSize) % queue->queCapacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n queue->nums[rear] = num;\n queue->queSize++;\n}\n\n/* \u51fa\u961f */\nint pop(ArrayQueue *queue) {\n int num = peek(queue);\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n queue->front = (queue->front + 1) % queue->queCapacity;\n queue->queSize--;\n return num;\n}\n// \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\n return struct {\n const Self = @This();\n\n nums: []T = undefined, // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4 \n cap: usize = 0, // \u961f\u5217\u5bb9\u91cf\n front: usize = 0, // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n queSize: usize = 0, // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.cap = cap;\n self.nums = try self.mem_allocator.alloc(T, self.cap);\n @memset(self.nums, @as(T, 0));\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\n pub fn capacity(self: *Self) usize {\n return self.cap;\n }\n\n // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.queSize;\n }\n\n // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.queSize == 0;\n }\n\n // \u5165\u961f\n pub fn push(self: *Self, num: T) !void {\n if (self.size() == self.capacity()) {\n std.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n var rear = (self.front + self.queSize) % self.capacity();\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n self.nums[rear] = num;\n self.queSize += 1;\n } \n\n // \u51fa\u961f\n pub fn pop(self: *Self) T {\n var num = self.peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n self.front = (self.front + 1) % self.capacity();\n self.queSize -= 1;\n return num;\n } \n\n // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n pub fn peek(self: *Self) T {\n if (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n return self.nums[self.front];\n } \n\n // \u8fd4\u56de\u6570\u7ec4\n pub fn toArray(self: *Self) ![]T {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n var res = try self.mem_allocator.alloc(T, self.size());\n @memset(res, @as(T, 0));\n var i: usize = 0;\n var j: usize = self.front;\n while (i < self.size()) : ({ i += 1; j += 1; }) {\n res[i] = self.nums[j % self.capacity()];\n }\n return res;\n }\n };\n}\n\u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u7136\u5177\u6709\u5c40\u9650\u6027\uff1a\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u95ee\u9898\u4e0d\u96be\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u52a8\u6001\u6570\u7ec4\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002
\u4e24\u79cd\u5b9e\u73b0\u7684\u5bf9\u6bd4\u7ed3\u8bba\u4e0e\u6808\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
"},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3 \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"- \u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c06\u52a0\u5165\u961f\u5217\u4e2d\uff0c\u7cfb\u7edf\u968f\u540e\u4f1a\u6839\u636e\u987a\u5e8f\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u671f\u95f4\uff0c\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u8ba2\u5355\uff0c\u9ad8\u5e76\u53d1\u6210\u4e3a\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u653b\u514b\u7684\u95ee\u9898\u3002
- \u5404\u7c7b\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u529f\u80fd\u7684\u573a\u666f\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\uff0c\u961f\u5217\u5728\u8fd9\u4e9b\u573a\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7ef4\u62a4\u5904\u7406\u987a\u5e8f\u3002
\u300c\u6808 stack\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u903b\u8f91\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002
\u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u60f3\u53d6\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u79fb\u8d70\u3002\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u5404\u79cd\u7c7b\u578b\u7684\u5143\u7d20\uff08\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u8fd9\u79cd\u6570\u636e\u7ed3\u6784\u3002
\u5982\u56fe 5-1 \u6240\u793a\uff0c\u6211\u4eec\u628a\u5806\u53e0\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u201c\u6808\u9876\u201d\uff0c\u5e95\u90e8\u79f0\u4e3a\u201c\u6808\u5e95\u201d\u3002\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u5165\u6808\u201d\uff0c\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u51fa\u6808\u201d\u3002
\u56fe 5-1 \u00a0 \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219
"},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1 \u00a0 \u6808\u5e38\u7528\u64cd\u4f5c","text":"\u6808\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-1 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u4eec\u4ee5\u5e38\u89c1\u7684
push()\u3001pop()\u3001peek()\u547d\u540d\u4e3a\u4f8b\u3002\u8868 5-1 \u00a0 \u6808\u7684\u64cd\u4f5c\u6548\u7387
\u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) pop() \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) peek() \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\)\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u6808\u7c7b\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8bed\u8a00\u53ef\u80fd\u6ca1\u6709\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u8bed\u8a00\u7684\u201c\u6570\u7ec4\u201d\u6216\u201c\u94fe\u8868\u201d\u5f53\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u5728\u7a0b\u5e8f\u903b\u8f91\u4e0a\u5ffd\u7565\u4e0e\u6808\u65e0\u5173\u7684\u64cd\u4f5c\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig stack.py
stack.cpp# \u521d\u59cb\u5316\u6808\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: list[int] = []\n\n# \u5143\u7d20\u5165\u6808\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npeek: int = stack[-1]\n\n# \u5143\u7d20\u51fa\u6808\npop: int = stack.pop()\n\n# \u83b7\u53d6\u6808\u7684\u957f\u5ea6\nsize: int = len(stack)\n\n# \u5224\u65ad\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(stack) == 0\n
stack.java/* \u521d\u59cb\u5316\u6808 */\nstack<int> stack;\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n
stack.cs/* \u521d\u59cb\u5316\u6808 */\nStack<Integer> stack = new Stack<>();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n
stack_test.go/* \u521d\u59cb\u5316\u6808 */\nStack<int> stack = new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count == 0;\n
stack.swift/* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n
stack.js/* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n
stack.ts/* \u521d\u59cb\u5316\u6808 */\n// Javascript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
stack.dart/* \u521d\u59cb\u5316\u6808 */\n// Typescript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
stack.rs/* \u521d\u59cb\u5316\u6808 */\n// Dart \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nList<int> stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.last;\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.removeLast();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.isEmpty;\n
stack.c/* \u521d\u59cb\u5316\u6808 */\n// \u628a Vec \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nlet mut stack: Vec<i32> = Vec::new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet top = stack.last().unwrap();\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.pop().unwrap();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.len();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = stack.is_empty();\n
stack.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u6808\n
"},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2 \u00a0 \u6808\u7684\u5b9e\u73b0","text":"\n\u4e3a\u4e86\u6df1\u5165\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u81ea\u5df1\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002
\u6808\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u539f\u5219\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u548c\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u6808\u53ef\u4ee5\u89c6\u4e3a\u4e00\u79cd\u53d7\u9650\u5236\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u5176\u5bf9\u5916\u8868\u73b0\u7684\u903b\u8f91\u7b26\u5408\u6808\u7684\u7279\u6027\u3002
"},{"location":"chapter_stack_and_queue/stack/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u5934\u8282\u70b9\u89c6\u4e3a\u6808\u9876\uff0c\u5c3e\u8282\u70b9\u89c6\u4e3a\u6808\u5e95\u3002
\u5982\u56fe 5-2 \u6240\u793a\uff0c\u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u5143\u7d20\u63d2\u5165\u94fe\u8868\u5934\u90e8\uff0c\u8fd9\u79cd\u8282\u70b9\u63d2\u5165\u65b9\u6cd5\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06\u5934\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002
LinkedListStackpush()pop()\u56fe 5-2 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c
\u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_stack.py
linkedlist_stack.cppclass LinkedListStack:\n \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._peek: ListNode | None = None\n self._size: int = 0\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n return not self._peek\n\n def push(self, val: int):\n \"\"\"\u5165\u6808\"\"\"\n node = ListNode(val)\n node.next = self._peek\n self._peek = node\n self._size += 1\n\n def pop(self) -> int:\n \"\"\"\u51fa\u6808\"\"\"\n num = self.peek()\n self._peek = self._peek.next\n self._size -= 1\n return num\n\n def peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u6808\u4e3a\u7a7a\")\n return self._peek.val\n\n def to_list(self) -> list[int]:\n \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n arr = []\n node = self._peek\n while node:\n arr.append(node.val)\n node = node.next\n arr.reverse()\n return arr\n
linkedlist_stack.java/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n private:\n ListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n int stkSize; // \u6808\u7684\u957f\u5ea6\n\n public:\n LinkedListStack() {\n stackTop = nullptr;\n stkSize = 0;\n }\n\n ~LinkedListStack() {\n // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n freeMemoryLinkedList(stackTop);\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n int size() {\n return stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u6808 */\n void push(int num) {\n ListNode *node = new ListNode(num);\n node->next = stackTop;\n stackTop = node;\n stkSize++;\n }\n\n /* \u51fa\u6808 */\n int pop() {\n int num = top();\n ListNode *tmp = stackTop;\n stackTop = stackTop->next;\n // \u91ca\u653e\u5185\u5b58\n delete tmp;\n stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n int top() {\n if (isEmpty())\n throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n return stackTop->val;\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n vector<int> toVector() {\n ListNode *node = stackTop;\n vector<int> res(size());\n for (int i = res.size() - 1; i >= 0; i--) {\n res[i] = node->val;\n node = node->next;\n }\n return res;\n }\n};\n
linkedlist_stack.cs/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n private ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n private int stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n public LinkedListStack() {\n stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n public int size() {\n return stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u6808 */\n public void push(int num) {\n ListNode node = new ListNode(num);\n node.next = stackPeek;\n stackPeek = node;\n stkSize++;\n }\n\n /* \u51fa\u6808 */\n public int pop() {\n int num = peek();\n stackPeek = stackPeek.next;\n stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n public int peek() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return stackPeek.val;\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] toArray() {\n ListNode node = stackPeek;\n int[] res = new int[size()];\n for (int i = res.length - 1; i >= 0; i--) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_stack.go/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n ListNode? stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n int stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n public LinkedListStack() {\n stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n public int Size() {\n return stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return Size() == 0;\n }\n\n /* \u5165\u6808 */\n public void Push(int num) {\n ListNode node = new(num) {\n next = stackPeek\n };\n stackPeek = node;\n stkSize++;\n }\n\n /* \u51fa\u6808 */\n public int Pop() {\n int num = Peek();\n stackPeek = stackPeek!.next;\n stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n public int Peek() {\n if (IsEmpty())\n throw new Exception();\n return stackPeek!.val;\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] ToArray() {\n if (stackPeek == null)\n return [];\n\n ListNode? node = stackPeek;\n int[] res = new int[Size()];\n for (int i = res.Length - 1; i >= 0; i--) {\n res[i] = node!.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_stack.swift/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\n data *list.List\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\n return &linkedListStack{\n data: list.New(),\n }\n}\n\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\n s.data.PushBack(value)\n}\n\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Back()\n s.data.Remove(e)\n return e.Value\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Back()\n return e.Value\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\n return s.data.Len()\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\n return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\n return s.data\n}\n
linkedlist_stack.js/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n private var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n private var _size = 0 // \u6808\u7684\u957f\u5ea6\n\n init() {}\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n func size() -> Int {\n _size\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n size() == 0\n }\n\n /* \u5165\u6808 */\n func push(num: Int) {\n let node = ListNode(x: num)\n node.next = _peek\n _peek = node\n _size += 1\n }\n\n /* \u51fa\u6808 */\n @discardableResult\n func pop() -> Int {\n let num = peek()\n _peek = _peek?.next\n _size -= 1\n return num\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n func peek() -> Int {\n if isEmpty() {\n fatalError(\"\u6808\u4e3a\u7a7a\")\n }\n return _peek!.val\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n func toArray() -> [Int] {\n var node = _peek\n var res = Array(repeating: 0, count: _size)\n for i in sequence(first: res.count - 1, next: { $0 >= 0 + 1 ? $0 - 1 : nil }) {\n res[i] = node!.val\n node = node?.next\n }\n return res\n }\n}\n
linkedlist_stack.ts/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n #stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n #stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n constructor() {\n this.#stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n get size() {\n return this.#stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.size === 0;\n }\n\n /* \u5165\u6808 */\n push(num) {\n const node = new ListNode(num);\n node.next = this.#stackPeek;\n this.#stackPeek = node;\n this.#stkSize++;\n }\n\n /* \u51fa\u6808 */\n pop() {\n const num = this.peek();\n this.#stackPeek = this.#stackPeek.next;\n this.#stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n peek() {\n if (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n return this.#stackPeek.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n toArray() {\n let node = this.#stackPeek;\n const res = new Array(this.size);\n for (let i = res.length - 1; i >= 0; i--) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_stack.dart/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n private stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n private stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\n\n constructor() {\n this.stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n get size(): number {\n return this.stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.size === 0;\n }\n\n /* \u5165\u6808 */\n push(num: number): void {\n const node = new ListNode(num);\n node.next = this.stackPeek;\n this.stackPeek = node;\n this.stkSize++;\n }\n\n /* \u51fa\u6808 */\n pop(): number {\n const num = this.peek();\n if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n this.stackPeek = this.stackPeek.next;\n this.stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n peek(): number {\n if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n return this.stackPeek.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n toArray(): number[] {\n let node = this.stackPeek;\n const res = new Array<number>(this.size);\n for (let i = res.length - 1; i >= 0; i--) {\n res[i] = node!.val;\n node = node!.next;\n }\n return res;\n }\n}\n
linkedlist_stack.rs/* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n ListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n int _stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n LinkedListStack() {\n _stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n int size() {\n return _stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _stkSize == 0;\n }\n\n /* \u5165\u6808 */\n void push(int _num) {\n final ListNode node = ListNode(_num);\n node.next = _stackPeek;\n _stackPeek = node;\n _stkSize++;\n }\n\n /* \u51fa\u6808 */\n int pop() {\n final int _num = peek();\n _stackPeek = _stackPeek!.next;\n _stkSize--;\n return _num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n int peek() {\n if (_stackPeek == null) {\n throw Exception(\"\u6808\u4e3a\u7a7a\");\n }\n return _stackPeek!.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\n List<int> toList() {\n ListNode? node = _stackPeek;\n List<int> list = [];\n while (node != null) {\n list.add(node.val);\n node = node.next;\n }\n list = list.reversed.toList();\n return list;\n }\n}\n
linkedlist_stack.c/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\n#[allow(dead_code)]\npub struct LinkedListStack<T> {\n stack_peek: Option<Rc<RefCell<ListNode<T>>>>, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n stk_size: usize, // \u6808\u7684\u957f\u5ea6\n}\n\nimpl<T: Copy> LinkedListStack<T> {\n pub fn new() -> Self {\n Self {\n stack_peek: None,\n stk_size: 0,\n }\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n pub fn size(&self) -> usize {\n return self.stk_size;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n pub fn is_empty(&self) -> bool {\n return self.size() == 0;\n }\n\n /* \u5165\u6808 */\n pub fn push(&mut self, num: T) {\n let node = ListNode::new(num);\n node.borrow_mut().next = self.stack_peek.take();\n self.stack_peek = Some(node);\n self.stk_size += 1;\n }\n\n /* \u51fa\u6808 */\n pub fn pop(&mut self) -> Option<T> {\n self.stack_peek.take().map(|old_head| {\n match old_head.borrow_mut().next.take() {\n Some(new_head) => {\n self.stack_peek = Some(new_head);\n }\n None => {\n self.stack_peek = None;\n }\n }\n self.stk_size -= 1;\n Rc::try_unwrap(old_head).ok().unwrap().into_inner().val\n })\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n pub fn peek(&self) -> Option<&Rc<RefCell<ListNode<T>>>> {\n self.stack_peek.as_ref()\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n pub fn to_array(&self, head: Option<&Rc<RefCell<ListNode<T>>>>) -> Vec<T> {\n if let Some(node) = head {\n let mut nums = self.to_array(node.borrow().next.as_ref());\n nums.push(node.borrow().val);\n return nums;\n }\n return Vec::new();\n }\n}\n
linkedlist_stack.zig/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n ListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n int size; // \u6808\u7684\u957f\u5ea6\n} LinkedListStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListStack *newLinkedListStack() {\n LinkedListStack *s = malloc(sizeof(LinkedListStack));\n s->top = NULL;\n s->size = 0;\n return s;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(LinkedListStack *s) {\n while (s->top) {\n ListNode *n = s->top->next;\n free(s->top);\n s->top = n;\n }\n free(s);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(LinkedListStack *s) {\n return s->size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(LinkedListStack *s) {\n return size(s) == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(LinkedListStack *s, int num) {\n ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n node->next = s->top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\n node->val = num; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\n s->top = node; // \u66f4\u65b0\u6808\u9876\n s->size++; // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(LinkedListStack *s) {\n if (s->size == 0) {\n printf(\"\u6808\u4e3a\u7a7a\\n\");\n return INT_MAX;\n }\n return s->top->val;\n}\n\n/* \u51fa\u6808 */\nint pop(LinkedListStack *s) {\n int val = peek(s);\n ListNode *tmp = s->top;\n s->top = s->top->next;\n // \u91ca\u653e\u5185\u5b58\n free(tmp);\n s->size--;\n return val;\n}\n
"},{"location":"chapter_stack_and_queue/stack/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\n return struct {\n const Self = @This();\n\n stack_top: ?*inc.ListNode(T) = null, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n stk_size: usize = 0, // \u6808\u7684\u957f\u5ea6\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.stack_top = null;\n self.stk_size = 0;\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.stk_size;\n }\n\n // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.size() == 0;\n }\n\n // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n pub fn peek(self: *Self) T {\n if (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\n return self.stack_top.?.val;\n } \n\n // \u5165\u6808\n pub fn push(self: *Self, num: T) !void {\n var node = try self.mem_allocator.create(inc.ListNode(T));\n node.init(num);\n node.next = self.stack_top;\n self.stack_top = node;\n self.stk_size += 1;\n } \n\n // \u51fa\u6808\n pub fn pop(self: *Self) T {\n var num = self.peek();\n self.stack_top = self.stack_top.?.next;\n self.stk_size -= 1;\n return num;\n } \n\n // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\n pub fn toArray(self: *Self) ![]T {\n var node = self.stack_top;\n var res = try self.mem_allocator.alloc(T, self.size());\n @memset(res, @as(T, 0));\n var i: usize = 0;\n while (i < res.len) : (i += 1) {\n res[res.len - i - 1] = node.?.val;\n node = node.?.next;\n }\n return res;\n }\n };\n}\n\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u4f5c\u4e3a\u6808\u9876\u3002\u5982\u56fe 5-3 \u6240\u793a\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u5206\u522b\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u4e0e\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002
ArrayStackpush()pop()\u56fe 5-3 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c
\u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u4f1a\u6e90\u6e90\u4e0d\u65ad\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u6570\u7ec4\uff0c\u8fd9\u6837\u5c31\u65e0\u987b\u81ea\u884c\u5904\u7406\u6570\u7ec4\u6269\u5bb9\u95ee\u9898\u3002\u4ee5\u4e0b\u4e3a\u793a\u4f8b\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array_stack.py
array_stack.cppclass ArrayStack:\n \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._stack: list[int] = []\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n return len(self._stack)\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n return self._stack == []\n\n def push(self, item: int):\n \"\"\"\u5165\u6808\"\"\"\n self._stack.append(item)\n\n def pop(self) -> int:\n \"\"\"\u51fa\u6808\"\"\"\n if self.is_empty():\n raise IndexError(\"\u6808\u4e3a\u7a7a\")\n return self._stack.pop()\n\n def peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u6808\u4e3a\u7a7a\")\n return self._stack[-1]\n\n def to_list(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n return self._stack\n
array_stack.java/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n private:\n vector<int> stack;\n\n public:\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n int size() {\n return stack.size();\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return stack.size() == 0;\n }\n\n /* \u5165\u6808 */\n void push(int num) {\n stack.push_back(num);\n }\n\n /* \u51fa\u6808 */\n int pop() {\n int num = top();\n stack.pop_back();\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n int top() {\n if (isEmpty())\n throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n return stack.back();\n }\n\n /* \u8fd4\u56de Vector */\n vector<int> toVector() {\n return stack;\n }\n};\n
array_stack.cs/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n private ArrayList<Integer> stack;\n\n public ArrayStack() {\n // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n stack = new ArrayList<>();\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n public int size() {\n return stack.size();\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u6808 */\n public void push(int num) {\n stack.add(num);\n }\n\n /* \u51fa\u6808 */\n public int pop() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return stack.remove(size() - 1);\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n public int peek() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return stack.get(size() - 1);\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public Object[] toArray() {\n return stack.toArray();\n }\n}\n
array_stack.go/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n List<int> stack;\n public ArrayStack() {\n // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n stack = [];\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n public int Size() {\n return stack.Count;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return Size() == 0;\n }\n\n /* \u5165\u6808 */\n public void Push(int num) {\n stack.Add(num);\n }\n\n /* \u51fa\u6808 */\n public int Pop() {\n if (IsEmpty())\n throw new Exception();\n var val = Peek();\n stack.RemoveAt(Size() - 1);\n return val;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n public int Peek() {\n if (IsEmpty())\n throw new Exception();\n return stack[Size() - 1];\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] ToArray() {\n return [.. stack];\n }\n}\n
array_stack.swift/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\n data []int // \u6570\u636e\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\n return &arrayStack{\n // \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\n data: make([]int, 0, 16),\n }\n}\n\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\n return len(s.data)\n}\n\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\n return s.size() == 0\n}\n\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n // \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\n s.data = append(s.data, v)\n}\n\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\n val := s.peek()\n s.data = s.data[:len(s.data)-1]\n return val\n}\n\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\n if s.isEmpty() {\n return nil\n }\n val := s.data[len(s.data)-1]\n return val\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\n return s.data\n}\n
array_stack.js/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n private var stack: [Int]\n\n init() {\n // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n stack = []\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n func size() -> Int {\n stack.count\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n stack.isEmpty\n }\n\n /* \u5165\u6808 */\n func push(num: Int) {\n stack.append(num)\n }\n\n /* \u51fa\u6808 */\n @discardableResult\n func pop() -> Int {\n if isEmpty() {\n fatalError(\"\u6808\u4e3a\u7a7a\")\n }\n return stack.removeLast()\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n func peek() -> Int {\n if isEmpty() {\n fatalError(\"\u6808\u4e3a\u7a7a\")\n }\n return stack.last!\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n func toArray() -> [Int] {\n stack\n }\n}\n
array_stack.ts/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n #stack;\n constructor() {\n this.#stack = [];\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n get size() {\n return this.#stack.length;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.#stack.length === 0;\n }\n\n /* \u5165\u6808 */\n push(num) {\n this.#stack.push(num);\n }\n\n /* \u51fa\u6808 */\n pop() {\n if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n return this.#stack.pop();\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n top() {\n if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n return this.#stack[this.#stack.length - 1];\n }\n\n /* \u8fd4\u56de Array */\n toArray() {\n return this.#stack;\n }\n}\n
array_stack.dart/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n private stack: number[];\n constructor() {\n this.stack = [];\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n get size(): number {\n return this.stack.length;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.stack.length === 0;\n }\n\n /* \u5165\u6808 */\n push(num: number): void {\n this.stack.push(num);\n }\n\n /* \u51fa\u6808 */\n pop(): number | undefined {\n if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n return this.stack.pop();\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n top(): number | undefined {\n if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n return this.stack[this.stack.length - 1];\n }\n\n /* \u8fd4\u56de Array */\n toArray() {\n return this.stack;\n }\n}\n
array_stack.rs/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n late List<int> _stack;\n ArrayStack() {\n _stack = [];\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n int size() {\n return _stack.length;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _stack.isEmpty;\n }\n\n /* \u5165\u6808 */\n void push(int _num) {\n _stack.add(_num);\n }\n\n /* \u51fa\u6808 */\n int pop() {\n if (isEmpty()) {\n throw Exception(\"\u6808\u4e3a\u7a7a\");\n }\n return _stack.removeLast();\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n int peek() {\n if (isEmpty()) {\n throw Exception(\"\u6808\u4e3a\u7a7a\");\n }\n return _stack.last;\n }\n\n /* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n List<int> toArray() => _stack;\n}\n
array_stack.c/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct ArrayStack<T> {\n stack: Vec<T>,\n}\n\nimpl<T> ArrayStack<T> {\n /* \u521d\u59cb\u5316\u6808 */\n fn new() -> ArrayStack<T> {\n ArrayStack::<T> { stack: Vec::<T>::new() }\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n fn size(&self) -> usize {\n self.stack.len()\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n fn is_empty(&self) -> bool {\n self.size() == 0\n }\n\n /* \u5165\u6808 */\n fn push(&mut self, num: T) {\n self.stack.push(num);\n }\n\n /* \u51fa\u6808 */\n fn pop(&mut self) -> Option<T> {\n match self.stack.pop() {\n Some(num) => Some(num),\n None => None,\n }\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n fn peek(&self) -> Option<&T> {\n if self.is_empty() { panic!(\"\u6808\u4e3a\u7a7a\") };\n self.stack.last()\n }\n\n /* \u8fd4\u56de &Vec */\n fn to_array(&self) -> &Vec<T> {\n &self.stack\n }\n}\n
array_stack.zig/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n int *data;\n int size;\n} ArrayStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayStack *newArrayStack() {\n ArrayStack *stack = malloc(sizeof(ArrayStack));\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\n stack->data = malloc(sizeof(int) * MAX_SIZE);\n stack->size = 0;\n return stack;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayStack(ArrayStack *stack) {\n free(stack->data);\n free(stack);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(ArrayStack *stack) {\n return stack->size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(ArrayStack *stack) {\n return stack->size == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(ArrayStack *stack, int num) {\n if (stack->size == MAX_SIZE) {\n printf(\"\u6808\u5df2\u6ee1\\n\");\n return;\n }\n stack->data[stack->size] = num;\n stack->size++;\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(ArrayStack *stack) {\n if (stack->size == 0) {\n printf(\"\u6808\u4e3a\u7a7a\\n\");\n return INT_MAX;\n }\n return stack->data[stack->size - 1];\n}\n\n/* \u51fa\u6808 */\nint pop(ArrayStack *stack) {\n int val = peek(stack);\n stack->size--;\n return val;\n}\n
"},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3 \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":"// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\n return struct {\n const Self = @This();\n\n stack: ?std.ArrayList(T) = null, \n\n // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) void {\n if (self.stack == null) {\n self.stack = std.ArrayList(T).init(allocator);\n }\n }\n\n // \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.stack == null) return;\n self.stack.?.deinit();\n }\n\n // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.stack.?.items.len;\n }\n\n // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.size() == 0;\n }\n\n // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n pub fn peek(self: *Self) T {\n if (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\n return self.stack.?.items[self.size() - 1];\n } \n\n // \u5165\u6808\n pub fn push(self: *Self, num: T) !void {\n try self.stack.?.append(num);\n } \n\n // \u51fa\u6808\n pub fn pop(self: *Self) T {\n var num = self.stack.?.pop();\n return num;\n } \n\n // \u8fd4\u56de ArrayList\n pub fn toList(self: *Self) std.ArrayList(T) {\n return self.stack.?;\n }\n };\n}\n\u652f\u6301\u64cd\u4f5c
\u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u3002\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u8d85\u51fa\u4e86\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002
\u65f6\u95f4\u6548\u7387
\u5728\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u548c\u51fa\u6808\u64cd\u4f5c\u90fd\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u8fdb\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f83\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u5bfc\u81f4\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d8\u4e3a \\(O(n)\\) \u3002
\u5728\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u6548\u7387\u964d\u4f4e\u7684\u95ee\u9898\u3002\u4f46\u662f\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u8282\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5bf9\u8f83\u4f4e\u3002\u4e0d\u8fc7\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u672c\u8eab\u5c31\u662f\u8282\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u9ad8\u6548\u7387\u3002
\u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\u65f6\uff0c\u4f8b\u5982
int\u6216double\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002- \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u6548\u7387\u4f1a\u964d\u4f4e\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002
- \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002
\u7a7a\u95f4\u6548\u7387
\u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u4e3a\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\uff1b\u5e76\u4e14\uff0c\u6269\u5bb9\u673a\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\u7684\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002
\u7136\u800c\uff0c\u7531\u4e8e\u94fe\u8868\u8282\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64\u94fe\u8868\u8282\u70b9\u5360\u7528\u7684\u7a7a\u95f4\u76f8\u5bf9\u8f83\u5927\u3002
\u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u8282\u7701\u5185\u5b58\uff0c\u9700\u8981\u9488\u5bf9\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002
"},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4 \u00a0 \u6808\u5178\u578b\u5e94\u7528","text":"- \u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u4f1a\u5bf9\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u540e\u9000\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u4e2a\u7f51\u9875\u3002\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002
- \u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u4e8e\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u9636\u6bb5\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u5219\u4f1a\u4e0d\u65ad\u6267\u884c\u51fa\u6808\u64cd\u4f5c\u3002
- \u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002
- \u5728\u65f6\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u8f83\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u6269\u5bb9\u8fc7\u7a0b\u4e2d\uff0c\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6808\u7684\u94fe\u8868\u5b9e\u73b0\u5177\u6709\u66f4\u4e3a\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002
- \u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u53ef\u80fd\u5bfc\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u94fe\u8868\u8282\u70b9\u6240\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u6bd4\u6570\u7ec4\u5143\u7d20\u66f4\u5927\u3002
- \u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u540c\u6837\u53ef\u4ee5\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u7684\u5bf9\u6bd4\u4e0a\uff0c\u961f\u5217\u7684\u7ed3\u8bba\u4e0e\u524d\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u4f3c\u3002
- \u53cc\u5411\u961f\u5217\u662f\u4e00\u79cd\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u961f\u5217\uff0c\u5b83\u5141\u8bb8\u5728\u4e24\u7aef\u8fdb\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u3002
\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u662f\u5426\u662f\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\uff1f
\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u529f\u80fd\u672c\u8d28\u4e0a\u662f\u201c\u6808\u201d\u7684\u4f53\u73b0\u3002\u5f53\u7528\u6237\u8bbf\u95ee\u4e00\u4e2a\u65b0\u9875\u9762\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u88ab\u6dfb\u52a0\u5230\u6808\u9876\uff1b\u5f53\u7528\u6237\u70b9\u51fb\u540e\u9000\u6309\u94ae\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u4ece\u6808\u9876\u5f39\u51fa\u3002\u4f7f\u7528\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u65b9\u4fbf\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u5728\u201c\u53cc\u5411\u961f\u5217\u201d\u7ae0\u8282\u6709\u63d0\u5230\u3002
\u5728\u51fa\u6808\u540e\uff0c\u662f\u5426\u9700\u8981\u91ca\u653e\u51fa\u6808\u8282\u70b9\u7684\u5185\u5b58\uff1f
\u5982\u679c\u540e\u7eed\u4ecd\u9700\u8981\u4f7f\u7528\u5f39\u51fa\u8282\u70b9\uff0c\u5219\u4e0d\u9700\u8981\u91ca\u653e\u5185\u5b58\u3002\u82e5\u4e4b\u540e\u4e0d\u9700\u8981\u7528\u5230\uff0c
Java\u548cPython\u7b49\u8bed\u8a00\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\uff1b\u5728C\u548cC++\u4e2d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\u3002\u53cc\u5411\u961f\u5217\u50cf\u662f\u4e24\u4e2a\u6808\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u4e48\uff1f
\u53cc\u5411\u961f\u5217\u5c31\u50cf\u662f\u6808\u548c\u961f\u5217\u7684\u7ec4\u5408\uff0c\u6216\u4e24\u4e2a\u6808\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73b0\u7684\u662f\u6808 + \u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u6808\u4e0e\u961f\u5217\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u66f4\u52a0\u7075\u6d3b\u3002
\u64a4\u9500\uff08undo\uff09\u548c\u53cd\u64a4\u9500\uff08redo\uff09\u5177\u4f53\u662f\u5982\u4f55\u5b9e\u73b0\u7684\uff1f
\u4f7f\u7528\u4e24\u4e2a\u6808\uff0c\u6808
A\u7528\u4e8e\u64a4\u9500\uff0c\u6808B\u7528\u4e8e\u53cd\u64a4\u9500\u3002- \u6bcf\u5f53\u7528\u6237\u6267\u884c\u4e00\u4e2a\u64cd\u4f5c\uff0c\u5c06\u8fd9\u4e2a\u64cd\u4f5c\u538b\u5165\u6808
A\uff0c\u5e76\u6e05\u7a7a\u6808B\u3002 - \u5f53\u7528\u6237\u6267\u884c\u201c\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808
A\u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808B\u3002 - \u5f53\u7528\u6237\u6267\u884c\u201c\u53cd\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808
B\u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808A\u3002
Abstract
\u53c2\u5929\u5927\u6811\u5145\u6ee1\u751f\u547d\u529b\uff0c\u6839\u6df1\u53f6\u8302\uff0c\u5206\u679d\u6276\u758f\u3002
\u5b83\u4e3a\u6211\u4eec\u5c55\u73b0\u4e86\u6570\u636e\u5206\u6cbb\u7684\u751f\u52a8\u5f62\u6001\u3002
"},{"location":"chapter_tree/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 7.1 \u00a0 \u4e8c\u53c9\u6811
- 7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386
- 7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a
- 7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811
- 7.5 \u00a0 AVL \u6811 *
- 7.6 \u00a0 \u5c0f\u7ed3
\u5728\u94fe\u8868\u8868\u793a\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u5b58\u50a8\u5355\u5143\u4e3a\u8282\u70b9
TreeNode\uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u76f8\u8fde\u63a5\u3002\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u94fe\u8868\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002\u90a3\u4e48\uff0c\u6211\u4eec\u80fd\u5426\u7528\u6570\u7ec4\u6765\u8868\u793a\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002
"},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1 \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"\u5148\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\u3002\u7ed9\u5b9a\u4e00\u68f5\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u6240\u6709\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u5b58\u50a8\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u5219\u6bcf\u4e2a\u8282\u70b9\u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u6570\u7ec4\u7d22\u5f15\u3002
\u6839\u636e\u5c42\u5e8f\u904d\u5386\u7684\u7279\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u7236\u8282\u70b9\u7d22\u5f15\u4e0e\u5b50\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u201c\u6620\u5c04\u516c\u5f0f\u201d\uff1a\u82e5\u67d0\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002\u56fe 7-12 \u5c55\u793a\u4e86\u5404\u4e2a\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\u3002
\u56fe 7-12 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a
\u6620\u5c04\u516c\u5f0f\u7684\u89d2\u8272\u76f8\u5f53\u4e8e\u94fe\u8868\u4e2d\u7684\u6307\u9488\u3002\u7ed9\u5b9a\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u8282\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u901a\u8fc7\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5b83\u7684\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3002
"},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2 \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u53c9\u6811","text":"\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u4e00\u4e2a\u7279\u4f8b\uff0c\u5728\u4e8c\u53c9\u6811\u7684\u4e2d\u95f4\u5c42\u901a\u5e38\u5b58\u5728\u8bb8\u591a \\(\\text{None}\\) \u3002\u7531\u4e8e\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b \\(\\text{None}\\) \uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u6cd5\u4ec5\u51ed\u8be5\u5e8f\u5217\u6765\u63a8\u6d4b \\(\\text{None}\\) \u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\u3002\u8fd9\u610f\u5473\u7740\u5b58\u5728\u591a\u79cd\u4e8c\u53c9\u6811\u7ed3\u6784\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002
\u5982\u56fe 7-13 \u6240\u793a\uff0c\u7ed9\u5b9a\u4e00\u68f5\u975e\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u4e0a\u8ff0\u6570\u7ec4\u8868\u793a\u65b9\u6cd5\u5df2\u7ecf\u5931\u6548\u3002
\u56fe 7-13 \u00a0 \u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027
\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u663e\u5f0f\u5730\u5199\u51fa\u6240\u6709 \\(\\text{None}\\) \u3002\u5982\u56fe 7-14 \u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig# \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\n# \u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c INT_MAX \u6807\u8bb0\u7a7a\u4f4d\nvector<int> tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nList<int?> tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 None \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n\n\u56fe 7-14 \u00a0 \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a
\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u56de\u987e\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u5b9a\u4e49\uff0c\\(\\text{None}\\) \u53ea\u51fa\u73b0\u5728\u6700\u5e95\u5c42\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u56e0\u6b64\u6240\u6709 \\(\\text{None}\\) \u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002
\u8fd9\u610f\u5473\u7740\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u6240\u6709 \\(\\text{None}\\) \uff0c\u975e\u5e38\u65b9\u4fbf\u3002\u56fe 7-15 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u4f8b\u5b50\u3002
\u56fe 7-15 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a
\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u68f5\u57fa\u4e8e\u6570\u7ec4\u8868\u793a\u7684\u4e8c\u53c9\u6811\uff0c\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u64cd\u4f5c\u3002
- \u7ed9\u5b9a\u67d0\u8282\u70b9\uff0c\u83b7\u53d6\u5b83\u7684\u503c\u3001\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3001\u7236\u8282\u70b9\u3002
- \u83b7\u53d6\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3001\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002
array_binary_tree.cppclass ArrayBinaryTree:\n \"\"\"\u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b\"\"\"\n\n def __init__(self, arr: list[int | None]):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._tree = list(arr)\n\n def size(self):\n \"\"\"\u8282\u70b9\u6570\u91cf\"\"\"\n return len(self._tree)\n\n def val(self, i: int) -> int:\n \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c\"\"\"\n # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if i < 0 or i >= self.size():\n return None\n return self._tree[i]\n\n def left(self, i: int) -> int | None:\n \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n return 2 * i + 1\n\n def right(self, i: int) -> int | None:\n \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n return 2 * i + 2\n\n def parent(self, i: int) -> int | None:\n \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n return (i - 1) // 2\n\n def level_order(self) -> list[int]:\n \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n self.res = []\n # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for i in range(self.size()):\n if self.val(i) is not None:\n self.res.append(self.val(i))\n return self.res\n\n def dfs(self, i: int, order: str):\n \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n if self.val(i) is None:\n return\n # \u524d\u5e8f\u904d\u5386\n if order == \"pre\":\n self.res.append(self.val(i))\n self.dfs(self.left(i), order)\n # \u4e2d\u5e8f\u904d\u5386\n if order == \"in\":\n self.res.append(self.val(i))\n self.dfs(self.right(i), order)\n # \u540e\u5e8f\u904d\u5386\n if order == \"post\":\n self.res.append(self.val(i))\n\n def pre_order(self) -> list[int]:\n \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n self.res = []\n self.dfs(0, order=\"pre\")\n return self.res\n\n def in_order(self) -> list[int]:\n \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n self.res = []\n self.dfs(0, order=\"in\")\n return self.res\n\n def post_order(self) -> list[int]:\n \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n self.res = []\n self.dfs(0, order=\"post\")\n return self.res\n
array_binary_tree.java/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n ArrayBinaryTree(vector<int> arr) {\n tree = arr;\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n int size() {\n return tree.size();\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n int val(int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= size())\n return INT_MAX;\n return tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n int left(int i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n int right(int i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n int parent(int i) {\n return (i - 1) / 2;\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n vector<int> levelOrder() {\n vector<int> res;\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size(); i++) {\n if (val(i) != INT_MAX)\n res.push_back(val(i));\n }\n return res;\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n vector<int> preOrder() {\n vector<int> res;\n dfs(0, \"pre\", res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n vector<int> inOrder() {\n vector<int> res;\n dfs(0, \"in\", res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n vector<int> postOrder() {\n vector<int> res;\n dfs(0, \"post\", res);\n return res;\n }\n\n private:\n vector<int> tree;\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n void dfs(int i, string order, vector<int> &res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (val(i) == INT_MAX)\n return;\n // \u524d\u5e8f\u904d\u5386\n if (order == \"pre\")\n res.push_back(val(i));\n dfs(left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order == \"in\")\n res.push_back(val(i));\n dfs(right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order == \"post\")\n res.push_back(val(i));\n }\n};\n
array_binary_tree.cs/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n private List<Integer> tree;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public ArrayBinaryTree(List<Integer> arr) {\n tree = new ArrayList<>(arr);\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n public int size() {\n return tree.size();\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n public Integer val(int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= size())\n return null;\n return tree.get(i);\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n public Integer left(int i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n public Integer right(int i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n public Integer parent(int i) {\n return (i - 1) / 2;\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n public List<Integer> levelOrder() {\n List<Integer> res = new ArrayList<>();\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size(); i++) {\n if (val(i) != null)\n res.add(val(i));\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n private void dfs(Integer i, String order, List<Integer> res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (val(i) == null)\n return;\n // \u524d\u5e8f\u904d\u5386\n if (\"pre\".equals(order))\n res.add(val(i));\n dfs(left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (\"in\".equals(order))\n res.add(val(i));\n dfs(right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (\"post\".equals(order))\n res.add(val(i));\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n public List<Integer> preOrder() {\n List<Integer> res = new ArrayList<>();\n dfs(0, \"pre\", res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n public List<Integer> inOrder() {\n List<Integer> res = new ArrayList<>();\n dfs(0, \"in\", res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n public List<Integer> postOrder() {\n List<Integer> res = new ArrayList<>();\n dfs(0, \"post\", res);\n return res;\n }\n}\n
array_binary_tree.go/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(List<int?> arr) {\n List<int?> tree = new(arr);\n\n /* \u8282\u70b9\u6570\u91cf */\n public int Size() {\n return tree.Count;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n public int? Val(int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= Size())\n return null;\n return tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n public int Left(int i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n public int Right(int i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n public int Parent(int i) {\n return (i - 1) / 2;\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n public List<int> LevelOrder() {\n List<int> res = [];\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < Size(); i++) {\n if (Val(i).HasValue)\n res.Add(Val(i)!.Value);\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n void DFS(int i, string order, List<int> res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (!Val(i).HasValue)\n return;\n // \u524d\u5e8f\u904d\u5386\n if (order == \"pre\")\n res.Add(Val(i)!.Value);\n DFS(Left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order == \"in\")\n res.Add(Val(i)!.Value);\n DFS(Right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order == \"post\")\n res.Add(Val(i)!.Value);\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n public List<int> PreOrder() {\n List<int> res = [];\n DFS(0, \"pre\", res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n public List<int> InOrder() {\n List<int> res = [];\n DFS(0, \"in\", res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n public List<int> PostOrder() {\n List<int> res = [];\n DFS(0, \"post\", res);\n return res;\n }\n}\n
array_binary_tree.swift/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\ntype arrayBinaryTree struct {\n tree []any\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n return &arrayBinaryTree{\n tree: arr,\n }\n}\n\n/* \u8282\u70b9\u6570\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\n return len(abt.tree)\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if i < 0 || i >= abt.size() {\n return nil\n }\n return abt.tree[i]\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n return 2*i + 1\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n return (i - 1) / 2\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n var res []any\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for i := 0; i < abt.size(); i++ {\n if abt.val(i) != nil {\n res = append(res, abt.val(i))\n }\n }\n return res\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if abt.val(i) == nil {\n return\n }\n // \u524d\u5e8f\u904d\u5386\n if order == \"pre\" {\n *res = append(*res, abt.val(i))\n }\n abt.dfs(abt.left(i), order, res)\n // \u4e2d\u5e8f\u904d\u5386\n if order == \"in\" {\n *res = append(*res, abt.val(i))\n }\n abt.dfs(abt.right(i), order, res)\n // \u540e\u5e8f\u904d\u5386\n if order == \"post\" {\n *res = append(*res, abt.val(i))\n }\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n var res []any\n abt.dfs(0, \"pre\", &res)\n return res\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n var res []any\n abt.dfs(0, \"in\", &res)\n return res\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n var res []any\n abt.dfs(0, \"post\", &res)\n return res\n}\n
array_binary_tree.js/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n private var tree: [Int?]\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init(arr: [Int?]) {\n tree = arr\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n func size() -> Int {\n tree.count\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n func val(i: Int) -> Int? {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if i < 0 || i >= size() {\n return nil\n }\n return tree[i]\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n func left(i: Int) -> Int {\n 2 * i + 1\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n func right(i: Int) -> Int {\n 2 * i + 2\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n func parent(i: Int) -> Int {\n (i - 1) / 2\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n func levelOrder() -> [Int] {\n var res: [Int] = []\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for i in stride(from: 0, to: size(), by: 1) {\n if let val = val(i: i) {\n res.append(val)\n }\n }\n return res\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n private func dfs(i: Int, order: String, res: inout [Int]) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n guard let val = val(i: i) else {\n return\n }\n // \u524d\u5e8f\u904d\u5386\n if order == \"pre\" {\n res.append(val)\n }\n dfs(i: left(i: i), order: order, res: &res)\n // \u4e2d\u5e8f\u904d\u5386\n if order == \"in\" {\n res.append(val)\n }\n dfs(i: right(i: i), order: order, res: &res)\n // \u540e\u5e8f\u904d\u5386\n if order == \"post\" {\n res.append(val)\n }\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n func preOrder() -> [Int] {\n var res: [Int] = []\n dfs(i: 0, order: \"pre\", res: &res)\n return res\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n func inOrder() -> [Int] {\n var res: [Int] = []\n dfs(i: 0, order: \"in\", res: &res)\n return res\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n func postOrder() -> [Int] {\n var res: [Int] = []\n dfs(i: 0, order: \"post\", res: &res)\n return res\n }\n}\n
array_binary_tree.ts/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n #tree;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(arr) {\n this.#tree = arr;\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n size() {\n return this.#tree.length;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n val(i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= this.size()) return null;\n return this.#tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n left(i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n right(i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n parent(i) {\n return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n levelOrder() {\n let res = [];\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (let i = 0; i < this.size(); i++) {\n if (this.val(i) !== null) res.push(this.val(i));\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n #dfs(i, order, res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (this.val(i) === null) return;\n // \u524d\u5e8f\u904d\u5386\n if (order === 'pre') res.push(this.val(i));\n this.#dfs(this.left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order === 'in') res.push(this.val(i));\n this.#dfs(this.right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order === 'post') res.push(this.val(i));\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n preOrder() {\n const res = [];\n this.#dfs(0, 'pre', res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n inOrder() {\n const res = [];\n this.#dfs(0, 'in', res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n postOrder() {\n const res = [];\n this.#dfs(0, 'post', res);\n return res;\n }\n}\n
array_binary_tree.dart/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n #tree: (number | null)[];\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(arr: (number | null)[]) {\n this.#tree = arr;\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n size(): number {\n return this.#tree.length;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n val(i: number): number | null {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= this.size()) return null;\n return this.#tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n left(i: number): number {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n right(i: number): number {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n parent(i: number): number {\n return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n levelOrder(): number[] {\n let res = [];\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (let i = 0; i < this.size(); i++) {\n if (this.val(i) !== null) res.push(this.val(i));\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n #dfs(i: number, order: Order, res: (number | null)[]): void {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (this.val(i) === null) return;\n // \u524d\u5e8f\u904d\u5386\n if (order === 'pre') res.push(this.val(i));\n this.#dfs(this.left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order === 'in') res.push(this.val(i));\n this.#dfs(this.right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order === 'post') res.push(this.val(i));\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n preOrder(): (number | null)[] {\n const res = [];\n this.#dfs(0, 'pre', res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n inOrder(): (number | null)[] {\n const res = [];\n this.#dfs(0, 'in', res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n postOrder(): (number | null)[] {\n const res = [];\n this.#dfs(0, 'post', res);\n return res;\n }\n}\n
array_binary_tree.rs/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n late List<int?> _tree;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n ArrayBinaryTree(this._tree);\n\n /* \u8282\u70b9\u6570\u91cf */\n int size() {\n return _tree.length;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n int? val(int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= size()) {\n return null;\n }\n return _tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n int? left(int i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n int? right(int i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n int? parent(int i) {\n return (i - 1) ~/ 2;\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n List<int> levelOrder() {\n List<int> res = [];\n for (int i = 0; i < size(); i++) {\n if (val(i) != null) {\n res.add(val(i)!);\n }\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n void dfs(int i, String order, List<int?> res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (val(i) == null) {\n return;\n }\n // \u524d\u5e8f\u904d\u5386\n if (order == 'pre') {\n res.add(val(i));\n }\n dfs(left(i)!, order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order == 'in') {\n res.add(val(i));\n }\n dfs(right(i)!, order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order == 'post') {\n res.add(val(i));\n }\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n List<int?> preOrder() {\n List<int?> res = [];\n dfs(0, 'pre', res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n List<int?> inOrder() {\n List<int?> res = [];\n dfs(0, 'in', res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n List<int?> postOrder() {\n List<int?> res = [];\n dfs(0, 'post', res);\n return res;\n }\n}\n
array_binary_tree.c/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nstruct ArrayBinaryTree {\n tree: Vec<Option<i32>>,\n}\n\nimpl ArrayBinaryTree {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new(arr: Vec<Option<i32>>) -> Self {\n Self { tree: arr }\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n fn size(&self) -> i32 {\n self.tree.len() as i32\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n fn val(&self, i: i32) -> Option<i32> {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if i < 0 || i >= self.size() {\n None\n } else {\n self.tree[i as usize]\n }\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n fn left(&self, i: i32) -> i32 {\n 2 * i + 1\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n fn right(&self, i: i32) -> i32 {\n 2 * i + 2\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n fn parent(&self, i: i32) -> i32 {\n (i - 1) / 2\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n fn level_order(&self) -> Vec<i32> {\n let mut res = vec![];\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for i in 0..self.size() {\n if let Some(val) = self.val(i) {\n res.push(val)\n }\n }\n res\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n fn dfs(&self, i: i32, order: &str, res: &mut Vec<i32>) {\n if self.val(i).is_none() {\n return;\n }\n let val = self.val(i).unwrap();\n // \u524d\u5e8f\u904d\u5386\n if order == \"pre\" {\n res.push(val);\n }\n self.dfs(self.left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if order == \"in\" {\n res.push(val);\n }\n self.dfs(self.right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if order == \"post\" {\n res.push(val);\n }\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n fn pre_order(&self) -> Vec<i32> {\n let mut res = vec![];\n self.dfs(0, \"pre\", &mut res);\n res\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n fn in_order(&self) -> Vec<i32> {\n let mut res = vec![];\n self.dfs(0, \"in\", &mut res);\n res\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n fn post_order(&self) -> Vec<i32> {\n let mut res = vec![];\n self.dfs(0, \"post\", &mut res);\n res\n }\n}\n
array_binary_tree.zig/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7ed3\u6784\u4f53 */\ntypedef struct {\n int *tree;\n int size;\n} ArrayBinaryTree;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n abt->tree = malloc(sizeof(int) * arrSize);\n memcpy(abt->tree, arr, sizeof(int) * arrSize);\n abt->size = arrSize;\n return abt;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n free(abt->tree);\n free(abt);\n}\n\n/* \u8282\u70b9\u6570\u91cf */\nint size(ArrayBinaryTree *abt) {\n return abt->size;\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= size(abt))\n return INT_MAX;\n return abt->tree[i];\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n int *res = (int *)malloc(sizeof(int) * size(abt));\n int index = 0;\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size(abt); i++) {\n if (val(abt, i) != INT_MAX)\n res[index++] = val(abt, i);\n }\n *returnSize = index;\n return res;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (val(abt, i) == INT_MAX)\n return;\n // \u524d\u5e8f\u904d\u5386\n if (strcmp(order, \"pre\") == 0)\n res[(*index)++] = val(abt, i);\n dfs(abt, left(i), order, res, index);\n // \u4e2d\u5e8f\u904d\u5386\n if (strcmp(order, \"in\") == 0)\n res[(*index)++] = val(abt, i);\n dfs(abt, right(i), order, res, index);\n // \u540e\u5e8f\u904d\u5386\n if (strcmp(order, \"post\") == 0)\n res[(*index)++] = val(abt, i);\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n int *res = (int *)malloc(sizeof(int) * size(abt));\n int index = 0;\n dfs(abt, 0, \"pre\", res, &index);\n *returnSize = index;\n return res;\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n int *res = (int *)malloc(sizeof(int) * size(abt));\n int index = 0;\n dfs(abt, 0, \"in\", res, &index);\n *returnSize = index;\n return res;\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n int *res = (int *)malloc(sizeof(int) * size(abt));\n int index = 0;\n dfs(abt, 0, \"post\", res, &index);\n *returnSize = index;\n return res;\n}\n
"},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"[class]{ArrayBinaryTree}-[func]{}\n\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\u4e3b\u8981\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002
- \u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u5bf9\u7f13\u5b58\u53cb\u597d\uff0c\u8bbf\u95ee\u4e0e\u904d\u5386\u901f\u5ea6\u8f83\u5feb\u3002
- \u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u6bd4\u8f83\u8282\u7701\u7a7a\u95f4\u3002
- \u5141\u8bb8\u968f\u673a\u8bbf\u95ee\u8282\u70b9\u3002
\u7136\u800c\uff0c\u6570\u7ec4\u8868\u793a\u4e5f\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002
- \u6570\u7ec4\u5b58\u50a8\u9700\u8981\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5b58\u50a8\u6570\u636e\u91cf\u8fc7\u5927\u7684\u6811\u3002
- \u589e\u5220\u8282\u70b9\u9700\u8981\u901a\u8fc7\u6570\u7ec4\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u5b9e\u73b0\uff0c\u6548\u7387\u8f83\u4f4e\u3002
- \u5f53\u4e8c\u53c9\u6811\u4e2d\u5b58\u5728\u5927\u91cf \\(\\text{None}\\) \u65f6\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u7684\u8282\u70b9\u6570\u636e\u6bd4\u91cd\u8f83\u4f4e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u8f83\u4f4e\u3002
\u5728\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\uff0c\u5728\u591a\u6b21\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u9000\u5316\u4e3a\u94fe\u8868\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c06\u4ece \\(O(\\log n)\\) \u6076\u5316\u4e3a \\(O(n)\\) \u3002
\u5982\u56fe 7-24 \u6240\u793a\uff0c\u7ecf\u8fc7\u4e24\u6b21\u5220\u9664\u8282\u70b9\u64cd\u4f5c\uff0c\u8fd9\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\u4fbf\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002
\u56fe 7-24 \u00a0 AVL \u6811\u5728\u5220\u9664\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316
\u518d\u4f8b\u5982\uff0c\u5728\u56fe 7-25 \u6240\u793a\u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u8282\u70b9\u540e\uff0c\u6811\u5c06\u4e25\u91cd\u5411\u5de6\u503e\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u6076\u5316\u3002
\u56fe 7-25 \u00a0 AVL \u6811\u5728\u63d2\u5165\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316
1962 \u5e74 G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u8bba\u6587 \"An algorithm for the organization of information\" \u4e2d\u63d0\u51fa\u4e86\u300cAVL \u6811\u300d\u3002\u8bba\u6587\u4e2d\u8be6\u7ec6\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u6301\u7eed\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4e0d\u4f1a\u9000\u5316\uff0c\u4ece\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u9700\u8981\u9891\u7e41\u8fdb\u884c\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u573a\u666f\u4e2d\uff0cAVL \u6811\u80fd\u59cb\u7ec8\u4fdd\u6301\u9ad8\u6548\u7684\u6570\u636e\u64cd\u4f5c\u6027\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002
"},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1 \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"AVL \u6811\u65e2\u662f\u4e8c\u53c9\u641c\u7d22\u6811\u4e5f\u662f\u5e73\u8861\u4e8c\u53c9\u6811\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u7c7b\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u4e5f\u88ab\u79f0\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 balanced binary search tree\u300d\u3002
"},{"location":"chapter_tree/avl_tree/#1","title":"1. \u00a0 \u8282\u70b9\u9ad8\u5ea6","text":"\u7531\u4e8e AVL \u6811\u7684\u76f8\u5173\u64cd\u4f5c\u9700\u8981\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e3a\u8282\u70b9\u7c7b\u6dfb\u52a0
PythonC++JavaC#GoSwiftJSTSDartRustCZigheight\u53d8\u91cf\uff1aclass TreeNode:\n \"\"\"AVL \u6811\u8282\u70b9\u7c7b\"\"\"\n def __init__(self, val: int):\n self.val: int = val # \u8282\u70b9\u503c\n self.height: int = 0 # \u8282\u70b9\u9ad8\u5ea6\n self.left: TreeNode | None = None # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n self.right: TreeNode | None = None # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n/* AVL \u6811\u8282\u70b9\u7c7b */\nstruct TreeNode {\n int val{}; // \u8282\u70b9\u503c\n int height = 0; // \u8282\u70b9\u9ad8\u5ea6\n TreeNode *left{}; // \u5de6\u5b50\u8282\u70b9\n TreeNode *right{}; // \u53f3\u5b50\u8282\u70b9\n TreeNode() = default;\n explicit TreeNode(int x) : val(x){}\n};\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n public int val; // \u8282\u70b9\u503c\n public int height; // \u8282\u70b9\u9ad8\u5ea6\n public TreeNode left; // \u5de6\u5b50\u8282\u70b9\n public TreeNode right; // \u53f3\u5b50\u8282\u70b9\n public TreeNode(int x) { val = x; }\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n public int? val = x; // \u8282\u70b9\u503c\n public int height; // \u8282\u70b9\u9ad8\u5ea6\n public TreeNode? left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n Val int // \u8282\u70b9\u503c\n Height int // \u8282\u70b9\u9ad8\u5ea6\n Left *TreeNode // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n Right *TreeNode // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n var val: Int // \u8282\u70b9\u503c\n var height: Int // \u8282\u70b9\u9ad8\u5ea6\n var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\n var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\n\n init(x: Int) {\n val = x\n height = 0\n }\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n val; // \u8282\u70b9\u503c\n height; //\u8282\u70b9\u9ad8\u5ea6\n left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n constructor(val, left, right, height) {\n this.val = val === undefined ? 0 : val;\n this.height = height === undefined ? 0 : height;\n this.left = left === undefined ? null : left;\n this.right = right === undefined ? null : right;\n }\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n val: number; // \u8282\u70b9\u503c\n height: number; // \u8282\u70b9\u9ad8\u5ea6\n left: TreeNode | null; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n right: TreeNode | null; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n this.val = val === undefined ? 0 : val;\n this.height = height === undefined ? 0 : height; \n this.left = left === undefined ? null : left; \n this.right = right === undefined ? null : right; \n }\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n int val; // \u8282\u70b9\u503c\n int height; // \u8282\u70b9\u9ad8\u5ea6\n TreeNode? left; // \u5de6\u5b50\u8282\u70b9\n TreeNode? right; // \u53f3\u5b50\u8282\u70b9\n TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n val: i32, // \u8282\u70b9\u503c\n height: i32, // \u8282\u70b9\u9ad8\u5ea6\n left: Option<Rc<RefCell<TreeNode>>>, // \u5de6\u5b50\u8282\u70b9\n right: Option<Rc<RefCell<TreeNode>>>, // \u53f3\u5b50\u8282\u70b9\n}\n\nimpl TreeNode {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new(val: i32) -> Rc<RefCell<Self>> {\n Rc::new(RefCell::new(Self {\n val,\n height: 0,\n left: None,\n right: None\n }))\n }\n}\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nTreeNode struct TreeNode {\n int val;\n int height;\n struct TreeNode *left;\n struct TreeNode *right;\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n TreeNode *node;\n\n node = (TreeNode *)malloc(sizeof(TreeNode));\n node->val = val;\n node->height = 0;\n node->left = NULL;\n node->right = NULL;\n return node;\n}\n\n\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u662f\u6307\u4ece\u8be5\u8282\u70b9\u5230\u5176\u6700\u8fdc\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u6240\u7ecf\u8fc7\u7684\u201c\u8fb9\u201d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\u7684\u662f\uff0c\u53f6\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(0\\) \uff0c\u800c\u7a7a\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(-1\\) \u3002\u6211\u4eec\u5c06\u521b\u5efa\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u548c\u66f4\u65b0\u8282\u70b9\u7684\u9ad8\u5ea6\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef height(self, node: TreeNode | None) -> int:\n \"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n # \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n if node is not None:\n return node.height\n return -1\n\ndef update_height(self, node: TreeNode | None):\n \"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n # \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height = max([self.height(node.left), self.height(node.right)]) + 1\n
avl_tree.java/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node == nullptr ? -1 : node->height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node->height = max(height(node->left), height(node->right)) + 1;\n}\n
avl_tree.cs/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n
avl_tree.go/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n
avl_tree.swift/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n if node != nil {\n return node.Height\n }\n return -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n lh := t.height(node.Left)\n rh := t.height(node.Right)\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n if lh > rh {\n node.Height = lh + 1\n } else {\n node.Height = rh + 1\n }\n}\n
avl_tree.js/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -> Int {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n node == nil ? -1 : node!.height\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n
avl_tree.ts/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height =\n Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
avl_tree.dart/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height =\n Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
avl_tree.rs/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node!.height = max(height(node.left), height(node.right)) + 1;\n}\n
avl_tree.c/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -> i32 {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n match node {\n Some(node) => node.borrow().height,\n None => -1,\n }\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\n if let Some(node) = node {\n let left = node.borrow().left.clone();\n let right = node.borrow().right.clone();\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n }\n}\n
avl_tree.zig/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n if (node != NULL) {\n return node->height;\n }\n return -1;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n int lh = height(node->left);\n int rh = height(node->right);\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n if (lh > rh) {\n node->height = lh + 1;\n } else {\n node->height = rh + 1;\n }\n}\n
"},{"location":"chapter_tree/avl_tree/#2","title":"2. \u00a0 \u8282\u70b9\u5e73\u8861\u56e0\u5b50","text":"// \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n _ = self;\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return if (node == null) -1 else node.?.height;\n}\n\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n\u8282\u70b9\u7684\u300c\u5e73\u8861\u56e0\u5b50 balance factor\u300d\u5b9a\u4e49\u4e3a\u8282\u70b9\u5de6\u5b50\u6811\u7684\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\uff0c\u540c\u65f6\u89c4\u5b9a\u7a7a\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a \\(0\\) \u3002\u6211\u4eec\u540c\u6837\u5c06\u83b7\u53d6\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef balance_factor(self, node: TreeNode | None) -> int:\n \"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n # \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if node is None:\n return 0\n # \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return self.height(node.left) - self.height(node.right)\n
avl_tree.java/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == nullptr)\n return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node->left) - height(node->right);\n}\n
avl_tree.cs/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == null)\n return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node.left) - height(node.right);\n}\n
avl_tree.go/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return Height(node.left) - Height(node.right);\n}\n
avl_tree.swift/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if node == nil {\n return 0\n }\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return t.height(node.Left) - t.height(node.Right)\n}\n
avl_tree.js/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -> Int {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n guard let node = node else { return 0 }\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node: node.left) - height(node: node.right)\n}\n
avl_tree.ts/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node === null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return this.height(node.left) - this.height(node.right);\n}\n
avl_tree.dart/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node === null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return this.height(node.left) - this.height(node.right);\n}\n
avl_tree.rs/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node.left) - height(node.right);\n}\n
avl_tree.c/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -> i32 {\n match node {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n None => 0,\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n Some(node) => {\n Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n }\n }\n}\n
avl_tree.zig/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == NULL) {\n return 0;\n }\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node->left) - height(node->right);\n}\n// \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return self.height(node.?.left) - self.height(node.?.right);\n}\nNote
\u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002
"},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2 \u00a0 AVL \u6811\u65cb\u8f6c","text":"AVL \u6811\u7684\u7279\u70b9\u5728\u4e8e\u201c\u65cb\u8f6c\u201d\u64cd\u4f5c\uff0c\u5b83\u80fd\u591f\u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7684\u6027\u8d28\uff0c\u4e5f\u80fd\u4f7f\u6811\u91cd\u65b0\u53d8\u4e3a\u201c\u5e73\u8861\u4e8c\u53c9\u6811\u201d\u3002
\u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7edd\u5bf9\u503c \\(> 1\\) \u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5931\u8861\u8282\u70b9\u201d\u3002\u6839\u636e\u8282\u70b9\u5931\u8861\u60c5\u51b5\u7684\u4e0d\u540c\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u56db\u79cd\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u4e0b\u9762\u8be6\u7ec6\u4ecb\u7ecd\u8fd9\u4e9b\u65cb\u8f6c\u64cd\u4f5c\u3002
"},{"location":"chapter_tree/avl_tree/#1_1","title":"1. \u00a0 \u53f3\u65cb","text":"\u5982\u56fe 7-26 \u6240\u793a\uff0c\u8282\u70b9\u4e0b\u65b9\u4e3a\u5e73\u8861\u56e0\u5b50\u3002\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u8282\u70b9\u662f\u201c\u8282\u70b9 3\u201d\u3002\u6211\u4eec\u5173\u6ce8\u4ee5\u8be5\u5931\u8861\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\uff0c\u5c06\u8be5\u8282\u70b9\u8bb0\u4e3a
<1><2><3><4>node\uff0c\u5176\u5de6\u5b50\u8282\u70b9\u8bb0\u4e3achild\uff0c\u6267\u884c\u201c\u53f3\u65cb\u201d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u5b50\u6811\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u3002\u56fe 7-26 \u00a0 \u53f3\u65cb\u64cd\u4f5c\u6b65\u9aa4
\u5982\u56fe 7-27 \u6240\u793a\uff0c\u5f53\u8282\u70b9
child\u6709\u53f3\u5b50\u8282\u70b9\uff08\u8bb0\u4e3agrandChild\uff09\u65f6\uff0c\u9700\u8981\u5728\u53f3\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06grandChild\u4f5c\u4e3anode\u7684\u5de6\u5b50\u8282\u70b9\u3002\u56fe 7-27 \u00a0 \u6709 grandChild \u7684\u53f3\u65cb\u64cd\u4f5c
\u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u4e0a\u9700\u8981\u901a\u8fc7\u4fee\u6539\u8282\u70b9\u6307\u9488\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef right_rotate(self, node: TreeNode | None) -> TreeNode | None:\n \"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\n child = node.left\n grand_child = child.right\n # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node\n node.left = grand_child\n # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.update_height(node)\n self.update_height(child)\n # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n
avl_tree.java/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n TreeNode *child = node->left;\n TreeNode *grandChild = child->right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child->right = node;\n node->left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.cs/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\n TreeNode child = node.left;\n TreeNode grandChild = child.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.go/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\n TreeNode? child = node?.left;\n TreeNode? grandChild = child?.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n UpdateHeight(node);\n UpdateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.swift/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n child := node.Left\n grandChild := child.Right\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.Right = node\n node.Left = grandChild\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n t.updateHeight(node)\n t.updateHeight(child)\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n}\n
avl_tree.js/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -> TreeNode? {\n let child = node?.left\n let grandChild = child?.right\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child?.right = node\n node?.left = grandChild\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node: node)\n updateHeight(node: child)\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n}\n
avl_tree.ts/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\n const child = node.left;\n const grandChild = child.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n this.#updateHeight(node);\n this.#updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.dart/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\n const child = node.left;\n const grandChild = child.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n this.updateHeight(node);\n this.updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.rs/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\n TreeNode? child = node!.left;\n TreeNode? grandChild = child!.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.c/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {\n match node {\n Some(node) => {\n let child = node.borrow().left.clone().unwrap();\n let grand_child = child.borrow().right.clone();\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.borrow_mut().right = Some(node.clone());\n node.borrow_mut().left = grand_child;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n Self::update_height(Some(node));\n Self::update_height(Some(child.clone()));\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n Some(child)\n }\n None => None,\n }\n}\n
avl_tree.zig/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n TreeNode *child, *grandChild;\n child = node->left;\n grandChild = child->right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child->right = node;\n node->left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
"},{"location":"chapter_tree/avl_tree/#2_1","title":"2. \u00a0 \u5de6\u65cb","text":"// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n var child = node.?.left;\n var grandChild = child.?.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.?.right = node;\n node.?.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.updateHeight(node);\n self.updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n\u76f8\u5e94\u5730\uff0c\u5982\u679c\u8003\u8651\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u5219\u9700\u8981\u6267\u884c\u56fe 7-28 \u6240\u793a\u7684\u201c\u5de6\u65cb\u201d\u64cd\u4f5c\u3002
\u56fe 7-28 \u00a0 \u5de6\u65cb\u64cd\u4f5c
\u540c\u7406\uff0c\u5982\u56fe 7-29 \u6240\u793a\uff0c\u5f53\u8282\u70b9
child\u6709\u5de6\u5b50\u8282\u70b9\uff08\u8bb0\u4e3agrandChild\uff09\u65f6\uff0c\u9700\u8981\u5728\u5de6\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06grandChild\u4f5c\u4e3anode\u7684\u53f3\u5b50\u8282\u70b9\u3002\u56fe 7-29 \u00a0 \u6709 grandChild \u7684\u5de6\u65cb\u64cd\u4f5c
\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u903b\u8f91\u4e0a\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u5b83\u4eec\u5206\u522b\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u57fa\u4e8e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u53f3\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\u4e2d\u7684\u6240\u6709\u7684
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.pyleft\u66ff\u6362\u4e3aright\uff0c\u5c06\u6240\u6709\u7684right\u66ff\u6362\u4e3aleft\uff0c\u5373\u53ef\u5f97\u5230\u5de6\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a
avl_tree.cppdef left_rotate(self, node: TreeNode | None) -> TreeNode | None:\n \"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\n child = node.right\n grand_child = child.left\n # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node\n node.right = grand_child\n # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.update_height(node)\n self.update_height(child)\n # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n
avl_tree.java/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n TreeNode *child = node->right;\n TreeNode *grandChild = child->left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child->left = node;\n node->right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.cs/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\n TreeNode child = node.right;\n TreeNode grandChild = child.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.go/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\n TreeNode? child = node?.right;\n TreeNode? grandChild = child?.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n UpdateHeight(node);\n UpdateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.swift/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n child := node.Right\n grandChild := child.Left\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.Left = node\n node.Right = grandChild\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n t.updateHeight(node)\n t.updateHeight(child)\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n}\n
avl_tree.js/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -> TreeNode? {\n let child = node?.right\n let grandChild = child?.left\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child?.left = node\n node?.right = grandChild\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node: node)\n updateHeight(node: child)\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n}\n
avl_tree.ts/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\n const child = node.right;\n const grandChild = child.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n this.#updateHeight(node);\n this.#updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.dart/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\n const child = node.right;\n const grandChild = child.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n this.updateHeight(node);\n this.updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.rs/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\n TreeNode? child = node!.right;\n TreeNode? grandChild = child!.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.c/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {\n match node {\n Some(node) => {\n let child = node.borrow().right.clone().unwrap();\n let grand_child = child.borrow().left.clone();\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.borrow_mut().left = Some(node.clone());\n node.borrow_mut().right = grand_child;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n Self::update_height(Some(node));\n Self::update_height(Some(child.clone()));\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n Some(child)\n }\n None => None,\n }\n}\n
avl_tree.zig/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n TreeNode *child, *grandChild;\n child = node->right;\n grandChild = child->left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child->left = node;\n node->right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
"},{"location":"chapter_tree/avl_tree/#3","title":"3. \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb","text":"// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n var child = node.?.right;\n var grandChild = child.?.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.?.left = node;\n node.?.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.updateHeight(node);\n self.updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n\u5bf9\u4e8e\u56fe 7-30 \u4e2d\u7684\u5931\u8861\u8282\u70b9 3 \uff0c\u4ec5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\u3002\u6b64\u65f6\u9700\u8981\u5148\u5bf9
child\u6267\u884c\u201c\u5de6\u65cb\u201d\uff0c\u518d\u5bf9node\u6267\u884c\u201c\u53f3\u65cb\u201d\u3002\u56fe 7-30 \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb
"},{"location":"chapter_tree/avl_tree/#4","title":"4. \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb","text":"\u5982\u56fe 7-31 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\u60c5\u51b5\uff0c\u9700\u8981\u5148\u5bf9
child\u6267\u884c\u201c\u53f3\u65cb\u201d\uff0c\u518d\u5bf9node\u6267\u884c\u201c\u5de6\u65cb\u201d\u3002\u56fe 7-31 \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb
"},{"location":"chapter_tree/avl_tree/#5","title":"5. \u00a0 \u65cb\u8f6c\u7684\u9009\u62e9","text":"\u56fe 7-32 \u5c55\u793a\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0\u6848\u4f8b\u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u8981\u91c7\u7528\u53f3\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5de6\u65cb\u7684\u64cd\u4f5c\u3002
\u56fe 7-32 \u00a0 AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5
\u5982\u4e0b\u8868\u6240\u793a\uff0c\u6211\u4eec\u901a\u8fc7\u5224\u65ad\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f83\u9ad8\u4e00\u4fa7\u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8d1f\u53f7\uff0c\u6765\u786e\u5b9a\u5931\u8861\u8282\u70b9\u5c5e\u4e8e\u56fe 7-32 \u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002
\u8868 7-3 \u00a0 \u56db\u79cd\u65cb\u8f6c\u60c5\u51b5\u7684\u9009\u62e9\u6761\u4ef6
\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(> 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(> 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(<0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(< -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(< -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(>0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb\u4e3a\u4e86\u4fbf\u4e8e\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u6709\u4e86\u8fd9\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5c31\u80fd\u5bf9\u5404\u79cd\u5931\u8861\u60c5\u51b5\u8fdb\u884c\u65cb\u8f6c\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef rotate(self, node: TreeNode | None) -> TreeNode | None:\n \"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n # \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n balance_factor = self.balance_factor(node)\n # \u5de6\u504f\u6811\n if balance_factor > 1:\n if self.balance_factor(node.left) >= 0:\n # \u53f3\u65cb\n return self.right_rotate(node)\n else:\n # \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = self.left_rotate(node.left)\n return self.right_rotate(node)\n # \u53f3\u504f\u6811\n elif balance_factor < -1:\n if self.balance_factor(node.right) <= 0:\n # \u5de6\u65cb\n return self.left_rotate(node)\n else:\n # \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = self.right_rotate(node.right)\n return self.left_rotate(node)\n # \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n
avl_tree.java/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int _balanceFactor = balanceFactor(node);\n // \u5de6\u504f\u6811\n if (_balanceFactor > 1) {\n if (balanceFactor(node->left) >= 0) {\n // \u53f3\u65cb\n return rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node->left = leftRotate(node->left);\n return rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (_balanceFactor < -1) {\n if (balanceFactor(node->right) <= 0) {\n // \u5de6\u65cb\n return leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node->right = rightRotate(node->right);\n return leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.cs/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int balanceFactor = balanceFactor(node);\n // \u5de6\u504f\u6811\n if (balanceFactor > 1) {\n if (balanceFactor(node.left) >= 0) {\n // \u53f3\u65cb\n return rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = leftRotate(node.left);\n return rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balanceFactor < -1) {\n if (balanceFactor(node.right) <= 0) {\n // \u5de6\u65cb\n return leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = rightRotate(node.right);\n return leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.go/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int balanceFactorInt = BalanceFactor(node);\n // \u5de6\u504f\u6811\n if (balanceFactorInt > 1) {\n if (BalanceFactor(node?.left) >= 0) {\n // \u53f3\u65cb\n return RightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node!.left = LeftRotate(node!.left);\n return RightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balanceFactorInt < -1) {\n if (BalanceFactor(node?.right) <= 0) {\n // \u5de6\u65cb\n return LeftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node!.right = RightRotate(node!.right);\n return LeftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.swift/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n // Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\n bf := t.balanceFactor(node)\n // \u5de6\u504f\u6811\n if bf > 1 {\n if t.balanceFactor(node.Left) >= 0 {\n // \u53f3\u65cb\n return t.rightRotate(node)\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.Left = t.leftRotate(node.Left)\n return t.rightRotate(node)\n }\n }\n // \u53f3\u504f\u6811\n if bf < -1 {\n if t.balanceFactor(node.Right) <= 0 {\n // \u5de6\u65cb\n return t.leftRotate(node)\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.Right = t.rightRotate(node.Right)\n return t.leftRotate(node)\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n}\n
avl_tree.js/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -> TreeNode? {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n let balanceFactor = balanceFactor(node: node)\n // \u5de6\u504f\u6811\n if balanceFactor > 1 {\n if self.balanceFactor(node: node?.left) >= 0 {\n // \u53f3\u65cb\n return rightRotate(node: node)\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node?.left = leftRotate(node: node?.left)\n return rightRotate(node: node)\n }\n }\n // \u53f3\u504f\u6811\n if balanceFactor < -1 {\n if self.balanceFactor(node: node?.right) <= 0 {\n // \u5de6\u65cb\n return leftRotate(node: node)\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node?.right = rightRotate(node: node?.right)\n return leftRotate(node: node)\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n}\n
avl_tree.ts/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n const balanceFactor = this.balanceFactor(node);\n // \u5de6\u504f\u6811\n if (balanceFactor > 1) {\n if (this.balanceFactor(node.left) >= 0) {\n // \u53f3\u65cb\n return this.#rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = this.#leftRotate(node.left);\n return this.#rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balanceFactor < -1) {\n if (this.balanceFactor(node.right) <= 0) {\n // \u5de6\u65cb\n return this.#leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = this.#rightRotate(node.right);\n return this.#leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.dart/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n const balanceFactor = this.balanceFactor(node);\n // \u5de6\u504f\u6811\n if (balanceFactor > 1) {\n if (this.balanceFactor(node.left) >= 0) {\n // \u53f3\u65cb\n return this.rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = this.leftRotate(node.left);\n return this.rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balanceFactor < -1) {\n if (this.balanceFactor(node.right) <= 0) {\n // \u5de6\u65cb\n return this.leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = this.rightRotate(node.right);\n return this.leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.rs/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int factor = balanceFactor(node);\n // \u5de6\u504f\u6811\n if (factor > 1) {\n if (balanceFactor(node!.left) >= 0) {\n // \u53f3\u65cb\n return rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = leftRotate(node.left);\n return rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (factor < -1) {\n if (balanceFactor(node!.right) <= 0) {\n // \u5de6\u65cb\n return leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = rightRotate(node.right);\n return leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.c/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n let balance_factor = Self::balance_factor(node.clone());\n // \u5de6\u504f\u6811\n if balance_factor > 1 {\n let node = node.unwrap();\n if Self::balance_factor(node.borrow().left.clone()) >= 0 {\n // \u53f3\u65cb\n Self::right_rotate(Some(node))\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n let left = node.borrow().left.clone();\n node.borrow_mut().left = Self::left_rotate(left);\n Self::right_rotate(Some(node))\n }\n }\n // \u53f3\u504f\u6811\n else if balance_factor < -1 {\n let node = node.unwrap();\n if Self::balance_factor(node.borrow().right.clone()) <= 0 {\n // \u5de6\u65cb\n Self::left_rotate(Some(node))\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n let right = node.borrow().right.clone();\n node.borrow_mut().right = Self::right_rotate(right);\n Self::left_rotate(Some(node))\n }\n } else {\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n node\n }\n}\n
avl_tree.zig/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int bf = balanceFactor(node);\n // \u5de6\u504f\u6811\n if (bf > 1) {\n if (balanceFactor(node->left) >= 0) {\n // \u53f3\u65cb\n return rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node->left = leftRotate(node->left);\n return rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (bf < -1) {\n if (balanceFactor(node->right) <= 0) {\n // \u5de6\u65cb\n return leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node->right = rightRotate(node->right);\n return leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
"},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3 \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#1_2","title":"1. \u00a0 \u63d2\u5165\u8282\u70b9","text":"// \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n var balance_factor = self.balanceFactor(node);\n // \u5de6\u504f\u6811\n if (balance_factor > 1) {\n if (self.balanceFactor(node.?.left) >= 0) {\n // \u53f3\u65cb\n return self.rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.?.left = self.leftRotate(node.?.left);\n return self.rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balance_factor < -1) {\n if (self.balanceFactor(node.?.right) <= 0) {\n // \u5de6\u65cb\n return self.leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.?.right = self.rightRotate(node.?.right);\n return self.leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\nAVL \u6811\u7684\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u5728\u4e3b\u4f53\u4e0a\u7c7b\u4f3c\u3002\u552f\u4e00\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5728 AVL \u6811\u4e2d\u63d2\u5165\u8282\u70b9\u540e\uff0c\u4ece\u8be5\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u53ef\u80fd\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u5931\u8861\u8282\u70b9\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8fd9\u4e2a\u8282\u70b9\u5f00\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef insert(self, val):\n \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -> TreeNode:\n \"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n if node is None:\n return TreeNode(val)\n # 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\n if val < node.val:\n node.left = self.insert_helper(node.left, val)\n elif val > node.val:\n node.right = self.insert_helper(node.right, val)\n else:\n # \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.update_height(node)\n # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n return self.rotate(node)\n
avl_tree.java/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n if (node == nullptr)\n return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node->val)\n node->left = insertHelper(node->left, val);\n else if (val > node->val)\n node->right = insertHelper(node->right, val);\n else\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.cs/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\n if (node == null)\n return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val)\n node.left = insertHelper(node.left, val);\n else if (val > node.val)\n node.right = insertHelper(node.right, val);\n else\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.go/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int val) {\n root = InsertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n if (node == null) return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val)\n node.left = InsertHelper(node.left, val);\n else if (val > node.val)\n node.right = InsertHelper(node.right, val);\n else\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n UpdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = Rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.swift/* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\n t.root = t.insertHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n if node == nil {\n return NewTreeNode(val)\n }\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if val < node.Val.(int) {\n node.Left = t.insertHelper(node.Left, val)\n } else if val > node.Val.(int) {\n node.Right = t.insertHelper(node.Right, val)\n } else {\n // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n }\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n t.updateHeight(node)\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = t.rotate(node)\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node\n}\n
avl_tree.js/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(val: Int) {\n root = insertHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -> TreeNode? {\n var node = node\n if node == nil {\n return TreeNode(x: val)\n }\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if val < node!.val {\n node?.left = insertHelper(node: node?.left, val: val)\n } else if val > node!.val {\n node?.right = insertHelper(node: node?.right, val: val)\n } else {\n return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n }\n updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node: node)\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node\n}\n
avl_tree.ts/* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\n this.root = this.#insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\n if (node === null) return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val) node.left = this.#insertHelper(node.left, val);\n else if (val > node.val)\n node.right = this.#insertHelper(node.right, val);\n else return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = this.#rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.dart/* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\n this.root = this.insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n if (node === null) return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val) {\n node.left = this.insertHelper(node.left, val);\n } else if (val > node.val) {\n node.right = this.insertHelper(node.right, val);\n } else {\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n }\n this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = this.rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.rs/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\n if (node == null) return TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val)\n node.left = insertHelper(node.left, val);\n else if (val > node.val)\n node.right = insertHelper(node.right, val);\n else\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.c/* \u63d2\u5165\u8282\u70b9 */\nfn insert(&mut self, val: i32) {\n self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -> OptionTreeNodeRc {\n match node {\n Some(mut node) => {\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n match {\n let node_val = node.borrow().val;\n node_val\n }\n .cmp(&val)\n {\n Ordering::Greater => {\n let left = node.borrow().left.clone();\n node.borrow_mut().left = Self::insert_helper(left, val);\n }\n Ordering::Less => {\n let right = node.borrow().right.clone();\n node.borrow_mut().right = Self::insert_helper(right, val);\n }\n Ordering::Equal => {\n return Some(node); // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n }\n }\n Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = Self::rotate(Some(node)).unwrap();\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n Some(node)\n }\n None => Some(TreeNode::new(val)),\n }\n}\n
avl_tree.zig/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(AVLTree *tree, int val) {\n tree->root = insertHelper(tree->root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n if (node == NULL) {\n return newTreeNode(val);\n }\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node->val) {\n node->left = insertHelper(node->left, val);\n } else if (val > node->val) {\n node->right = insertHelper(node->right, val);\n } else {\n // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n }\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
"},{"location":"chapter_tree/avl_tree/#2_2","title":"2. \u00a0 \u5220\u9664\u8282\u70b9","text":"// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\n self.root = (try self.insertHelper(self.root, val)).?;\n}\n\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\n var node = node_;\n if (node == null) {\n var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\n tmp_node.init(val);\n return tmp_node;\n }\n // 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\n if (val < node.?.val) {\n node.?.left = try self.insertHelper(node.?.left, val);\n } else if (val > node.?.val) {\n node.?.right = try self.insertHelper(node.?.right, val);\n } else {\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n }\n self.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n node = self.rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n\u7c7b\u4f3c\u5730\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5220\u9664\u8282\u70b9\u65b9\u6cd5\u7684\u57fa\u7840\u4e0a\uff0c\u9700\u8981\u4ece\u5e95\u81f3\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef remove(self, val: int):\n \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -> TreeNode | None:\n \"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n if node is None:\n return None\n # 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\n if val < node.val:\n node.left = self.remove_helper(node.left, val)\n elif val > node.val:\n node.right = self.remove_helper(node.right, val)\n else:\n if node.left is None or node.right is None:\n child = node.left or node.right\n # \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if child is None:\n return None\n # \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else:\n node = child\n else:\n # \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n temp = node.right\n while temp.left is not None:\n temp = temp.left\n node.right = self.remove_helper(node.right, temp.val)\n node.val = temp.val\n # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.update_height(node)\n # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n return self.rotate(node)\n
avl_tree.java/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n if (node == nullptr)\n return nullptr;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node->val)\n node->left = removeHelper(node->left, val);\n else if (val > node->val)\n node->right = removeHelper(node->right, val);\n else {\n if (node->left == nullptr || node->right == nullptr) {\n TreeNode *child = node->left != nullptr ? node->left : node->right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == nullptr) {\n delete node;\n return nullptr;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else {\n delete node;\n node = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode *temp = node->right;\n while (temp->left != nullptr) {\n temp = temp->left;\n }\n int tempVal = temp->val;\n node->right = removeHelper(node->right, temp->val);\n node->val = tempVal;\n }\n }\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.cs/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\n if (node == null)\n return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val)\n node.left = removeHelper(node.left, val);\n else if (val > node.val)\n node.right = removeHelper(node.right, val);\n else {\n if (node.left == null || node.right == null) {\n TreeNode child = node.left != null ? node.left : node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == null)\n return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else\n node = child;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode temp = node.right;\n while (temp.left != null) {\n temp = temp.left;\n }\n node.right = removeHelper(node.right, temp.val);\n node.val = temp.val;\n }\n }\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.go/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int val) {\n root = RemoveHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n if (node == null) return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val)\n node.left = RemoveHelper(node.left, val);\n else if (val > node.val)\n node.right = RemoveHelper(node.right, val);\n else {\n if (node.left == null || node.right == null) {\n TreeNode? child = node.left ?? node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == null)\n return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else\n node = child;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode? temp = node.right;\n while (temp.left != null) {\n temp = temp.left;\n }\n node.right = RemoveHelper(node.right, temp.val!.Value);\n node.val = temp.val;\n }\n }\n UpdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = Rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.swift/* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\n t.root = t.removeHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n if node == nil {\n return nil\n }\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if val < node.Val.(int) {\n node.Left = t.removeHelper(node.Left, val)\n } else if val > node.Val.(int) {\n node.Right = t.removeHelper(node.Right, val)\n } else {\n if node.Left == nil || node.Right == nil {\n child := node.Left\n if node.Right != nil {\n child = node.Right\n }\n if child == nil {\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n return nil\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n node = child\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n temp := node.Right\n for temp.Left != nil {\n temp = temp.Left\n }\n node.Right = t.removeHelper(node.Right, temp.Val.(int))\n node.Val = temp.Val\n }\n }\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n t.updateHeight(node)\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = t.rotate(node)\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node\n}\n
avl_tree.js/* \u5220\u9664\u8282\u70b9 */\nfunc remove(val: Int) {\n root = removeHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -> TreeNode? {\n var node = node\n if node == nil {\n return nil\n }\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if val < node!.val {\n node?.left = removeHelper(node: node?.left, val: val)\n } else if val > node!.val {\n node?.right = removeHelper(node: node?.right, val: val)\n } else {\n if node?.left == nil || node?.right == nil {\n let child = node?.left != nil ? node?.left : node?.right\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if child == nil {\n return nil\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else {\n node = child\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n var temp = node?.right\n while temp?.left != nil {\n temp = temp?.left\n }\n node?.right = removeHelper(node: node?.right, val: temp!.val)\n node?.val = temp!.val\n }\n }\n updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node: node)\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node\n}\n
avl_tree.ts/* \u5220\u9664\u8282\u70b9 */\nremove(val) {\n this.root = this.#removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\n if (node === null) return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val) node.left = this.#removeHelper(node.left, val);\n else if (val > node.val)\n node.right = this.#removeHelper(node.right, val);\n else {\n if (node.left === null || node.right === null) {\n const child = node.left !== null ? node.left : node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child === null) return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else node = child;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n let temp = node.right;\n while (temp.left !== null) {\n temp = temp.left;\n }\n node.right = this.#removeHelper(node.right, temp.val);\n node.val = temp.val;\n }\n }\n this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = this.#rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.dart/* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\n this.root = this.removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n if (node === null) return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val) {\n node.left = this.removeHelper(node.left, val);\n } else if (val > node.val) {\n node.right = this.removeHelper(node.right, val);\n } else {\n if (node.left === null || node.right === null) {\n const child = node.left !== null ? node.left : node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child === null) {\n return null;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n node = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n let temp = node.right;\n while (temp.left !== null) {\n temp = temp.left;\n }\n node.right = this.removeHelper(node.right, temp.val);\n node.val = temp.val;\n }\n }\n this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = this.rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.rs/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\n if (node == null) return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val)\n node.left = removeHelper(node.left, val);\n else if (val > node.val)\n node.right = removeHelper(node.right, val);\n else {\n if (node.left == null || node.right == null) {\n TreeNode? child = node.left ?? node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == null)\n return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else\n node = child;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode? temp = node.right;\n while (temp!.left != null) {\n temp = temp.left;\n }\n node.right = removeHelper(node.right, temp.val);\n node.val = temp.val;\n }\n }\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.c/* \u5220\u9664\u8282\u70b9 */\nfn remove(&self, val: i32) {\n Self::remove_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -> OptionTreeNodeRc {\n match node {\n Some(mut node) => {\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if val < node.borrow().val {\n let left = node.borrow().left.clone();\n node.borrow_mut().left = Self::remove_helper(left, val);\n } else if val > node.borrow().val {\n let right = node.borrow().right.clone();\n node.borrow_mut().right = Self::remove_helper(right, val);\n } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n let child = if node.borrow().left.is_some() {\n node.borrow().left.clone()\n } else {\n node.borrow().right.clone()\n };\n match child {\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n None => {\n return None;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n Some(child) => node = child,\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n let mut temp = node.borrow().right.clone().unwrap();\n loop {\n let temp_left = temp.borrow().left.clone();\n if temp_left.is_none() {\n break;\n }\n temp = temp_left.unwrap();\n }\n let right = node.borrow().right.clone();\n node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n node.borrow_mut().val = temp.borrow().val;\n }\n Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = Self::rotate(Some(node)).unwrap();\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n Some(node)\n }\n None => None,\n }\n}\n
avl_tree.zig/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(AVLTree *tree, int val) {\n TreeNode *root = removeHelper(tree->root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n TreeNode *child, *grandChild;\n if (node == NULL) {\n return NULL;\n }\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node->val) {\n node->left = removeHelper(node->left, val);\n } else if (val > node->val) {\n node->right = removeHelper(node->right, val);\n } else {\n if (node->left == NULL || node->right == NULL) {\n child = node->left;\n if (node->right != NULL) {\n child = node->right;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == NULL) {\n return NULL;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n node = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode *temp = node->right;\n while (temp->left != NULL) {\n temp = temp->left;\n }\n int tempVal = temp->val;\n node->right = removeHelper(node->right, temp->val);\n node->val = tempVal;\n }\n }\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
"},{"location":"chapter_tree/avl_tree/#3_1","title":"3. \u00a0 \u67e5\u627e\u8282\u70b9","text":"// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\n self.root = self.removeHelper(self.root, val).?;\n}\n\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\n var node = node_;\n if (node == null) return null;\n // 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\n if (val < node.?.val) {\n node.?.left = self.removeHelper(node.?.left, val);\n } else if (val > node.?.val) {\n node.?.right = self.removeHelper(node.?.right, val);\n } else {\n if (node.?.left == null or node.?.right == null) {\n var child = if (node.?.left != null) node.?.left else node.?.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == null) {\n return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n } else {\n node = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n var temp = node.?.right;\n while (temp.?.left != null) {\n temp = temp.?.left;\n }\n node.?.right = self.removeHelper(node.?.right, temp.?.val);\n node.?.val = temp.?.val;\n }\n }\n self.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n node = self.rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\nAVL \u6811\u7684\u8282\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
"},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4 \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"- \u7ec4\u7ec7\u548c\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u7684\u573a\u666f\u3002
- \u7528\u4e8e\u6784\u5efa\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\u3002
- \u7ea2\u9ed1\u6811\u5728\u8bb8\u591a\u5e94\u7528\u4e2d\u6bd4 AVL \u6811\u66f4\u53d7\u6b22\u8fce\u3002\u8fd9\u662f\u56e0\u4e3a\u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u76f8\u5bf9\u5bbd\u677e\uff0c\u5728\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u5176\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u7684\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002
\u5982\u56fe 7-16 \u6240\u793a\uff0c\u300c\u4e8c\u53c9\u641c\u7d22\u6811 binary search tree\u300d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u3002
- \u5bf9\u4e8e\u6839\u8282\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c \\(<\\) \u6839\u8282\u70b9\u7684\u503c \\(<\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c\u3002
- \u4efb\u610f\u8282\u70b9\u7684\u5de6\u3001\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u540c\u6837\u6ee1\u8db3\u6761\u4ef6
1.\u3002
\u56fe 7-16 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811
"},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":"\u6211\u4eec\u5c06\u4e8c\u53c9\u641c\u7d22\u6811\u5c01\u88c5\u4e3a\u4e00\u4e2a\u7c7b
"},{"location":"chapter_tree/binary_search_tree/#1","title":"1. \u00a0 \u67e5\u627e\u8282\u70b9","text":"BinarySearchTree\uff0c\u5e76\u58f0\u660e\u4e00\u4e2a\u6210\u5458\u53d8\u91cfroot\uff0c\u6307\u5411\u6811\u7684\u6839\u8282\u70b9\u3002\u7ed9\u5b9a\u76ee\u6807\u8282\u70b9\u503c
num\uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u5982\u56fe 7-17 \u6240\u793a\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u8282\u70b9cur\uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9root\u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u8282\u70b9\u503ccur.val\u548cnum\u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb\u3002- \u82e5
cur.val < num\uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728cur\u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884ccur = cur.right\u3002 - \u82e5
cur.val > num\uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728cur\u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884ccur = cur.left\u3002 - \u82e5
cur.val = num\uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u8282\u70b9\u3002
\u56fe 7-17 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u8282\u70b9\u793a\u4f8b
\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py
binary_search_tree.cppdef search(self, num: int) -> TreeNode | None:\n \"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\n cur = self._root\n # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while cur is not None:\n # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur.val < num:\n cur = cur.right\n # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n elif cur.val > num:\n cur = cur.left\n # \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else:\n break\n return cur\n
binary_search_tree.java/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\n TreeNode *cur = root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != nullptr) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur->val < num)\n cur = cur->right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur->val > num)\n cur = cur->left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else\n break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.cs/* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\n TreeNode cur = root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > num)\n cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else\n break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.go/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? Search(int num) {\n TreeNode? cur = root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur =\n cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > num)\n cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else\n break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.swift/* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n node := bst.root\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n for node != nil {\n if node.Val.(int) < num {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n node = node.Right\n } else if node.Val.(int) > num {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n node = node.Left\n } else {\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n break\n }\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return node\n}\n
binary_search_tree.js/* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -> TreeNode? {\n var cur = root\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while cur != nil {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur!.val < num {\n cur = cur?.right\n }\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if cur!.val > num {\n cur = cur?.left\n }\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else {\n break\n }\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur\n}\n
binary_search_tree.ts/* \u67e5\u627e\u8282\u70b9 */\nsearch(num) {\n let cur = this.root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > num) cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.dart/* \u67e5\u627e\u8282\u70b9 */\nsearch(num: number): TreeNode | null {\n let cur = this.root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > num) cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.rs/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int _num) {\n TreeNode? cur = _root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < _num)\n cur = cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > _num)\n cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else\n break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.c/* \u67e5\u627e\u8282\u70b9 */\npub fn search(&self, num: i32) -> Option<TreeNodeRc> {\n let mut cur = self.root.clone();\n\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while let Some(node) = cur.clone() {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if node.borrow().val < num {\n cur = node.borrow().right.clone();\n }\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if node.borrow().val > num {\n cur = node.borrow().left.clone();\n }\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else {\n break;\n }\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n cur\n}\n
binary_search_tree.zig/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(BinarySearchTree *bst, int num) {\n TreeNode *cur = bst->root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != NULL) {\n if (cur->val < num) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n cur = cur->right;\n } else if (cur->val > num) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n cur = cur->left;\n } else {\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
"},{"location":"chapter_tree/binary_search_tree/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"// \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\n var cur = self.root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.?.val < num) {\n cur = cur.?.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n } else if (cur.?.val > num) {\n cur = cur.?.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n } else {\n break;\n }\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n\u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20
num\uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 < \u6839\u8282\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-18 \u6240\u793a\u3002- \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u76f8\u4f3c\uff0c\u4ece\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u8282\u70b9\u503c\u548c
num\u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\uff08\u904d\u5386\u81f3 \\(\\text{None}\\) \uff09\u65f6\u8df3\u51fa\u5faa\u73af\u3002 - \u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u8282\u70b9\uff1a\u521d\u59cb\u5316\u8282\u70b9
num\uff0c\u5c06\u8be5\u8282\u70b9\u7f6e\u4e8e \\(\\text{None}\\) \u7684\u4f4d\u7f6e\u3002
\u56fe 7-18 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u8282\u70b9
\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002
- \u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u8282\u70b9\uff0c\u5426\u5219\u5c06\u8fdd\u53cd\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u8282\u70b9\u5728\u6811\u4e2d\u5df2\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002
- \u4e3a\u4e86\u5b9e\u73b0\u63d2\u5165\u8282\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u8282\u70b9
pre\u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u8282\u70b9\u3002\u8fd9\u6837\u5728\u904d\u5386\u81f3 \\(\\text{None}\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u8282\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u3002
binary_search_tree.cppdef insert(self, num: int):\n \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n # \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if self._root is None:\n self._root = TreeNode(num)\n return\n # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n cur, pre = self._root, None\n while cur is not None:\n # \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if cur.val == num:\n return\n pre = cur\n # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur.val < num:\n cur = cur.right\n # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else:\n cur = cur.left\n # \u63d2\u5165\u8282\u70b9\n node = TreeNode(num)\n if pre.val < num:\n pre.right = node\n else:\n pre.left = node\n
binary_search_tree.java/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (root == nullptr) {\n root = new TreeNode(num);\n return;\n }\n TreeNode *cur = root, *pre = nullptr;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != nullptr) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur->val == num)\n return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur->val < num)\n cur = cur->right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur->left;\n }\n // \u63d2\u5165\u8282\u70b9\n TreeNode *node = new TreeNode(num);\n if (pre->val < num)\n pre->right = node;\n else\n pre->left = node;\n}\n
binary_search_tree.cs/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (root == null) {\n root = new TreeNode(num);\n return;\n }\n TreeNode cur = root, pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val == num)\n return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u63d2\u5165\u8282\u70b9\n TreeNode node = new TreeNode(num);\n if (pre.val < num)\n pre.right = node;\n else\n pre.left = node;\n}\n
binary_search_tree.go/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (root == null) {\n root = new TreeNode(num);\n return;\n }\n TreeNode? cur = root, pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val == num)\n return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n\n // \u63d2\u5165\u8282\u70b9\n TreeNode node = new(num);\n if (pre != null) {\n if (pre.val < num)\n pre.right = node;\n else\n pre.left = node;\n }\n}\n
binary_search_tree.swift/* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\n cur := bst.root\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if cur == nil {\n bst.root = NewTreeNode(num)\n return\n }\n // \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n var pre *TreeNode = nil\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n for cur != nil {\n if cur.Val == num {\n return\n }\n pre = cur\n if cur.Val.(int) < num {\n cur = cur.Right\n } else {\n cur = cur.Left\n }\n }\n // \u63d2\u5165\u8282\u70b9\n node := NewTreeNode(num)\n if pre.Val.(int) < num {\n pre.Right = node\n } else {\n pre.Left = node\n }\n}\n
binary_search_tree.js/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if root == nil {\n root = TreeNode(x: num)\n return\n }\n var cur = root\n var pre: TreeNode?\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while cur != nil {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if cur!.val == num {\n return\n }\n pre = cur\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur!.val < num {\n cur = cur?.right\n }\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else {\n cur = cur?.left\n }\n }\n // \u63d2\u5165\u8282\u70b9\n let node = TreeNode(x: num)\n if pre!.val < num {\n pre?.right = node\n } else {\n pre?.left = node\n }\n}\n
binary_search_tree.ts/* \u63d2\u5165\u8282\u70b9 */\ninsert(num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (this.root === null) {\n this.root = new TreeNode(num);\n return;\n }\n let cur = this.root,\n pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val === num) return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else cur = cur.left;\n }\n // \u63d2\u5165\u8282\u70b9\n const node = new TreeNode(num);\n if (pre.val < num) pre.right = node;\n else pre.left = node;\n}\n
binary_search_tree.dart/* \u63d2\u5165\u8282\u70b9 */\ninsert(num: number): void {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (this.root === null) {\n this.root = new TreeNode(num);\n return;\n }\n let cur: TreeNode | null = this.root,\n pre: TreeNode | null = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val === num) return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else cur = cur.left;\n }\n // \u63d2\u5165\u8282\u70b9\n const node = new TreeNode(num);\n if (pre!.val < num) pre!.right = node;\n else pre!.left = node;\n}\n
binary_search_tree.rs/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int _num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (_root == null) {\n _root = TreeNode(_num);\n return;\n }\n TreeNode? cur = _root;\n TreeNode? pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val == _num) return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < _num)\n cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u63d2\u5165\u8282\u70b9\n TreeNode? node = TreeNode(_num);\n if (pre!.val < _num)\n pre.right = node;\n else\n pre.left = node;\n}\n
binary_search_tree.c/* \u63d2\u5165\u8282\u70b9 */\npub fn insert(&mut self, num: i32) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if self.root.is_none() {\n self.root = Some(TreeNode::new(num));\n return;\n }\n let mut cur = self.root.clone();\n let mut pre = None;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while let Some(node) = cur.clone() {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if node.borrow().val == num {\n return;\n }\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n pre = cur.clone();\n if node.borrow().val < num {\n cur = node.borrow().right.clone();\n }\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else {\n cur = node.borrow().left.clone();\n }\n }\n // \u63d2\u5165\u8282\u70b9\n let node = TreeNode::new(num);\n let pre = pre.unwrap();\n if pre.borrow().val < num {\n pre.borrow_mut().right = Some(Rc::clone(&node));\n } else {\n pre.borrow_mut().left = Some(Rc::clone(&node));\n }\n}\n
binary_search_tree.zig/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(BinarySearchTree *bst, int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (bst->root == NULL) {\n bst->root = newTreeNode(num);\n return;\n }\n TreeNode *cur = bst->root, *pre = NULL;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != NULL) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur->val == num) {\n return;\n }\n pre = cur;\n if (cur->val < num) {\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n cur = cur->right;\n } else {\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n cur = cur->left;\n }\n }\n // \u63d2\u5165\u8282\u70b9\n TreeNode *node = newTreeNode(num);\n if (pre->val < num) {\n pre->right = node;\n } else {\n pre->left = node;\n }\n}\n// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (self.root == null) {\n self.root = try self.mem_allocator.create(inc.TreeNode(T));\n return;\n }\n var cur = self.root;\n var pre: ?*inc.TreeNode(T) = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.?.val == num) return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.?.val < num) {\n cur = cur.?.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n } else {\n cur = cur.?.left;\n }\n }\n // \u63d2\u5165\u8282\u70b9\n var node = try self.mem_allocator.create(inc.TreeNode(T));\n node.init(num);\n if (pre.?.val < num) {\n pre.?.right = node;\n } else {\n pre.?.left = node;\n }\n}\n\u4e0e\u67e5\u627e\u8282\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u8282\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002
"},{"location":"chapter_tree/binary_search_tree/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"\u5148\u5728\u4e8c\u53c9\u6811\u4e2d\u67e5\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u518d\u5c06\u5176\u5220\u9664\u3002
\u4e0e\u63d2\u5165\u8282\u70b9\u7c7b\u4f3c\uff0c\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 < \u6839\u8282\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u4ecd\u7136\u6ee1\u8db3\u3002
\u56e0\u6b64\uff0c\u6211\u4eec\u6839\u636e\u76ee\u6807\u8282\u70b9\u7684\u5b50\u8282\u70b9\u6570\u91cf\uff0c\u5206 0\u30011 \u548c 2 \u4e09\u79cd\u60c5\u51b5\uff0c\u6267\u884c\u5bf9\u5e94\u7684\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u3002
\u5982\u56fe 7-19 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \u65f6\uff0c\u8868\u793a\u8be5\u8282\u70b9\u662f\u53f6\u8282\u70b9\uff0c\u53ef\u4ee5\u76f4\u63a5\u5220\u9664\u3002
\u56fe 7-19 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 0 \uff09
\u5982\u56fe 7-20 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u8282\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u8282\u70b9\u5373\u53ef\u3002
\u56fe 7-20 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 1 \uff09
\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(2\\) \u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u5220\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u4e2a\u8282\u70b9\u66ff\u6362\u8be5\u8282\u70b9\u3002\u7531\u4e8e\u8981\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 \\(<\\) \u6839\u8282\u70b9 \\(<\\) \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u4e2a\u8282\u70b9\u53ef\u4ee5\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\u6216\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u3002
\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\uff08\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\uff0c\u5219\u5220\u9664\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-21 \u6240\u793a\u3002
- \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\u5728\u201c\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff0c\u8bb0\u4e3a
tmp\u3002 - \u7528
tmp\u7684\u503c\u8986\u76d6\u5f85\u5220\u9664\u8282\u70b9\u7684\u503c\uff0c\u5e76\u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u8282\u70b9tmp\u3002
\u56fe 7-21 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 2 \uff09
\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u540c\u6837\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py
binary_search_tree.cppdef remove(self, num: int):\n \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n # \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if self._root is None:\n return\n # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n cur, pre = self._root, None\n while cur is not None:\n # \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if cur.val == num:\n break\n pre = cur\n # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur.val < num:\n cur = cur.right\n # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else:\n cur = cur.left\n # \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if cur is None:\n return\n\n # \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if cur.left is None or cur.right is None:\n # \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n child = cur.left or cur.right\n # \u5220\u9664\u8282\u70b9 cur\n if cur != self._root:\n if pre.left == cur:\n pre.left = child\n else:\n pre.right = child\n else:\n # \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n self._root = child\n # \u5b50\u8282\u70b9\u6570\u91cf = 2\n else:\n # \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n tmp: TreeNode = cur.right\n while tmp.left is not None:\n tmp = tmp.left\n # \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n self.remove(tmp.val)\n # \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val\n
binary_search_tree.java/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (root == nullptr)\n return;\n TreeNode *cur = root, *pre = nullptr;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != nullptr) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur->val == num)\n break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur->val < num)\n cur = cur->right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur->left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == nullptr)\n return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur->left == nullptr || cur->right == nullptr) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n TreeNode *child = cur->left != nullptr ? cur->left : cur->right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur != root) {\n if (pre->left == cur)\n pre->left = child;\n else\n pre->right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n root = child;\n }\n // \u91ca\u653e\u5185\u5b58\n delete cur;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode *tmp = cur->right;\n while (tmp->left != nullptr) {\n tmp = tmp->left;\n }\n int tmpVal = tmp->val;\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n remove(tmp->val);\n // \u7528 tmp \u8986\u76d6 cur\n cur->val = tmpVal;\n }\n}\n
binary_search_tree.cs/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (root == null)\n return;\n TreeNode cur = root, pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val == num)\n break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == null)\n return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left == null || cur.right == null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n TreeNode child = cur.left != null ? cur.left : cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur != root) {\n if (pre.left == cur)\n pre.left = child;\n else\n pre.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode tmp = cur.right;\n while (tmp.left != null) {\n tmp = tmp.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n remove(tmp.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val;\n }\n}\n
binary_search_tree.go/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (root == null)\n return;\n TreeNode? cur = root, pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val == num)\n break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == null)\n return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left == null || cur.right == null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n TreeNode? child = cur.left ?? cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur != root) {\n if (pre!.left == cur)\n pre.left = child;\n else\n pre.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode? tmp = cur.right;\n while (tmp.left != null) {\n tmp = tmp.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n Remove(tmp.val!.Value);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val;\n }\n}\n
binary_search_tree.swift/* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\n cur := bst.root\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if cur == nil {\n return\n }\n // \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n var pre *TreeNode = nil\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n for cur != nil {\n if cur.Val == num {\n break\n }\n pre = cur\n if cur.Val.(int) < num {\n // \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\n cur = cur.Right\n } else {\n // \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\n cur = cur.Left\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if cur == nil {\n return\n }\n // \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\n if cur.Left == nil || cur.Right == nil {\n var child *TreeNode = nil\n // \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\n if cur.Left != nil {\n child = cur.Left\n } else {\n child = cur.Right\n }\n // \u5220\u9664\u8282\u70b9 cur\n if cur != bst.root {\n if pre.Left == cur {\n pre.Left = child\n } else {\n pre.Right = child\n }\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n bst.root = child\n }\n // \u5b50\u8282\u70b9\u6570\u4e3a 2\n } else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n tmp := cur.Right\n for tmp.Left != nil {\n tmp = tmp.Left\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n bst.remove(tmp.Val.(int))\n // \u7528 tmp \u8986\u76d6 cur\n cur.Val = tmp.Val\n }\n}\n
binary_search_tree.js/* \u5220\u9664\u8282\u70b9 */\nfunc remove(num: Int) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if root == nil {\n return\n }\n var cur = root\n var pre: TreeNode?\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while cur != nil {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if cur!.val == num {\n break\n }\n pre = cur\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur!.val < num {\n cur = cur?.right\n }\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else {\n cur = cur?.left\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if cur == nil {\n return\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if cur?.left == nil || cur?.right == nil {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n let child = cur?.left != nil ? cur?.left : cur?.right\n // \u5220\u9664\u8282\u70b9 cur\n if cur !== root {\n if pre?.left === cur {\n pre?.left = child\n } else {\n pre?.right = child\n }\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n root = child\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n var tmp = cur?.right\n while tmp?.left != nil {\n tmp = tmp?.left\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n remove(num: tmp!.val)\n // \u7528 tmp \u8986\u76d6 cur\n cur?.val = tmp!.val\n }\n}\n
binary_search_tree.ts/* \u5220\u9664\u8282\u70b9 */\nremove(num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (this.root === null) return;\n let cur = this.root,\n pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val === num) break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur === null) return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left === null || cur.right === null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n const child = cur.left !== null ? cur.left : cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur !== this.root) {\n if (pre.left === cur) pre.left = child;\n else pre.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n this.root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n let tmp = cur.right;\n while (tmp.left !== null) {\n tmp = tmp.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n this.remove(tmp.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val;\n }\n}\n
binary_search_tree.dart/* \u5220\u9664\u8282\u70b9 */\nremove(num: number): void {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (this.root === null) return;\n let cur: TreeNode | null = this.root,\n pre: TreeNode | null = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val === num) break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur === null) return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left === null || cur.right === null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n const child: TreeNode | null =\n cur.left !== null ? cur.left : cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur !== this.root) {\n if (pre!.left === cur) pre!.left = child;\n else pre!.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n this.root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n let tmp: TreeNode | null = cur.right;\n while (tmp!.left !== null) {\n tmp = tmp!.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n this.remove(tmp!.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp!.val;\n }\n}\n
binary_search_tree.rs/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int _num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (_root == null) return;\n TreeNode? cur = _root;\n TreeNode? pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val == _num) break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < _num)\n cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur == null) return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left == null || cur.right == null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n TreeNode? child = cur.left ?? cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur != _root) {\n if (pre!.left == cur)\n pre.left = child;\n else\n pre.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n _root = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode? tmp = cur.right;\n while (tmp!.left != null) {\n tmp = tmp.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n remove(tmp.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val;\n }\n}\n
binary_search_tree.c/* \u5220\u9664\u8282\u70b9 */\npub fn remove(&mut self, num: i32) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if self.root.is_none() { \n return; \n }\n let mut cur = self.root.clone();\n let mut pre = None;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while let Some(node) = cur.clone() {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if node.borrow().val == num {\n break;\n }\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n pre = cur.clone();\n if node.borrow().val < num {\n cur = node.borrow().right.clone();\n }\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else {\n cur = node.borrow().left.clone();\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if cur.is_none() {\n return;\n }\n let cur = cur.unwrap();\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if cur.borrow().left.is_none() || cur.borrow().right.is_none() {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n let child = cur.borrow().left.clone().or_else(|| cur.borrow().right.clone());\n let pre = pre.unwrap();\n let left = pre.borrow().left.clone().unwrap();\n // \u5220\u9664\u8282\u70b9 cur\n if !Rc::ptr_eq(&cur, self.root.as_ref().unwrap()) {\n if Rc::ptr_eq(&left, &cur) {\n pre.borrow_mut().left = child;\n } else {\n pre.borrow_mut().right = child;\n }\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n self.root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n let mut tmp = cur.borrow().right.clone();\n while let Some(node) = tmp.clone() {\n if node.borrow().left.is_some() {\n tmp = node.borrow().left.clone();\n } else {\n break;\n }\n }\n let tmpval = tmp.unwrap().borrow().val;\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n self.remove(tmpval);\n // \u7528 tmp \u8986\u76d6 cur\n cur.borrow_mut().val = tmpval;\n }\n}\n
binary_search_tree.zig/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(BinarySearchTree *bst, int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (bst->root == NULL)\n return;\n TreeNode *cur = bst->root, *pre = NULL;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != NULL) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur->val == num)\n break;\n pre = cur;\n if (cur->val < num) {\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\n cur = cur->right;\n } else {\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\n cur = cur->left;\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == NULL)\n return;\n // \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\n if (cur->left == NULL || cur->right == NULL) {\n /* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n TreeNode *child = cur->left != NULL ? cur->left : cur->right;\n // \u5220\u9664\u8282\u70b9 cur\n if (pre->left == cur) {\n pre->left = child;\n } else {\n pre->right = child;\n }\n } else {\n /* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode *tmp = cur->right;\n while (tmp->left != NULL) {\n tmp = tmp->left;\n }\n int tmpVal = tmp->val;\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n removeItem(bst, tmp->val);\n // \u7528 tmp \u8986\u76d6 cur\n cur->val = tmpVal;\n }\n}\n
"},{"location":"chapter_tree/binary_search_tree/#4","title":"4. \u00a0 \u4e2d\u5e8f\u904d\u5386\u6709\u5e8f","text":"// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (self.root == null) return;\n var cur = self.root;\n var pre: ?*inc.TreeNode(T) = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.?.val == num) break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.?.val < num) {\n cur = cur.?.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n } else {\n cur = cur.?.left;\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == null) return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.?.left == null or cur.?.right == null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n var child = if (cur.?.left != null) cur.?.left else cur.?.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (pre.?.left == cur) {\n pre.?.left = child;\n } else {\n pre.?.right = child;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n } else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n var tmp = cur.?.right;\n while (tmp.?.left != null) {\n tmp = tmp.?.left;\n }\n var tmp_val = tmp.?.val;\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n self.remove(tmp.?.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.?.val = tmp_val;\n }\n}\n\u5982\u56fe 7-22 \u6240\u793a\uff0c\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u987a\u5e8f\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u6ee1\u8db3\u201c\u5de6\u5b50\u8282\u70b9 \\(<\\) \u6839\u8282\u70b9 \\(<\\) \u53f3\u5b50\u8282\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002
\u8fd9\u610f\u5473\u7740\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u8282\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002
\u5229\u7528\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u65e0\u987b\u8fdb\u884c\u989d\u5916\u7684\u6392\u5e8f\u64cd\u4f5c\uff0c\u975e\u5e38\u9ad8\u6548\u3002
\u56fe 7-22 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217
"},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528\u6570\u7ec4\u6216\u4e8c\u53c9\u641c\u7d22\u6811\u5b58\u50a8\u3002\u89c2\u5bdf\u8868 7-2 \uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5177\u6709\u7a33\u5b9a\u4e14\u9ad8\u6548\u7684\u6027\u80fd\u3002\u53ea\u6709\u5728\u9ad8\u9891\u6dfb\u52a0\u3001\u4f4e\u9891\u67e5\u627e\u5220\u9664\u6570\u636e\u7684\u573a\u666f\u4e0b\uff0c\u6570\u7ec4\u6bd4\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002
\u8868 7-2 \u00a0 \u6570\u7ec4\u4e0e\u641c\u7d22\u6811\u7684\u6548\u7387\u5bf9\u6bd4
\u65e0\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\)\u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u8282\u70b9\u3002
\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u4e0d\u65ad\u5730\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u56fe 7-23 \u6240\u793a\u7684\u94fe\u8868\uff0c\u8fd9\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4f1a\u9000\u5316\u4e3a \\(O(n)\\) \u3002
\u56fe 7-23 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316
"},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"- \u7528\u4f5c\u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002
- \u4f5c\u4e3a\u67d0\u4e9b\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
- \u7528\u4e8e\u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72b6\u6001\u3002
\u300c\u4e8c\u53c9\u6811 binary tree\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u5355\u5143\u662f\u8282\u70b9\uff0c\u6bcf\u4e2a\u8282\u70b9\u5305\u542b\u503c\u3001\u5de6\u5b50\u8282\u70b9\u5f15\u7528\u548c\u53f3\u5b50\u8282\u70b9\u5f15\u7528\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZigclass TreeNode:\n \"\"\"\u4e8c\u53c9\u6811\u8282\u70b9\u7c7b\"\"\"\n def __init__(self, val: int):\n self.val: int = val # \u8282\u70b9\u503c\n self.left: TreeNode | None = None # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n self.right: TreeNode | None = None # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n int val; // \u8282\u70b9\u503c\n TreeNode *left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n int val; // \u8282\u70b9\u503c\n TreeNode left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n TreeNode right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n TreeNode(int x) { val = x; }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n public int? val = x; // \u8282\u70b9\u503c\n public TreeNode? left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n Val int\n Left *TreeNode\n Right *TreeNode\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\n return &TreeNode{\n Left: nil, // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n Right: nil, // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n Val: v, // \u8282\u70b9\u503c\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n var val: Int // \u8282\u70b9\u503c\n var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n\n init(x: Int) {\n val = x\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n val; // \u8282\u70b9\u503c\n left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n constructor(val, left, right) {\n this.val = val === undefined ? 0 : val;\n this.left = left === undefined ? null : left;\n this.right = right === undefined ? null : right;\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n val: number;\n left: TreeNode | null;\n right: TreeNode | null;\n\n constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.left = left === undefined ? null : left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n this.right = right === undefined ? null : right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n int val; // \u8282\u70b9\u503c\n TreeNode? left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n TreeNode(this.val, [this.left, this.right]);\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n val: i32, // \u8282\u70b9\u503c\n left: Option<Rc<RefCell<TreeNode>>>, // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n right: Option<Rc<RefCell<TreeNode>>>, // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n\nimpl TreeNode {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new(val: i32) -> Rc<RefCell<Self>> {\n Rc::new(RefCell::new(Self {\n val,\n left: None,\n right: None\n }))\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct TreeNode {\n int val; // \u8282\u70b9\u503c\n int height; // \u8282\u70b9\u9ad8\u5ea6\n struct TreeNode *left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n struct TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n TreeNode *node;\n\n node = (TreeNode *)malloc(sizeof(TreeNode));\n node->val = val;\n node->height = 0;\n node->left = NULL;\n node->right = NULL;\n return node;\n}\n\n\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u5206\u522b\u6307\u5411\u300c\u5de6\u5b50\u8282\u70b9 left-child node\u300d\u548c\u300c\u53f3\u5b50\u8282\u70b9 right-child node\u300d\uff0c\u8be5\u8282\u70b9\u88ab\u79f0\u4e3a\u8fd9\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u300c\u7236\u8282\u70b9 parent node\u300d\u3002\u5f53\u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u65f6\uff0c\u6211\u4eec\u5c06\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u300c\u5de6\u5b50\u6811 left subtree\u300d\uff0c\u540c\u7406\u53ef\u5f97\u300c\u53f3\u5b50\u6811 right subtree\u300d\u3002
\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u9664\u53f6\u8282\u70b9\u5916\uff0c\u5176\u4ed6\u6240\u6709\u8282\u70b9\u90fd\u5305\u542b\u5b50\u8282\u70b9\u548c\u975e\u7a7a\u5b50\u6811\u3002\u5982\u56fe 7-1 \u6240\u793a\uff0c\u5982\u679c\u5c06\u201c\u8282\u70b9 2\u201d\u89c6\u4e3a\u7236\u8282\u70b9\uff0c\u5219\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u5206\u522b\u662f\u201c\u8282\u70b9 4\u201d\u548c\u201c\u8282\u70b9 5\u201d\uff0c\u5de6\u5b50\u6811\u662f\u201c\u8282\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\uff0c\u53f3\u5b50\u6811\u662f\u201c\u8282\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\u3002
\u56fe 7-1 \u00a0 \u7236\u8282\u70b9\u3001\u5b50\u8282\u70b9\u3001\u5b50\u6811
"},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1 \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"\u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed\u5982\u56fe 7-2 \u6240\u793a\u3002
- \u300c\u6839\u8282\u70b9 root node\u300d\uff1a\u4f4d\u4e8e\u4e8c\u53c9\u6811\u9876\u5c42\u7684\u8282\u70b9\uff0c\u6ca1\u6709\u7236\u8282\u70b9\u3002
- \u300c\u53f6\u8282\u70b9 leaf node\u300d\uff1a\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u5747\u6307\u5411 \\(\\text{None}\\) \u3002
- \u300c\u8fb9 edge\u300d\uff1a\u8fde\u63a5\u4e24\u4e2a\u8282\u70b9\u7684\u7ebf\u6bb5\uff0c\u5373\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u3002
- \u8282\u70b9\u6240\u5728\u7684\u300c\u5c42 level\u300d\uff1a\u4ece\u9876\u81f3\u5e95\u9012\u589e\uff0c\u6839\u8282\u70b9\u6240\u5728\u5c42\u4e3a 1 \u3002
- \u8282\u70b9\u7684\u300c\u5ea6 degree\u300d\uff1a\u8282\u70b9\u7684\u5b50\u8282\u70b9\u7684\u6570\u91cf\u3002\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u53d6\u503c\u8303\u56f4\u662f 0\u30011\u30012 \u3002
- \u4e8c\u53c9\u6811\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002
- \u8282\u70b9\u7684\u300c\u6df1\u5ea6 depth\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002
- \u8282\u70b9\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u8ddd\u79bb\u8be5\u8282\u70b9\u6700\u8fdc\u7684\u53f6\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002
\u56fe 7-2 \u00a0 \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed
Tip
\u8bf7\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u201c\u9ad8\u5ea6\u201d\u548c\u201c\u6df1\u5ea6\u201d\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u53ef\u80fd\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8282\u70b9\u7684\u6570\u91cf\u201d\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002
"},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2 \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_tree/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e8c\u53c9\u6811","text":"\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u8282\u70b9\uff0c\u7136\u540e\u6784\u5efa\u5f15\u7528\uff08\u6307\u9488\uff09\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree.py
binary_tree.cpp# \u521d\u59cb\u5316\u4e8c\u53c9\u6811\n# \u521d\u59cb\u5316\u8282\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
binary_tree.java/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
binary_tree.cs// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.go/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.swift/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n
binary_tree.js// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
binary_tree.ts/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n n2 = new TreeNode(2),\n n3 = new TreeNode(3),\n n4 = new TreeNode(4),\n n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.dart/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n n2 = new TreeNode(2),\n n3 = new TreeNode(3),\n n4 = new TreeNode(4),\n n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.rs/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.c// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n
binary_tree.zig/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
"},{"location":"chapter_tree/binary_tree/#2","title":"2. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9","text":"\n\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u6307\u9488\u6765\u5b9e\u73b0\u3002\u56fe 7-3 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u793a\u4f8b\u3002
\u56fe 7-3 \u00a0 \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree.py
binary_tree.cpp# \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\np = TreeNode(0)\n# \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n
binary_tree.java/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1->left = n2;\n
binary_tree.csTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.go/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.swift/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.Left = n2\n
binary_tree.jslet P = TreeNode(x: 0)\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n
binary_tree.ts/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.dart/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.rs/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.clet p = TreeNode::new(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// \u5220\u9664\u8282\u70b9 p\nn1.borrow_mut().left = Some(n2);\n
binary_tree.zig/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1->left = n2;\n\nNote
\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u8282\u70b9\u53ef\u80fd\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u800c\u5220\u9664\u8282\u70b9\u901a\u5e38\u610f\u5473\u7740\u5220\u9664\u8be5\u8282\u70b9\u53ca\u5176\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5b9e\u73b0\u6709\u5b9e\u9645\u610f\u4e49\u7684\u64cd\u4f5c\u3002
"},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3 \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#1_1","title":"1. \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"\u5982\u56fe 7-4 \u6240\u793a\uff0c\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811 perfect binary tree\u300d\u6240\u6709\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u53f6\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u7684\u5ea6\u90fd\u4e3a \\(2\\) \uff1b\u82e5\u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \uff0c\u5219\u8282\u70b9\u603b\u6570\u4e3a \\(2^{h+1} - 1\\) \uff0c\u5448\u73b0\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u73b0\u8c61\u3002
Tip
\u8bf7\u6ce8\u610f\uff0c\u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\u3002
\u56fe 7-4 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811
"},{"location":"chapter_tree/binary_tree/#2_1","title":"2. \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811","text":"\u5982\u56fe 7-5 \u6240\u793a\uff0c\u300c\u5b8c\u5168\u4e8c\u53c9\u6811 complete binary tree\u300d\u53ea\u6709\u6700\u5e95\u5c42\u7684\u8282\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u8282\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002
\u56fe 7-5 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811
"},{"location":"chapter_tree/binary_tree/#3","title":"3. \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"\u5982\u56fe 7-6 \u6240\u793a\uff0c\u300c\u5b8c\u6ee1\u4e8c\u53c9\u6811 full binary tree\u300d\u9664\u4e86\u53f6\u8282\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\u3002
\u56fe 7-6 \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811
"},{"location":"chapter_tree/binary_tree/#4","title":"4. \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811","text":"\u5982\u56fe 7-7 \u6240\u793a\uff0c\u300c\u5e73\u8861\u4e8c\u53c9\u6811 balanced binary tree\u300d\u4e2d\u4efb\u610f\u8282\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc7 1 \u3002
\u56fe 7-7 \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811
"},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4 \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"\u56fe 7-8 \u5c55\u793a\u4e86\u4e8c\u53c9\u6811\u7684\u7406\u60f3\u7ed3\u6784\u4e0e\u9000\u5316\u7ed3\u6784\u3002\u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff1b\u800c\u5f53\u6240\u6709\u8282\u70b9\u90fd\u504f\u5411\u4e00\u4fa7\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u201c\u94fe\u8868\u201d\u3002
- \u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u60c5\u51b5\uff0c\u53ef\u4ee5\u5145\u5206\u53d1\u6325\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\u3002
- \u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002
\u56fe 7-8 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784
\u5982\u8868 7-1 \u6240\u793a\uff0c\u5728\u6700\u4f73\u7ed3\u6784\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf\u3001\u8282\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u503c\u6216\u6781\u5c0f\u503c\u3002
\u8868 7-1 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784
\u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u8282\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u8282\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \u7684\u6811\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"\u4ece\u7269\u7406\u7ed3\u6784\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u5176\u904d\u5386\u65b9\u5f0f\u662f\u901a\u8fc7\u6307\u9488\u9010\u4e2a\u8bbf\u95ee\u8282\u70b9\u3002\u7136\u800c\uff0c\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u4f7f\u5f97\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u501f\u52a9\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002
\u4e8c\u53c9\u6811\u5e38\u89c1\u7684\u904d\u5386\u65b9\u5f0f\u5305\u62ec\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u7b49\u3002
"},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1 \u00a0 \u5c42\u5e8f\u904d\u5386","text":"\u5982\u56fe 7-9 \u6240\u793a\uff0c\u300c\u5c42\u5e8f\u904d\u5386 level-order traversal\u300d\u4ece\u9876\u90e8\u5230\u5e95\u90e8\u9010\u5c42\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u4e00\u5c42\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u8282\u70b9\u3002
\u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u5c5e\u4e8e\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal, BFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\u3002
\u56fe 7-9 \u00a0 \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386
"},{"location":"chapter_tree/binary_tree_traversal/#1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u901a\u5e38\u501f\u52a9\u201c\u961f\u5217\u201d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u9075\u5faa\u201c\u5148\u8fdb\u5148\u51fa\u201d\u7684\u89c4\u5219\uff0c\u800c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5219\u9075\u5faa\u201c\u9010\u5c42\u63a8\u8fdb\u201d\u7684\u89c4\u5219\uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree_bfs.py
binary_tree_bfs.cppdef level_order(root: TreeNode | None) -> list[int]:\n \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n # \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n queue: deque[TreeNode] = deque()\n queue.append(root)\n # \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n res = []\n while queue:\n node: TreeNode = queue.popleft() # \u961f\u5217\u51fa\u961f\n res.append(node.val) # \u4fdd\u5b58\u8282\u70b9\u503c\n if node.left is not None:\n queue.append(node.left) # \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if node.right is not None:\n queue.append(node.right) # \u53f3\u5b50\u8282\u70b9\u5165\u961f\n return res\n
binary_tree_bfs.java/* \u5c42\u5e8f\u904d\u5386 */\nvector<int> levelOrder(TreeNode *root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n queue<TreeNode *> queue;\n queue.push(root);\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n vector<int> vec;\n while (!queue.empty()) {\n TreeNode *node = queue.front();\n queue.pop(); // \u961f\u5217\u51fa\u961f\n vec.push_back(node->val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node->left != nullptr)\n queue.push(node->left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node->right != nullptr)\n queue.push(node->right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return vec;\n}\n
binary_tree_bfs.cs/* \u5c42\u5e8f\u904d\u5386 */\nList<Integer> levelOrder(TreeNode root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n Queue<TreeNode> queue = new LinkedList<>();\n queue.add(root);\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n List<Integer> list = new ArrayList<>();\n while (!queue.isEmpty()) {\n TreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\n list.add(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left != null)\n queue.offer(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node.right != null)\n queue.offer(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return list;\n}\n
binary_tree_bfs.go/* \u5c42\u5e8f\u904d\u5386 */\nList<int> LevelOrder(TreeNode root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n Queue<TreeNode> queue = new();\n queue.Enqueue(root);\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n List<int> list = [];\n while (queue.Count != 0) {\n TreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\n list.Add(node.val!.Value); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left != null)\n queue.Enqueue(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node.right != null)\n queue.Enqueue(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return list;\n}\n
binary_tree_bfs.swift/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []any {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n queue := list.New()\n queue.PushBack(root)\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n nums := make([]any, 0)\n for queue.Len() > 0 {\n // \u961f\u5217\u51fa\u961f\n node := queue.Remove(queue.Front()).(*TreeNode)\n // \u4fdd\u5b58\u8282\u70b9\u503c\n nums = append(nums, node.Val)\n if node.Left != nil {\n // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n queue.PushBack(node.Left)\n }\n if node.Right != nil {\n // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n queue.PushBack(node.Right)\n }\n }\n return nums\n}\n
binary_tree_bfs.js/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -> [Int] {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n var queue: [TreeNode] = [root]\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n var list: [Int] = []\n while !queue.isEmpty {\n let node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\n list.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\n if let left = node.left {\n queue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n }\n if let right = node.right {\n queue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n }\n return list\n}\n
binary_tree_bfs.ts/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n const queue = [root];\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n const list = [];\n while (queue.length) {\n let node = queue.shift(); // \u961f\u5217\u51fa\u961f\n list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return list;\n}\n
binary_tree_bfs.dart/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n const queue = [root];\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n const list: number[] = [];\n while (queue.length) {\n let node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\n list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left) {\n queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n }\n if (node.right) {\n queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n }\n return list;\n}\n
binary_tree_bfs.rs/* \u5c42\u5e8f\u904d\u5386 */\nList<int> levelOrder(TreeNode? root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n Queue<TreeNode?> queue = Queue();\n queue.add(root);\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n List<int> res = [];\n while (queue.isNotEmpty) {\n TreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\n res.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return res;\n}\n
binary_tree_bfs.c/* \u5c42\u5e8f\u904d\u5386 */\nfn level_order(root: &Rc<RefCell<TreeNode>>) -> Vec<i32> {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n let mut que = VecDeque::new();\n que.push_back(Rc::clone(&root));\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n let mut vec = Vec::new();\n\n while let Some(node) = que.pop_front() { // \u961f\u5217\u51fa\u961f\n vec.push(node.borrow().val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if let Some(left) = node.borrow().left.as_ref() {\n que.push_back(Rc::clone(left)); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n }\n if let Some(right) = node.borrow().right.as_ref() {\n que.push_back(Rc::clone(right)); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n };\n }\n vec\n}\n
binary_tree_bfs.zig/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n /* \u8f85\u52a9\u961f\u5217 */\n int front, rear;\n int index, *arr;\n TreeNode *node;\n TreeNode **queue;\n\n /* \u8f85\u52a9\u961f\u5217 */\n queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n // \u961f\u5217\u6307\u9488\n front = 0, rear = 0;\n // \u52a0\u5165\u6839\u8282\u70b9\n queue[rear++] = root;\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n /* \u8f85\u52a9\u6570\u7ec4 */\n arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n // \u6570\u7ec4\u6307\u9488\n index = 0;\n while (front < rear) {\n // \u961f\u5217\u51fa\u961f\n node = queue[front++];\n // \u4fdd\u5b58\u8282\u70b9\u503c\n arr[index++] = node->val;\n if (node->left != NULL) {\n // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n queue[rear++] = node->left;\n }\n if (node->right != NULL) {\n // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n queue[rear++] = node->right;\n }\n }\n // \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n *size = index;\n arr = realloc(arr, sizeof(int) * (*size));\n\n // \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\n free(queue);\n return arr;\n}\n
"},{"location":"chapter_tree/binary_tree_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"// \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n const L = std.TailQueue(*inc.TreeNode(T));\n var queue = L{};\n var root_node = try mem_allocator.create(L.Node);\n root_node.data = root;\n queue.append(root_node); \n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n var list = std.ArrayList(T).init(std.heap.page_allocator);\n while (queue.len > 0) {\n var queue_node = queue.popFirst().?; // \u961f\u5217\u51fa\u961f\n var node = queue_node.data;\n try list.append(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left != null) {\n var tmp_node = try mem_allocator.create(L.Node);\n tmp_node.data = node.left.?;\n queue.append(tmp_node); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n }\n if (node.right != null) {\n var tmp_node = try mem_allocator.create(L.Node);\n tmp_node.data = node.right.?;\n queue.append(tmp_node); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n } \n }\n return list;\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6ee1\u4e8c\u53c9\u6811\u65f6\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\((n + 1) / 2\\) \u4e2a\u8282\u70b9\uff0c\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002
\u76f8\u5e94\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal, DFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\u3002
\u56fe 7-10 \u5c55\u793a\u4e86\u5bf9\u4e8c\u53c9\u6811\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5de5\u4f5c\u539f\u7406\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u68f5\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u6bcf\u4e2a\u8282\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u3002
\u56fe 7-10 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386
"},{"location":"chapter_tree/binary_tree_traversal/#1_1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree_dfs.py
binary_tree_dfs.cppdef pre_order(root: TreeNode | None):\n \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n if root is None:\n return\n # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n res.append(root.val)\n pre_order(root=root.left)\n pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n if root is None:\n return\n # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n in_order(root=root.left)\n res.append(root.val)\n in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n if root is None:\n return\n # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n post_order(root=root.left)\n post_order(root=root.right)\n res.append(root.val)\n
binary_tree_dfs.java/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\n if (root == nullptr)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n vec.push_back(root->val);\n preOrder(root->left);\n preOrder(root->right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\n if (root == nullptr)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root->left);\n vec.push_back(root->val);\n inOrder(root->right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\n if (root == nullptr)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root->left);\n postOrder(root->right);\n vec.push_back(root->val);\n}\n
binary_tree_dfs.cs/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\n if (root == null)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.add(root.val);\n preOrder(root.left);\n preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\n if (root == null)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root.left);\n list.add(root.val);\n inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\n if (root == null)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root.left);\n postOrder(root.right);\n list.add(root.val);\n}\n
binary_tree_dfs.go/* \u524d\u5e8f\u904d\u5386 */\nvoid PreOrder(TreeNode? root) {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.Add(root.val!.Value);\n PreOrder(root.left);\n PreOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid InOrder(TreeNode? root) {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n InOrder(root.left);\n list.Add(root.val!.Value);\n InOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid PostOrder(TreeNode? root) {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n PostOrder(root.left);\n PostOrder(root.right);\n list.Add(root.val!.Value);\n}\n
binary_tree_dfs.swift/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\n if node == nil {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n nums = append(nums, node.Val)\n preOrder(node.Left)\n preOrder(node.Right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\n if node == nil {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(node.Left)\n nums = append(nums, node.Val)\n inOrder(node.Right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\n if node == nil {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(node.Left)\n postOrder(node.Right)\n nums = append(nums, node.Val)\n}\n
binary_tree_dfs.js/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.append(root.val)\n preOrder(root: root.left)\n preOrder(root: root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root: root.left)\n list.append(root.val)\n inOrder(root: root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root: root.left)\n postOrder(root: root.right)\n list.append(root.val)\n}\n
binary_tree_dfs.ts/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\n if (root === null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.push(root.val);\n preOrder(root.left);\n preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\n if (root === null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root.left);\n list.push(root.val);\n inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\n if (root === null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root.left);\n postOrder(root.right);\n list.push(root.val);\n}\n
binary_tree_dfs.dart/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\n if (root === null) {\n return;\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.push(root.val);\n preOrder(root.left);\n preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\n if (root === null) {\n return;\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root.left);\n list.push(root.val);\n inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\n if (root === null) {\n return;\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root.left);\n postOrder(root.right);\n list.push(root.val);\n}\n
binary_tree_dfs.rs/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\n if (node == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.add(node.val);\n preOrder(node.left);\n preOrder(node.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\n if (node == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(node.left);\n list.add(node.val);\n inOrder(node.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\n if (node == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(node.left);\n postOrder(node.right);\n list.add(node.val);\n}\n
binary_tree_dfs.c/* \u524d\u5e8f\u904d\u5386 */\nfn pre_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {\n let mut result = vec![];\n\n if let Some(node) = root {\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n result.push(node.borrow().val);\n result.append(&mut pre_order(node.borrow().left.as_ref()));\n result.append(&mut pre_order(node.borrow().right.as_ref()));\n }\n result\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfn in_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {\n let mut result = vec![];\n\n if let Some(node) = root {\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n result.append(&mut in_order(node.borrow().left.as_ref()));\n result.push(node.borrow().val);\n result.append(&mut in_order(node.borrow().right.as_ref()));\n }\n result\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfn post_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {\n let mut result = vec![];\n\n if let Some(node) = root {\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n result.append(&mut post_order(node.borrow().left.as_ref()));\n result.append(&mut post_order(node.borrow().right.as_ref()));\n result.push(node.borrow().val);\n }\n result\n}\n
binary_tree_dfs.zig/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\n if (root == NULL)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n arr[(*size)++] = root->val;\n preOrder(root->left, size);\n preOrder(root->right, size);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\n if (root == NULL)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root->left, size);\n arr[(*size)++] = root->val;\n inOrder(root->right, size);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\n if (root == NULL)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root->left, size);\n postOrder(root->right, size);\n arr[(*size)++] = root->val;\n}\n// \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n try list.append(root.?.val);\n try preOrder(T, root.?.left);\n try preOrder(T, root.?.right);\n}\n\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n try inOrder(T, root.?.left);\n try list.append(root.?.val);\n try inOrder(T, root.?.right);\n}\n\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n try postOrder(T, root.?.left);\n try postOrder(T, root.?.right);\n try list.append(root.?.val);\n}\nTip
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\u56fe 7-11 \u00a0 \u524d\u5e8f\u904d\u5386\u7684\u9012\u5f52\u8fc7\u7a0b
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- \u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u5176\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \u3002\u5f53\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
- AVL \u6811\uff0c\u4e5f\u79f0\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5b83\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\u786e\u4fdd\u5728\u4e0d\u65ad\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\u540e\u6811\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002
- AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u5411\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002
\u5bf9\u4e8e\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u53c9\u6811\uff0c\u6811\u7684\u9ad8\u5ea6\u548c\u6839\u8282\u70b9\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u5417\uff1f
\u662f\u7684\uff0c\u56e0\u4e3a\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\u3002
\u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\uff0c\u8fd9\u91cc\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u4e48\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d44\u6e90\u7684\u5b50\u8282\u70b9\u7684\u8d44\u6e90\u91ca\u653e\u5417\uff1f
\u62ff\u4e8c\u53c9\u641c\u7d22\u6811\u6765\u4e3e\u4f8b\uff0c\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u8981\u5206\u4e09\u79cd\u60c5\u51b5\u5904\u7406\uff0c\u5176\u4e2d\u6bcf\u79cd\u60c5\u51b5\u90fd\u9700\u8981\u8fdb\u884c\u591a\u4e2a\u6b65\u9aa4\u7684\u8282\u70b9\u64cd\u4f5c\u3002
\u4e3a\u4ec0\u4e48 DFS \u904d\u5386\u4e8c\u53c9\u6811\u6709\u524d\u3001\u4e2d\u3001\u540e\u4e09\u79cd\u987a\u5e8f\uff0c\u5206\u522b\u6709\u4ec0\u4e48\u7528\u5462\uff1f
\u4e0e\u987a\u5e8f\u548c\u9006\u5e8f\u904d\u5386\u6570\u7ec4\u7c7b\u4f3c\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u662f\u4e09\u79cd\u4e8c\u53c9\u6811\u904d\u5386\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5b83\u4eec\u5f97\u5230\u4e00\u4e2a\u7279\u5b9a\u987a\u5e8f\u7684\u904d\u5386\u7ed3\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\uff0c\u7531\u4e8e\u8282\u70b9\u5927\u5c0f\u6ee1\u8db3
\u5de6\u5b50\u8282\u70b9\u503c < \u6839\u8282\u70b9\u503c < \u53f3\u5b50\u8282\u70b9\u503c\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u8981\u6309\u7167\u5de6 $\\rightarrow$ \u6839 $\\rightarrow$ \u53f3\u7684\u4f18\u5148\u7ea7\u904d\u5386\u6811\uff0c\u5c31\u53ef\u4ee5\u83b7\u5f97\u6709\u5e8f\u7684\u8282\u70b9\u5e8f\u5217\u3002\u53f3\u65cb\u64cd\u4f5c\u662f\u5904\u7406\u5931\u8861\u8282\u70b9
node\u3001child\u3001grand_child\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u90a3node\u7684\u7236\u8282\u70b9\u548cnode\u539f\u6765\u7684\u8fde\u63a5\u4e0d\u9700\u8981\u7ef4\u62a4\u5417\uff1f\u53f3\u65cb\u64cd\u4f5c\u540e\u5c82\u4e0d\u662f\u65ad\u6389\u4e86\uff1f\u6211\u4eec\u9700\u8981\u4ece\u9012\u5f52\u7684\u89c6\u89d2\u6765\u770b\u8fd9\u4e2a\u95ee\u9898\u3002\u53f3\u65cb\u64cd\u4f5c
right_rotate(root)\u4f20\u5165\u7684\u662f\u5b50\u6811\u7684\u6839\u8282\u70b9\uff0c\u6700\u7ec8return child\u8fd4\u56de\u65cb\u8f6c\u4e4b\u540e\u7684\u5b50\u6811\u7684\u6839\u8282\u70b9\u3002\u5b50\u6811\u7684\u6839\u8282\u70b9\u548c\u5176\u7236\u8282\u70b9\u7684\u8fde\u63a5\u662f\u5728\u8be5\u51fd\u6570\u8fd4\u56de\u540e\u5b8c\u6210\u7684\uff0c\u4e0d\u5c5e\u4e8e\u53f3\u65cb\u64cd\u4f5c\u7684\u7ef4\u62a4\u8303\u56f4\u3002\u5728 C++ \u4e2d\uff0c\u51fd\u6570\u88ab\u5212\u5206\u5230
private\u548cpublic\u4e2d\uff0c\u8fd9\u65b9\u9762\u6709\u4ec0\u4e48\u8003\u91cf\u5417\uff1f\u4e3a\u4ec0\u4e48\u8981\u5c06height()\u51fd\u6570\u548cupdateHeight()\u51fd\u6570\u5206\u522b\u653e\u5728public\u548cprivate\u4e2d\u5462\uff1f\u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u8303\u56f4\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u7c7b\u5185\u90e8\u4f7f\u7528\uff0c\u90a3\u4e48\u5c31\u8bbe\u8ba1\u4e3a
private\u3002\u4f8b\u5982\uff0c\u7528\u6237\u5355\u72ec\u8c03\u7528updateHeight()\u662f\u6ca1\u6709\u610f\u4e49\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800cheight()\u662f\u8bbf\u95ee\u8282\u70b9\u9ad8\u5ea6\uff0c\u7c7b\u4f3c\u4e8evector.size()\uff0c\u56e0\u6b64\u8bbe\u7f6e\u6210public\u4ee5\u4fbf\u4f7f\u7528\u3002\u5982\u4f55\u4ece\u4e00\u7ec4\u8f93\u5165\u6570\u636e\u6784\u5efa\u4e00\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\uff1f\u6839\u8282\u70b9\u7684\u9009\u62e9\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f
\u662f\u7684\uff0c\u6784\u5efa\u6811\u7684\u65b9\u6cd5\u5df2\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4ee3\u7801\u4e2d\u7684
build_tree()\u65b9\u6cd5\u4e2d\u7ed9\u51fa\u3002\u81f3\u4e8e\u6839\u8282\u70b9\u7684\u9009\u62e9\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u8f93\u5165\u6570\u636e\u6392\u5e8f\uff0c\u7136\u540e\u5c06\u4e2d\u70b9\u5143\u7d20\u4f5c\u4e3a\u6839\u8282\u70b9\uff0c\u518d\u9012\u5f52\u5730\u6784\u5efa\u5de6\u53f3\u5b50\u6811\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8bc1\u6811\u7684\u5e73\u8861\u6027\u3002\u5728 Java \u4e2d\uff0c\u5b57\u7b26\u4e32\u5bf9\u6bd4\u662f\u5426\u4e00\u5b9a\u8981\u7528
equals()\u65b9\u6cd5\uff1f\u5728 Java \u4e2d\uff0c\u5bf9\u4e8e\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c
==\u7528\u4e8e\u5bf9\u6bd4\u4e24\u4e2a\u53d8\u91cf\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002\u5bf9\u4e8e\u5f15\u7528\u7c7b\u578b\uff0c\u4e24\u79cd\u7b26\u53f7\u7684\u5de5\u4f5c\u539f\u7406\u662f\u4e0d\u540c\u7684\u3002==\uff1a\u7528\u6765\u6bd4\u8f83\u4e24\u4e2a\u53d8\u91cf\u662f\u5426\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5373\u5b83\u4eec\u5728\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u662f\u5426\u76f8\u540c\u3002equals()\uff1a\u7528\u6765\u5bf9\u6bd4\u4e24\u4e2a\u5bf9\u8c61\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002
\u56e0\u6b64\uff0c\u5982\u679c\u8981\u5bf9\u6bd4\u503c\uff0c\u6211\u4eec\u5e94\u8be5\u4f7f\u7528
equals()\u3002\u7136\u800c\uff0c\u901a\u8fc7String a = \"hi\"; String b = \"hi\";\u521d\u59cb\u5316\u7684\u5b57\u7b26\u4e32\u90fd\u5b58\u50a8\u5728\u5b57\u7b26\u4e32\u5e38\u91cf\u6c60\u4e2d\uff0c\u5b83\u4eec\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u56e0\u6b64\u4e5f\u53ef\u4ee5\u7528a == b\u6765\u6bd4\u8f83\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5185\u5bb9\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u7684\u8282\u70b9\u6570\u91cf\u662f \\(2^h\\) \u5417\uff1f
\u662f\u7684\uff0c\u4f8b\u5982\u9ad8\u5ea6 \\(h = 2\\) \u7684\u6ee1\u4e8c\u53c9\u6811\uff0c\u5176\u8282\u70b9\u603b\u6570 \\(n = 7\\) \uff0c\u5219\u5e95\u5c42\u8282\u70b9\u6570\u91cf \\(4 = 2^h = (n + 1) / 2\\) \u3002
"}]} \ No newline at end of file +{"config":{"lang":["en"],"separator":"[\\s\\u200b\\u3000\\-\u3001\u3002\uff0c\uff0e\uff1f\uff01\uff1b]+","pipeline":["stemmer"]},"docs":[{"location":"","title":"Home","text":"\u300a Hello \u7b97\u6cd5 \u300b\u52a8\u753b\u56fe\u89e3\u3001\u4e00\u952e\u8fd0\u884c\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6559\u7a0b
\u5f00\u59cb\u9605\u8bfb \u4e0b\u8f7d\u4ee3\u7801 \u4e0b\u8f7d PDF
\u63a8\u8350\u8bedQuote
\u201c\u4e00\u672c\u901a\u4fd7\u6613\u61c2\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u4e66\uff0c\u5f15\u5bfc\u8bfb\u8005\u624b\u8111\u5e76\u7528\u5730\u5b66\u4e60\uff0c\u5f3a\u70c8\u63a8\u8350\u7b97\u6cd5\u521d\u5b66\u8005\u9605\u8bfb\u3002\u201d
\u2014\u2014 \u9093\u4fca\u8f89\uff0c\u6e05\u534e\u5927\u5b66\u8ba1\u7b97\u673a\u7cfb\u6559\u6388
Quote
\u201c\u5982\u679c\u6211\u5f53\u5e74\u5b66\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u65f6\u5019\u6709\u300aHello \u7b97\u6cd5\u300b\uff0c\u5b66\u8d77\u6765\u5e94\u8be5\u4f1a\u7b80\u5355 10 \u500d\uff01\u201d
\u2014\u2014 \u674e\u6c90\uff0c\u4e9a\u9a6c\u900a\u8d44\u6df1\u9996\u5e2d\u79d1\u5b66\u5bb6
\u52a8\u753b\u56fe\u89e3\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1
\"A picture is worth a thousand words.\"
\u201c\u4e00\u56fe\u80dc\u5343\u8a00\u201d
\u4e00\u952e\u8fd0\u884c\u5341\u4f59\u79cd\u7f16\u7a0b\u8bed\u8a00\u4ee3\u7801\u53ef\u76f4\u63a5\u8fd0\u884c
\"Talk is cheap. Show me the code.\"
\u201c\u5c11\u5439\u725b\uff0c\u770b\u4ee3\u7801\u201d
\u4e92\u52a9\u5b66\u4e60\u6b22\u8fce\u8ba8\u8bba\u4e0e\u63d0\u95ee\u8bfb\u8005\u95f4\u643a\u624b\u5171\u8fdb
\"\u8ffd\u98ce\u8d76\u6708\u83ab\u505c\u7559\"
\u201c\u5e73\u829c\u5c3d\u5904\u662f\u6625\u5c71\u201d
\u5e8f\u4e24\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u201c\u5251\u6307 Offer\u201d\u7cfb\u5217\u9898\u89e3\uff0c\u53d7\u5230\u4e86\u8bb8\u591a\u540c\u5b66\u7684\u559c\u7231\u548c\u652f\u6301\u3002\u5728\u4e0e\u8bfb\u8005\u4ea4\u6d41\u671f\u95f4\uff0c\u6211\u6700\u5e38\u6536\u5230\u7684\u4e00\u4e2a\u95ee\u9898\u662f\u201c\u5982\u4f55\u5165\u95e8\u7b97\u6cd5\u201d\u3002\u9010\u6e10\u5730\uff0c\u6211\u5bf9\u8fd9\u4e2a\u95ee\u9898\u4ea7\u751f\u4e86\u6d53\u539a\u7684\u5174\u8da3\u3002
\u4e24\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u9898\u4f3c\u4e4e\u662f\u6700\u53d7\u6b22\u8fce\u7684\u65b9\u6cd5\uff0c\u7b80\u5355\u76f4\u63a5\u4e14\u6709\u6548\u3002\u7136\u800c\u5237\u9898\u5c31\u5982\u540c\u73a9\u201c\u626b\u96f7\u201d\u6e38\u620f\uff0c\u81ea\u5b66\u80fd\u529b\u5f3a\u7684\u540c\u5b66\u80fd\u591f\u987a\u5229\u5c06\u5730\u96f7\u9010\u4e2a\u6392\u6389\uff0c\u800c\u57fa\u7840\u4e0d\u8db3\u7684\u540c\u5b66\u5f88\u53ef\u80fd\u88ab\u70b8\u7684\u6ee1\u5934\u662f\u5305\uff0c\u5e76\u5728\u632b\u6298\u4e2d\u6b65\u6b65\u9000\u7f29\u3002\u901a\u8bfb\u6559\u6750\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u505a\u6cd5\uff0c\u4f46\u5bf9\u4e8e\u9762\u5411\u6c42\u804c\u7684\u540c\u5b66\u6765\u8bf4\uff0c\u6bd5\u4e1a\u5b63\u3001\u6295\u9012\u7b80\u5386\u3001\u51c6\u5907\u7b14\u8bd5\u9762\u8bd5\u5df2\u7ecf\u6d88\u8017\u4e86\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u5543\u539a\u91cd\u7684\u4e66\u5f80\u5f80\u53d8\u6210\u4e86\u4e00\u9879\u8270\u5de8\u7684\u6311\u6218\u3002
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\u6211\u6df1\u6df1\u8d5e\u540c\u8d39\u66fc\u6559\u6388\u6240\u8a00\uff1a\u201cKnowledge isn't free. You have to pay attention.\u201d\u4ece\u8fd9\u4e2a\u610f\u4e49\u4e0a\u770b\uff0c\u8fd9\u672c\u4e66\u5e76\u975e\u5b8c\u5168\u201c\u514d\u8d39\u201d\u3002\u4e3a\u4e86\u4e0d\u8f9c\u8d1f\u4f60\u4e3a\u672c\u4e66\u6240\u4ed8\u51fa\u7684\u5b9d\u8d35\u201c\u6ce8\u610f\u529b\u201d\uff0c\u6211\u4f1a\u5c3d\u6211\u6240\u80fd\uff0c\u6295\u5165\u6700\u5927\u7684\u201c\u6ce8\u610f\u529b\u201d\u6765\u5b8c\u6210\u8fd9\u672c\u4e66\u7684\u521b\u4f5c\u3002
\u4f5c\u8005\u9773\u5b87\u680b (Krahets)\uff0c\u5927\u5382\u9ad8\u7ea7\u7b97\u6cd5\u5de5\u7a0b\u5e08\uff0c\u4e0a\u6d77\u4ea4\u901a\u5927\u5b66\u7855\u58eb\u3002\u529b\u6263\uff08LeetCode\uff09\u5168\u7f51\u9605\u8bfb\u91cf\u6700\u9ad8\u535a\u4e3b\uff0c\u53d1\u8868\u7684\u300a\u56fe\u89e3\u7b97\u6cd5\u6570\u636e\u7ed3\u6784\u300b\u5df2\u88ab\u8ba2\u9605 30 \u4e07\u672c\u3002
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\u672c\u4e66\u7684\u4ee3\u7801\u5ba1\u9605\u5de5\u4f5c\u7531 Gonglja\u3001gvenusleo\u3001hpstory\u3001justin\u2010tse\u3001krahets\u3001night-cruise\u3001nuomi1\u3001Reanon \u548c sjinzh \u5b8c\u6210\uff08\u6309\u7167\u9996\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\uff09\u3002\u611f\u8c22\u4ed6\u4eec\u4ed8\u51fa\u7684\u65f6\u95f4\u4e0e\u7cbe\u529b\uff0c\u6b63\u662f\u4ed6\u4eec\u786e\u4fdd\u4e86\u5404\u8bed\u8a00\u4ee3\u7801\u7684\u89c4\u8303\u4e0e\u7edf\u4e00\u3002
GongljaC, C++ gvenusleoDart hpstoryC# justin-tseJS, TS krahetsJava, Python night-cruiseRust nuomi1Swift ReanonGo, C sjinzhRust, Zig"},{"location":"chapter_appendix/","title":"\u7b2c 16 \u7ae0 \u00a0 \u9644\u5f55","text":""},{"location":"chapter_appendix/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5
- 16.2 \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c
- 16.3 \u00a0 \u672f\u8bed\u8868
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\u6240\u6709\u64b0\u7a3f\u4eba\u7684 GitHub ID \u5c06\u5728\u672c\u4e66\u4ed3\u5e93\u3001\u7f51\u9875\u7248\u548c PDF \u7248\u7684\u4e3b\u9875\u4e0a\u8fdb\u884c\u5c55\u793a\uff0c\u4ee5\u611f\u8c22\u4ed6\u4eec\u5bf9\u5f00\u6e90\u793e\u533a\u7684\u65e0\u79c1\u5949\u732e\u3002
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"},{"location":"chapter_appendix/contribution/#1","title":"1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"\u5982\u56fe 16-3 \u6240\u793a\uff0c\u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u201c\u7f16\u8f91\u56fe\u6807\u201d\u3002\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u672c\u6216\u4ee3\u7801\u3002
- \u70b9\u51fb\u201c\u7f16\u8f91\u56fe\u6807\u201d\uff0c\u5982\u679c\u9047\u5230\u201c\u9700\u8981 Fork \u6b64\u4ed3\u5e93\u201d\u7684\u63d0\u793a\uff0c\u8bf7\u540c\u610f\u8be5\u64cd\u4f5c\u3002
- \u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u68c0\u67e5\u5185\u5bb9\u7684\u6b63\u786e\u6027\uff0c\u5e76\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7edf\u4e00\u3002
- \u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u4fee\u6539\u8bf4\u660e\uff0c\u7136\u540e\u70b9\u51fb\u201cPropose file change\u201d\u6309\u94ae\u3002\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002
\u56fe 16-3 \u00a0 \u9875\u9762\u7f16\u8f91\u6309\u952e
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"},{"location":"chapter_appendix/contribution/#2","title":"2. \u00a0 \u5185\u5bb9\u521b\u4f5c","text":"\u5982\u679c\u60a8\u6709\u5174\u8da3\u53c2\u4e0e\u6b64\u5f00\u6e90\u9879\u76ee\uff0c\u5305\u62ec\u5c06\u4ee3\u7801\u7ffb\u8bd1\u6210\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u3001\u6269\u5c55\u6587\u7ae0\u5185\u5bb9\u7b49\uff0c\u90a3\u4e48\u9700\u8981\u5b9e\u65bd\u4ee5\u4e0b Pull Request \u5de5\u4f5c\u6d41\u7a0b\u3002
- \u767b\u5f55 GitHub \uff0c\u5c06\u672c\u4e66\u7684\u4ee3\u7801\u4ed3\u5e93 Fork \u5230\u4e2a\u4eba\u8d26\u53f7\u4e0b\u3002
- \u8fdb\u5165\u60a8\u7684 Fork \u4ed3\u5e93\u7f51\u9875\uff0c\u4f7f\u7528
git clone\u547d\u4ee4\u5c06\u4ed3\u5e93\u514b\u9686\u81f3\u672c\u5730\u3002 - \u5728\u672c\u5730\u8fdb\u884c\u5185\u5bb9\u521b\u4f5c\uff0c\u5e76\u8fdb\u884c\u5b8c\u6574\u6d4b\u8bd5\uff0c\u9a8c\u8bc1\u4ee3\u7801\u7684\u6b63\u786e\u6027\u3002
- \u5c06\u672c\u5730\u6240\u505a\u66f4\u6539 Commit \uff0c\u7136\u540e Push \u81f3\u8fdc\u7a0b\u4ed3\u5e93\u3002
- \u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002
\u5728
hello-algo\u6839\u76ee\u5f55\u4e0b\uff0c\u6267\u884c\u4ee5\u4e0b Docker \u811a\u672c\uff0c\u5373\u53ef\u5728http://localhost:8000\u8bbf\u95ee\u672c\u9879\u76ee\uff1adocker-compose up -d\n\u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\uff1a
"},{"location":"chapter_appendix/installation/","title":"16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1611-ide","title":"16.1.1 \u00a0 \u5b89\u88c5 IDE","text":"docker-compose down\n\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u3001\u8f7b\u91cf\u7684 VS Code \u4f5c\u4e3a\u672c\u5730\u96c6\u6210\u5f00\u53d1\u73af\u5883\uff08IDE\uff09\u3002\u8bbf\u95ee VS Code \u5b98\u7f51\uff0c\u6839\u636e\u64cd\u4f5c\u7cfb\u7edf\u9009\u62e9\u76f8\u5e94\u7248\u672c\u7684 VS Code \u8fdb\u884c\u4e0b\u8f7d\u548c\u5b89\u88c5\u3002
\u56fe 16-1 \u00a0 \u4ece\u5b98\u7f51\u4e0b\u8f7d VS Code
VS Code \u62e5\u6709\u5f3a\u5927\u7684\u6269\u5c55\u5305\u751f\u6001\u7cfb\u7edf\uff0c\u652f\u6301\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u7684\u8fd0\u884c\u548c\u8c03\u8bd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5b89\u88c5\u201cPython Extension Pack\u201d\u6269\u5c55\u5305\u4e4b\u540e\uff0c\u5373\u53ef\u8fdb\u884c Python \u4ee3\u7801\u8c03\u8bd5\u3002\u5b89\u88c5\u6b65\u9aa4\u5982\u56fe 16-2 \u6240\u793a\u3002
\u56fe 16-2 \u00a0 \u5b89\u88c5 VS Code \u6269\u5c55\u5305
"},{"location":"chapter_appendix/installation/#1612","title":"16.1.2 \u00a0 \u5b89\u88c5\u8bed\u8a00\u73af\u5883","text":""},{"location":"chapter_appendix/installation/#1-python","title":"1. \u00a0 Python \u73af\u5883","text":"- \u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \uff0c\u9700\u8981 Python 3.10 \u6216\u66f4\u65b0\u7248\u672c\u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
python\uff0c\u5b89\u88c5 Python Extension Pack \u3002 - \uff08\u53ef\u9009\uff09\u5728\u547d\u4ee4\u884c\u8f93\u5165
pip install black\uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002
- Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff1bMacOS \u81ea\u5e26 Clang \uff0c\u65e0\u987b\u5b89\u88c5\u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
c++\uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002 - \uff08\u53ef\u9009\uff09\u6253\u5f00 Settings \u9875\u9762\uff0c\u641c\u7d22
Clang_format_fallback Style\u4ee3\u7801\u683c\u5f0f\u5316\u9009\u9879\uff0c\u8bbe\u7f6e\u4e3a{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }\u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 > JDK 9\uff09\u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
java\uff0c\u5b89\u88c5 Extension Pack for Java \u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 8.0 \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
C# Dev Kit\uff0c\u5b89\u88c5 C# Dev Kit \uff08\u914d\u7f6e\u6559\u7a0b\uff09\u3002 - \u4e5f\u53ef\u4f7f\u7528 Visual Studio\uff08\u5b89\u88c5\u6559\u7a0b\uff09\u3002
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- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
go\uff0c\u5b89\u88c5 Go \u3002 - \u6309\u5feb\u6377\u952e
Ctrl + Shift + P\u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9Go: Install/Update Tools\uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
swift\uff0c\u5b89\u88c5 Swift for Visual Studio Code \u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
javascript\uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002 - \uff08\u53ef\u9009\uff09\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
Prettier\uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 Dart \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
dart\uff0c\u5b89\u88c5 Dart \u3002
- \u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust \u3002
- \u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22
rust\uff0c\u5b89\u88c5 rust-analyzer \u3002
\u8868 16-1 \u5217\u51fa\u4e86\u4e66\u4e2d\u51fa\u73b0\u7684\u91cd\u8981\u672f\u8bed\u3002\u5efa\u8bae\u8bfb\u8005\u540c\u65f6\u8bb0\u4f4f\u5b83\u4eec\u7684\u4e2d\u82f1\u6587\u53eb\u6cd5\uff0c\u4ee5\u4fbf\u9605\u8bfb\u82f1\u6587\u6587\u732e\u3002
\u8868 16-1 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u540d\u8bcd
\u4e2d\u6587 English \u4e2d\u6587 English \u7b97\u6cd5 algorithm \u5c42\u5e8f\u904d\u5386 level-order traversal \u6570\u636e\u7ed3\u6784 data structure \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal \u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 asymptotic complexity analysis \u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal \u65f6\u95f4\u590d\u6742\u5ea6 time complexity \u4e8c\u53c9\u641c\u7d22\u6811 binary search tree \u7a7a\u95f4\u590d\u6742\u5ea6 space complexity \u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 balanced binary search tree \u8fed\u4ee3 iteration \u5e73\u8861\u56e0\u5b50 balance factor \u9012\u5f52 recursion \u5806 heap \u5c3e\u9012\u5f52 tail recursion \u5927\u9876\u5806 max heap \u9012\u5f52\u6811 recursion tree \u5c0f\u9876\u5806 min heap \u5927 \\(O\\) \u8bb0\u53f7 big-\\(O\\) notation \u4f18\u5148\u961f\u5217 priority queue \u6e10\u8fd1\u4e0a\u754c asymptotic upper bound \u5806\u5316 heapify \u539f\u7801 sign-magnitude \u56fe graph \u53cd\u7801 1\u2019s complement \u9876\u70b9 vertex \u8865\u7801 2\u2019s complement \u65e0\u5411\u56fe undirected graph \u6570\u7ec4 array \u6709\u5411\u56fe directed graph \u7d22\u5f15 index \u8fde\u901a\u56fe connected graph \u94fe\u8868 linked list \u975e\u8fde\u901a\u56fe disconnected graph \u94fe\u8868\u8282\u70b9 linked list node, list node \u6709\u6743\u56fe weighted graph \u5217\u8868 list \u90bb\u63a5 adjacency \u52a8\u6001\u6570\u7ec4 dynamic array \u8def\u5f84 path \u786c\u76d8 hard disk \u5165\u5ea6 in-degree \u5185\u5b58 random-access memory (RAM) \u51fa\u5ea6 out-degree \u7f13\u5b58 cache memory \u90bb\u63a5\u77e9\u9635 adjacency matrix \u7f13\u5b58\u672a\u547d\u4e2d cache miss \u90bb\u63a5\u8868 adjacency list \u7f13\u5b58\u547d\u4e2d\u7387 cache hit rate \u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 breadth-first search \u6808 stack \u6df1\u5ea6\u4f18\u5148\u641c\u7d22 depth-first search \u961f\u5217 queue \u4e8c\u5206\u67e5\u627e binary search \u53cc\u5411\u961f\u5217 double-ended queue \u641c\u7d22\u7b97\u6cd5 searching algorithm \u54c8\u5e0c\u8868 hash table \u6392\u5e8f\u7b97\u6cd5 sorting algorithm \u6876 bucket \u9009\u62e9\u6392\u5e8f selection sort \u54c8\u5e0c\u51fd\u6570 hash function \u5192\u6ce1\u6392\u5e8f bubble sort \u54c8\u5e0c\u51b2\u7a81 hash collision \u63d2\u5165\u6392\u5e8f insertion sort \u8d1f\u8f7d\u56e0\u5b50 load factor \u5feb\u901f\u6392\u5e8f quick sort \u94fe\u5f0f\u5730\u5740 separate chaining \u5f52\u5e76\u6392\u5e8f merge sort \u5f00\u653e\u5bfb\u5740 open addressing \u5806\u6392\u5e8f heap sort \u7ebf\u6027\u63a2\u6d4b linear probing \u6876\u6392\u5e8f bucket sort \u61d2\u5220\u9664 lazy deletion \u8ba1\u6570\u6392\u5e8f counting sort \u4e8c\u53c9\u6811 binary tree \u57fa\u6570\u6392\u5e8f radix sort \u6811\u8282\u70b9 tree node \u5206\u6cbb divide and conquer \u5de6\u5b50\u8282\u70b9 left-child node \u6c49\u8bfa\u5854\u95ee\u9898 hanota problem \u53f3\u5b50\u8282\u70b9 right-child node \u56de\u6eaf\u7b97\u6cd5 backtracking algorithm \u7236\u8282\u70b9 parent node \u7ea6\u675f constraint \u5de6\u5b50\u6811 left subtree \u89e3 solution \u53f3\u5b50\u6811 right subtree \u72b6\u6001 state \u6839\u8282\u70b9 root node \u526a\u679d pruning \u53f6\u8282\u70b9 leaf node \u5168\u6392\u5217\u95ee\u9898 permutations problem \u8fb9 edge \u5b50\u96c6\u548c\u95ee\u9898 subset-sum problem \u5c42 level N \u7687\u540e\u95ee\u9898 N-queens problem \u5ea6 degree \u52a8\u6001\u89c4\u5212 dynamic programming \u9ad8\u5ea6 height \u521d\u59cb\u72b6\u6001 initial state \u6df1\u5ea6 depth \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b state-trasition equation \u5b8c\u7f8e\u4e8c\u53c9\u6811 perfect binary tree \u80cc\u5305\u95ee\u9898 knapsack problem \u5b8c\u5168\u4e8c\u53c9\u6811 complete binary tree \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 edit distance problem \u5b8c\u6ee1\u4e8c\u53c9\u6811 full binary tree \u8d2a\u5fc3\u7b97\u6cd5 greedy algorithm \u5e73\u8861\u4e8c\u53c9\u6811 balanced binary tree AVL \u6811 AVL tree \u7ea2\u9ed1\u6811 red-black tree"},{"location":"chapter_array_and_linkedlist/","title":"\u7b2c 4 \u7ae0 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868","text":"Abstract
\u6570\u636e\u7ed3\u6784\u7684\u4e16\u754c\u5982\u540c\u4e00\u5835\u539a\u5b9e\u7684\u7816\u5899\u3002
\u6570\u7ec4\u7684\u7816\u5757\u6574\u9f50\u6392\u5217\uff0c\u9010\u4e2a\u7d27\u8d34\u3002\u94fe\u8868\u7684\u7816\u5757\u5206\u6563\u5404\u5904\uff0c\u8fde\u63a5\u7684\u85e4\u8513\u81ea\u7531\u5730\u7a7f\u68ad\u4e8e\u7816\u7f1d\u4e4b\u95f4\u3002
"},{"location":"chapter_array_and_linkedlist/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 4.1 \u00a0 \u6570\u7ec4
- 4.2 \u00a0 \u94fe\u8868
- 4.3 \u00a0 \u5217\u8868
- 4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *
- 4.5 \u00a0 \u5c0f\u7ed3
\u300c\u6570\u7ec4 array\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5c06\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002\u6211\u4eec\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u8be5\u5143\u7d20\u7684\u300c\u7d22\u5f15 index\u300d\u3002\u56fe 4-1 \u5c55\u793a\u4e86\u6570\u7ec4\u7684\u4e3b\u8981\u672f\u8bed\u548c\u6982\u5ff5\u3002
\u56fe 4-1 \u00a0 \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f
"},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1 \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/array/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u6570\u7ec4","text":"\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9700\u6c42\u9009\u7528\u6570\u7ec4\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u5f0f\uff1a\u65e0\u521d\u59cb\u503c\u3001\u7ed9\u5b9a\u521d\u59cb\u503c\u3002\u5728\u672a\u6307\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4f1a\u5c06\u6570\u7ec4\u5143\u7d20\u521d\u59cb\u5316\u4e3a \\(0\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cpp# \u521d\u59cb\u5316\u6570\u7ec4\narr: list[int] = [0] * 5 # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4] \n
array.java/* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\uff08\u9700\u8981\u624b\u52a8\u91ca\u653e\u7a7a\u95f4\uff09\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n
array.cs/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
array.go/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = [1, 3, 2, 5, 4];\n
array.swift/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n
array.js/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n
array.ts/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n
array.dart/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n
array.rs/* \u521d\u59cb\u5316\u6570\u7ec4 */\nList<int> arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList<int> nums = [1, 3, 2, 5, 4];\n
array.c/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: Vec<i32> = vec![0; 5]; // [0, 0, 0, 0, 0]\nlet nums: Vec<i32> = vec![1, 3, 2, 5, 4];\n
array.zig/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n
"},{"location":"chapter_array_and_linkedlist/array/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"// \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n\u6570\u7ec4\u5143\u7d20\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u8ba1\u7b97\u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u5185\u5b58\u5730\u5740\uff08\u9996\u5143\u7d20\u5185\u5b58\u5730\u5740\uff09\u548c\u67d0\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u56fe 4-2 \u6240\u793a\u7684\u516c\u5f0f\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u8be5\u5143\u7d20\u3002
\u56fe 4-2 \u00a0 \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97
\u89c2\u5bdf\u56fe 4-2 \uff0c\u6211\u4eec\u53d1\u73b0\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \uff0c\u8fd9\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\u4f1a\u66f4\u81ea\u7136\u3002\u4f46\u4ece\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u662f\u5185\u5b58\u5730\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u5b83\u7684\u7d22\u5f15\u4e3a \\(0\\) \u662f\u5408\u7406\u7684\u3002
\u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u5143\u7d20\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef random_access(nums: list[int]) -> int:\n \"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n # \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n random_index = random.randint(0, len(nums) - 1)\n # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n random_num = nums[random_index]\n return random_num\n
array.java/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = rand() % size;\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
array.cs/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int[] nums) {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
array.go/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint RandomAccess(int[] nums) {\n Random random = new();\n // \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = random.Next(nums.Length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
array.swift/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n randomIndex := rand.Intn(len(nums))\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n randomNum = nums[randomIndex]\n return\n}\n
array.js/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -> Int {\n // \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n let randomIndex = nums.indices.randomElement()!\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n let randomNum = nums[randomIndex]\n return randomNum\n}\n
array.ts/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums) {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n const random_index = Math.floor(Math.random() * nums.length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n const random_num = nums[random_index];\n return random_num;\n}\n
array.dart/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n const random_index = Math.floor(Math.random() * nums.length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n const random_num = nums[random_index];\n return random_num;\n}\n
array.rs/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(List<int> nums) {\n // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = Random().nextInt(nums.length);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
array.c/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfn random_access(nums: &[i32]) -> i32 {\n // \u5728\u533a\u95f4 [0, nums.len()) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n let random_index = rand::thread_rng().gen_range(0..nums.len());\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n let random_num = nums[random_index];\n random_num\n}\n
array.zig/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n int randomIndex = rand() % size;\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n int randomNum = nums[randomIndex];\n return randomNum;\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#3","title":"3. \u00a0 \u63d2\u5165\u5143\u7d20","text":"// \u968f\u673a\u8bbf\u95ee\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n // \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\n var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n var randomNum = nums[randomIndex];\n return randomNum;\n}\n\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u5b58\u653e\u4efb\u4f55\u6570\u636e\u3002\u5982\u56fe 4-3 \u6240\u793a\uff0c\u5982\u679c\u60f3\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u5219\u9700\u8981\u5c06\u8be5\u5143\u7d20\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u4e4b\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002
\u56fe 4-3 \u00a0 \u6570\u7ec4\u63d2\u5165\u5143\u7d20\u793a\u4f8b
\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u56fa\u5b9a\u7684\uff0c\u56e0\u6b64\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u4f1a\u5bfc\u81f4\u6570\u7ec4\u5c3e\u90e8\u5143\u7d20\u201c\u4e22\u5931\u201d\u3002\u6211\u4eec\u5c06\u8fd9\u4e2a\u95ee\u9898\u7684\u89e3\u51b3\u65b9\u6848\u7559\u5728\u201c\u5217\u8868\u201d\u7ae0\u8282\u4e2d\u8ba8\u8bba\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef insert(nums: list[int], num: int, index: int):\n \"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for i in range(len(nums) - 1, index, -1):\n nums[i] = nums[i - 1]\n # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num\n
array.java/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int i = size - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.cs/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int i = nums.length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.go/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int i = nums.Length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.swift/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for i := len(nums) - 1; i > index; i-- {\n nums[i] = nums[i-1]\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num\n}\n
array.js/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for i in nums.indices.dropFirst(index).reversed() {\n nums[i] = nums[i - 1]\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num\n}\n
array.ts/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (let i = nums.length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.dart/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (let i = nums.length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.rs/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List<int> nums, int _num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (var i = nums.length - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 _num \u8d4b\u7ed9 index \u5904\u5143\u7d20\n nums[index] = _num;\n}\n
array.c/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &mut Vec<i32>, num: i32, index: usize) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for i in (index + 1..nums.len()).rev() {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
array.zig/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int i = size - 1; i > index; i--) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#4","title":"4. \u00a0 \u5220\u9664\u5143\u7d20","text":"// \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n var i = nums.len - 1;\n while (i > index) : (i -= 1) {\n nums[i] = nums[i - 1];\n }\n // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n nums[index] = num;\n}\n\u540c\u7406\uff0c\u5982\u56fe 4-4 \u6240\u793a\uff0c\u82e5\u60f3\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002
\u56fe 4-4 \u00a0 \u6570\u7ec4\u5220\u9664\u5143\u7d20\u793a\u4f8b
\u8bf7\u6ce8\u610f\uff0c\u5220\u9664\u5143\u7d20\u5b8c\u6210\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u987b\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef remove(nums: list[int], index: int):\n \"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\"\"\"\n # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for i in range(index, len(nums) - 1):\n nums[i] = nums[i + 1]\n
array.java/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int i = index; i < size - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.cs/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int i = index; i < nums.length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.go/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int i = index; i < nums.Length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.swift/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for i := index; i < len(nums)-1; i++ {\n nums[i] = nums[i+1]\n }\n}\n
array.js/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for i in nums.indices.dropFirst(index).dropLast() {\n nums[i] = nums[i + 1]\n }\n}\n
array.ts/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (let i = index; i < nums.length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.dart/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (let i = index; i < nums.length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.rs/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(List<int> nums, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (var i = index; i < nums.length - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n
array.c/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfn remove(nums: &mut Vec<i32>, index: usize) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for i in index..nums.len() - 1 {\n nums[i] = nums[i + 1];\n }\n}\n
array.zig/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int i = index; i < size - 1; i++) {\n nums[i] = nums[i + 1];\n }\n}\n// \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n var i = index;\n while (i < nums.len - 1) : (i += 1) {\n nums[i] = nums[i + 1];\n }\n}\n\u603b\u7684\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\u3002
- \u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002
- \u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u4e22\u5931\u3002
- \u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u201c\u65e0\u610f\u4e49\u201d\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e2\u53ef\u4ee5\u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u83b7\u53d6\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef traverse(nums: list[int]):\n \"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\n count = 0\n # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for i in range(len(nums)):\n count += nums[i]\n # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for num in nums:\n count += num\n # \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n for i, num in enumerate(nums):\n count += nums[i]\n count += num\n
array.java/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size; i++) {\n count += nums[i];\n }\n}\n
array.cs/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < nums.length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (int num : nums) {\n count += num;\n }\n}\n
array.go/* \u904d\u5386\u6570\u7ec4 */\nvoid Traverse(int[] nums) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < nums.Length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n foreach (int num in nums) {\n count += num;\n }\n}\n
array.swift/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\n count := 0\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for i := 0; i < len(nums); i++ {\n count += nums[i]\n }\n count = 0\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for _, num := range nums {\n count += num\n }\n // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n for i, num := range nums {\n count += nums[i]\n count += num\n }\n}\n
array.js/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\n var count = 0\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for i in nums.indices {\n count += nums[i]\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for num in nums {\n count += num\n }\n}\n
array.ts/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\n let count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (let i = 0; i < nums.length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (const num of nums) {\n count += num;\n }\n}\n
array.dart/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\n let count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (let i = 0; i < nums.length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (const num of nums) {\n count += num;\n }\n}\n
array.rs/* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List<int> nums) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (var i = 0; i < nums.length; i++) {\n count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (int _num in nums) {\n count += _num;\n }\n // \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\n nums.forEach((_num) {\n count += _num;\n });\n}\n
array.c/* \u904d\u5386\u6570\u7ec4 */\nfn traverse(nums: &[i32]) {\n let mut _count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for i in 0..nums.len() {\n _count += nums[i];\n }\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for num in nums {\n _count += num;\n }\n}\n
array.zig/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n int count = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size; i++) {\n count += nums[i];\n }\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#6","title":"6. \u00a0 \u67e5\u627e\u5143\u7d20","text":"// \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\n var count: i32 = 0;\n // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n var i: i32 = 0;\n while (i < nums.len) : (i += 1) {\n count += nums[i];\n }\n count = 0;\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n for (nums) |num| {\n count += num;\n }\n}\n\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\u9700\u8981\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u5224\u65ad\u5143\u7d20\u503c\u662f\u5426\u5339\u914d\uff0c\u82e5\u5339\u914d\u5219\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002
\u56e0\u4e3a\u6570\u7ec4\u662f\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u6240\u4ee5\u4e0a\u8ff0\u67e5\u627e\u64cd\u4f5c\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u67e5\u627e\u201d\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef find(nums: list[int], target: int) -> int:\n \"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\n for i in range(len(nums)):\n if nums[i] == target:\n return i\n return -1\n
array.java/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n for (int i = 0; i < size; i++) {\n if (nums[i] == target)\n return i;\n }\n return -1;\n}\n
array.cs/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\n for (int i = 0; i < nums.length; i++) {\n if (nums[i] == target)\n return i;\n }\n return -1;\n}\n
array.go/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\n for (int i = 0; i < nums.Length; i++) {\n if (nums[i] == target)\n return i;\n }\n return -1;\n}\n
array.swift/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\n index = -1\n for i := 0; i < len(nums); i++ {\n if nums[i] == target {\n index = i\n break\n }\n }\n return\n}\n
array.js/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -> Int {\n for i in nums.indices {\n if nums[i] == target {\n return i\n }\n }\n return -1\n}\n
array.ts/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\n for (let i = 0; i < nums.length; i++) {\n if (nums[i] === target) return i;\n }\n return -1;\n}\n
array.dart/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\n for (let i = 0; i < nums.length; i++) {\n if (nums[i] === target) {\n return i;\n }\n }\n return -1;\n}\n
array.rs/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List<int> nums, int target) {\n for (var i = 0; i < nums.length; i++) {\n if (nums[i] == target) return i;\n }\n return -1;\n}\n
array.c/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &[i32], target: i32) -> Option<usize> {\n for i in 0..nums.len() {\n if nums[i] == target {\n return Some(i);\n }\n }\n None\n}\n
array.zig/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n for (int i = 0; i < size; i++) {\n if (nums[i] == target)\n return i;\n }\n return -1;\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#7","title":"7. \u00a0 \u6269\u5bb9\u6570\u7ec4","text":"// \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\n for (nums, 0..) |num, i| {\n if (num == target) return @intCast(i);\n }\n return -1;\n}\n\u5728\u590d\u6742\u7684\u7cfb\u7edf\u73af\u5883\u4e2d\uff0c\u7a0b\u5e8f\u96be\u4ee5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u4ece\u800c\u65e0\u6cd5\u5b89\u5168\u5730\u6269\u5c55\u6570\u7ec4\u5bb9\u91cf\u3002\u56e0\u6b64\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u4e0d\u53ef\u53d8\u7684\u3002
\u5982\u679c\u6211\u4eec\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u91cd\u65b0\u5efa\u7acb\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u590d\u5236\u5230\u65b0\u6570\u7ec4\u3002\u8fd9\u662f\u4e00\u4e2a \\(O(n)\\) \u7684\u64cd\u4f5c\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\u975e\u5e38\u8017\u65f6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py
array.cppdef extend(nums: list[int], enlarge: int) -> list[int]:\n \"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n res = [0] * (len(nums) + enlarge)\n # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for i in range(len(nums)):\n res[i] = nums[i]\n # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res\n
array.java/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n int *res = new int[size + enlarge];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < size; i++) {\n res[i] = nums[i];\n }\n // \u91ca\u653e\u5185\u5b58\n delete[] nums;\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.cs/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n int[] res = new int[nums.length + enlarge];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < nums.length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.go/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n int[] res = new int[nums.Length + enlarge];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < nums.Length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.swift/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n res := make([]int, len(nums)+enlarge)\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for i, num := range nums {\n res[i] = num\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res\n}\n
array.js/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -> [Int] {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n var res = Array(repeating: 0, count: nums.count + enlarge)\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for i in nums.indices {\n res[i] = nums[i]\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res\n}\n
array.ts/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n const res = new Array(nums.length + enlarge).fill(0);\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (let i = 0; i < nums.length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.dart/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n const res = new Array(nums.length + enlarge).fill(0);\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (let i = 0; i < nums.length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.rs/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList<int> extend(List<int> nums, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n List<int> res = List.filled(nums.length + enlarge, 0);\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (var i = 0; i < nums.length; i++) {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
array.c/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfn extend(nums: Vec<i32>, enlarge: usize) -> Vec<i32> {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n let mut res: Vec<i32> = vec![0; nums.len() + enlarge];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\n for i in 0..nums.len() {\n res[i] = nums[i];\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n res\n}\n
array.zig/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < size; i++) {\n res[i] = nums[i];\n }\n // \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\n for (int i = size; i < size + enlarge; i++) {\n res[i] = 0;\n }\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n
"},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2 \u00a0 \u6570\u7ec4\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"// \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n var res = try mem_allocator.alloc(i32, nums.len + enlarge);\n @memset(res, 0);\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n std.mem.copy(i32, res, nums);\n // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n return res;\n}\n\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u4e14\u5143\u7d20\u7c7b\u578b\u76f8\u540c\u3002\u8fd9\u79cd\u505a\u6cd5\u5305\u542b\u4e30\u5bcc\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u7cfb\u7edf\u53ef\u4ee5\u5229\u7528\u8fd9\u4e9b\u4fe1\u606f\u6765\u4f18\u5316\u6570\u636e\u7ed3\u6784\u7684\u64cd\u4f5c\u6548\u7387\u3002
- \u7a7a\u95f4\u6548\u7387\u9ad8\uff1a\u6570\u7ec4\u4e3a\u6570\u636e\u5206\u914d\u4e86\u8fde\u7eed\u7684\u5185\u5b58\u5757\uff0c\u65e0\u987b\u989d\u5916\u7684\u7ed3\u6784\u5f00\u9500\u3002
- \u652f\u6301\u968f\u673a\u8bbf\u95ee\uff1a\u6570\u7ec4\u5141\u8bb8\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u4efb\u4f55\u5143\u7d20\u3002
- \u7f13\u5b58\u5c40\u90e8\u6027\uff1a\u5f53\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u4ed6\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002
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- \u63d2\u5165\u4e0e\u5220\u9664\u6548\u7387\u4f4e\uff1a\u5f53\u6570\u7ec4\u4e2d\u5143\u7d20\u8f83\u591a\u65f6\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u9700\u8981\u79fb\u52a8\u5927\u91cf\u7684\u5143\u7d20\u3002
- \u957f\u5ea6\u4e0d\u53ef\u53d8\uff1a\u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u5c31\u56fa\u5b9a\u4e86\uff0c\u6269\u5bb9\u6570\u7ec4\u9700\u8981\u5c06\u6240\u6709\u6570\u636e\u590d\u5236\u5230\u65b0\u6570\u7ec4\uff0c\u5f00\u9500\u5f88\u5927\u3002
- \u7a7a\u95f4\u6d6a\u8d39\uff1a\u5982\u679c\u6570\u7ec4\u5206\u914d\u7684\u5927\u5c0f\u8d85\u8fc7\u5b9e\u9645\u6240\u9700\uff0c\u90a3\u4e48\u591a\u4f59\u7684\u7a7a\u95f4\u5c31\u88ab\u6d6a\u8d39\u4e86\u3002
\u6570\u7ec4\u662f\u4e00\u79cd\u57fa\u7840\u4e14\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\uff0c\u65e2\u9891\u7e41\u5e94\u7528\u5728\u5404\u7c7b\u7b97\u6cd5\u4e4b\u4e2d\uff0c\u4e5f\u53ef\u7528\u4e8e\u5b9e\u73b0\u5404\u79cd\u590d\u6742\u6570\u636e\u7ed3\u6784\u3002
- \u968f\u673a\u8bbf\u95ee\uff1a\u5982\u679c\u6211\u4eec\u60f3\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u968f\u673a\u62bd\u6837\u3002
- \u6392\u5e8f\u548c\u641c\u7d22\uff1a\u6570\u7ec4\u662f\u6392\u5e8f\u548c\u641c\u7d22\u7b97\u6cd5\u6700\u5e38\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u4e8c\u5206\u67e5\u627e\u7b49\u90fd\u4e3b\u8981\u5728\u6570\u7ec4\u4e0a\u8fdb\u884c\u3002
- \u67e5\u627e\u8868\uff1a\u5f53\u9700\u8981\u5feb\u901f\u67e5\u627e\u4e00\u4e2a\u5143\u7d20\u6216\u5176\u5bf9\u5e94\u5173\u7cfb\u65f6\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u4f5c\u4e3a\u67e5\u627e\u8868\u3002\u5047\u5982\u6211\u4eec\u60f3\u5b9e\u73b0\u5b57\u7b26\u5230 ASCII \u7801\u7684\u6620\u5c04\uff0c\u5219\u53ef\u4ee5\u5c06\u5b57\u7b26\u7684 ASCII \u7801\u503c\u4f5c\u4e3a\u7d22\u5f15\uff0c\u5bf9\u5e94\u7684\u5143\u7d20\u5b58\u653e\u5728\u6570\u7ec4\u4e2d\u7684\u5bf9\u5e94\u4f4d\u7f6e\u3002
- \u673a\u5668\u5b66\u4e60\uff1a\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002
- \u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff1a\u6570\u7ec4\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002\u4f8b\u5982\uff0c\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\u3002
\u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u5728\u4e00\u4e2a\u590d\u6742\u7684\u7cfb\u7edf\u8fd0\u884c\u73af\u5883\u4e0b\uff0c\u7a7a\u95f2\u7684\u5185\u5b58\u7a7a\u95f4\u53ef\u80fd\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5b58\u50a8\u6570\u7ec4\u7684\u5185\u5b58\u7a7a\u95f4\u5fc5\u987b\u662f\u8fde\u7eed\u7684\uff0c\u800c\u5f53\u6570\u7ec4\u975e\u5e38\u5927\u65f6\uff0c\u5185\u5b58\u53ef\u80fd\u65e0\u6cd5\u63d0\u4f9b\u5982\u6b64\u5927\u7684\u8fde\u7eed\u7a7a\u95f4\u3002\u6b64\u65f6\u94fe\u8868\u7684\u7075\u6d3b\u6027\u4f18\u52bf\u5c31\u4f53\u73b0\u51fa\u6765\u4e86\u3002
\u300c\u94fe\u8868 linked list\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u4e00\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u5404\u4e2a\u8282\u70b9\u901a\u8fc7\u201c\u5f15\u7528\u201d\u76f8\u8fde\u63a5\u3002\u5f15\u7528\u8bb0\u5f55\u4e86\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684\u5185\u5b58\u5730\u5740\uff0c\u901a\u8fc7\u5b83\u53ef\u4ee5\u4ece\u5f53\u524d\u8282\u70b9\u8bbf\u95ee\u5230\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002
\u94fe\u8868\u7684\u8bbe\u8ba1\u4f7f\u5f97\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\uff0c\u5b83\u4eec\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002
\u56fe 4-5 \u00a0 \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f
\u89c2\u5bdf\u56fe 4-5 \uff0c\u94fe\u8868\u7684\u7ec4\u6210\u5355\u4f4d\u662f\u300c\u8282\u70b9 node\u300d\u5bf9\u8c61\u3002\u6bcf\u4e2a\u8282\u70b9\u90fd\u5305\u542b\u4e24\u9879\u6570\u636e\uff1a\u8282\u70b9\u7684\u201c\u503c\u201d\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u201c\u5f15\u7528\u201d\u3002
- \u94fe\u8868\u7684\u9996\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5934\u8282\u70b9\u201d\uff0c\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5c3e\u8282\u70b9\u201d\u3002
- \u5c3e\u8282\u70b9\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5b83\u5728 Java\u3001C++ \u548c Python \u4e2d\u5206\u522b\u88ab\u8bb0\u4e3a \\(\\text{null}\\)\u3001\\(\\text{nullptr}\\) \u548c \\(\\text{None}\\) \u3002
- \u5728 C\u3001C++\u3001Go \u548c Rust \u7b49\u652f\u6301\u6307\u9488\u7684\u8bed\u8a00\u4e2d\uff0c\u4e0a\u8ff0\u7684\u201c\u5f15\u7528\u201d\u5e94\u88ab\u66ff\u6362\u4e3a\u201c\u6307\u9488\u201d\u3002
\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u94fe\u8868\u8282\u70b9
PythonC++JavaC#GoSwiftJSTSDartRustCZigListNode\u9664\u4e86\u5305\u542b\u503c\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u4e00\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u56e0\u6b64\u5728\u76f8\u540c\u6570\u636e\u91cf\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002class ListNode:\n \"\"\"\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n def __init__(self, val: int):\n self.val: int = val # \u8282\u70b9\u503c\n self.next: ListNode | None = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n ListNode(int x) : val(x), next(nullptr) {} // \u6784\u9020\u51fd\u6570\n};\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n ListNode(int x) { val = x; } // \u6784\u9020\u51fd\u6570\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) { //\u6784\u9020\u51fd\u6570\n int val = x; // \u8282\u70b9\u503c\n ListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\n Val int // \u8282\u70b9\u503c\n Next *ListNode // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\n return &ListNode{\n Val: val,\n Next: nil,\n }\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n var val: Int // \u8282\u70b9\u503c\n var next: ListNode? // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\n init(x: Int) { // \u6784\u9020\u51fd\u6570\n val = x\n }\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n constructor(val, next) {\n this.val = (val === undefined ? 0 : val); // \u8282\u70b9\u503c\n this.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n val: number;\n next: ListNode | null;\n constructor(val?: number, next?: ListNode | null) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = next === undefined ? null : next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n ListNode(this.val, [this.next]); // \u6784\u9020\u51fd\u6570\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\n#[derive(Debug)]\nstruct ListNode {\n val: i32, // \u8282\u70b9\u503c\n next: Option<Rc<RefCell<ListNode>>>, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n int val; // \u8282\u70b9\u503c\n struct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n ListNode *node;\n node = (ListNode *) malloc(sizeof(ListNode));\n node->val = val;\n node->next = NULL;\n return node;\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1 \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u94fe\u8868","text":"// \u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n return struct {\n const Self = @This();\n\n val: T = 0, // \u8282\u70b9\u503c\n next: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n\n // \u6784\u9020\u51fd\u6570\n pub fn init(self: *Self, x: i32) void {\n self.val = x;\n self.next = null;\n }\n };\n}\n\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u5173\u7cfb\u3002\u521d\u59cb\u5316\u5b8c\u6210\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u901a\u8fc7\u5f15\u7528\u6307\u5411
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.pynext\u4f9d\u6b21\u8bbf\u95ee\u6240\u6709\u8282\u70b9\u3002
linked_list.cpp# \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
linked_list.java/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
linked_list.cs/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.go/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.swift/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n
linked_list.js/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
linked_list.ts/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.dart/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.rs/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\\\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
linked_list.c/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n
linked_list.zig/* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n// \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = &n1;\nn1.next = &n2;\nn2.next = &n3;\nn3.next = &n4;\n\u6570\u7ec4\u6574\u4f53\u662f\u4e00\u4e2a\u53d8\u91cf\uff0c\u6bd4\u5982\u6570\u7ec4
"},{"location":"chapter_array_and_linkedlist/linked_list/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"nums\u5305\u542b\u5143\u7d20nums[0]\u548cnums[1]\u7b49\uff0c\u800c\u94fe\u8868\u662f\u7531\u591a\u4e2a\u72ec\u7acb\u7684\u8282\u70b9\u5bf9\u8c61\u7ec4\u6210\u7684\u3002\u6211\u4eec\u901a\u5e38\u5c06\u5934\u8282\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u6bd4\u5982\u4ee5\u4e0a\u4ee3\u7801\u4e2d\u7684\u94fe\u8868\u53ef\u8bb0\u4f5c\u94fe\u8868n0\u3002\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u8282\u70b9\u975e\u5e38\u5bb9\u6613\u3002\u5982\u56fe 4-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u5728\u76f8\u90bb\u7684\u4e24\u4e2a\u8282\u70b9
n0\u548cn1\u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u8282\u70b9P\uff0c\u5219\u53ea\u9700\u6539\u53d8\u4e24\u4e2a\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5728\u6570\u7ec4\u4e2d\u63d2\u5165\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u6548\u7387\u8f83\u4f4e\u3002
\u56fe 4-6 \u00a0 \u94fe\u8868\u63d2\u5165\u8282\u70b9\u793a\u4f8b
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py
linked_list.cppdef insert(n0: ListNode, P: ListNode):\n \"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\n n1 = n0.next\n P.next = n1\n n0.next = P\n
linked_list.java/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n ListNode *n1 = n0->next;\n P->next = n1;\n n0->next = P;\n}\n
linked_list.cs/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n ListNode n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.go/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid Insert(ListNode n0, ListNode P) {\n ListNode? n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.swift/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n n1 := n0.Next\n P.Next = n1\n n0.Next = P\n}\n
linked_list.js/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\n let n1 = n0.next\n P.next = n1\n n0.next = P\n}\n
linked_list.ts/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\n const n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.dart/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\n const n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.rs/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n ListNode? n1 = n0.next;\n P.next = n1;\n n0.next = P;\n}\n
linked_list.c/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\n#[allow(non_snake_case)]\npub fn insert<T>(n0: &Rc<RefCell<ListNode<T>>>, P: Rc<RefCell<ListNode<T>>>) {\n let n1 = n0.borrow_mut().next.take();\n P.borrow_mut().next = n1;\n n0.borrow_mut().next = Some(P);\n}\n
linked_list.zig/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n ListNode *n1 = n0->next;\n P->next = n1;\n n0->next = P;\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"// \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\n var n1 = n0.?.next;\n P.?.next = n1;\n n0.?.next = P;\n}\n\u5982\u56fe 4-7 \u6240\u793a\uff0c\u5728\u94fe\u8868\u4e2d\u5220\u9664\u8282\u70b9\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u53d8\u4e00\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u3002
\u8bf7\u6ce8\u610f\uff0c\u5c3d\u7ba1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\u8282\u70b9
P\u4ecd\u7136\u6307\u5411n1\uff0c\u4f46\u5b9e\u9645\u4e0a\u904d\u5386\u6b64\u94fe\u8868\u5df2\u7ecf\u65e0\u6cd5\u8bbf\u95ee\u5230P\uff0c\u8fd9\u610f\u5473\u7740P\u5df2\u7ecf\u4e0d\u518d\u5c5e\u4e8e\u8be5\u94fe\u8868\u4e86\u3002\u56fe 4-7 \u00a0 \u94fe\u8868\u5220\u9664\u8282\u70b9
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py
linked_list.cppdef remove(n0: ListNode):\n \"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n if not n0.next:\n return\n # n0 -> P -> n1\n P = n0.next\n n1 = P.next\n n0.next = n1\n
linked_list.java/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\n if (n0->next == nullptr)\n return;\n // n0 -> P -> n1\n ListNode *P = n0->next;\n ListNode *n1 = P->next;\n n0->next = n1;\n // \u91ca\u653e\u5185\u5b58\n delete P;\n}\n
linked_list.cs/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n if (n0.next == null)\n return;\n // n0 -> P -> n1\n ListNode P = n0.next;\n ListNode n1 = P.next;\n n0.next = n1;\n}\n
linked_list.go/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid Remove(ListNode n0) {\n if (n0.next == null)\n return;\n // n0 -> P -> n1\n ListNode P = n0.next;\n ListNode? n1 = P.next;\n n0.next = n1;\n}\n
linked_list.swift/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeItem(n0 *ListNode) {\n if n0.Next == nil {\n return\n }\n // n0 -> P -> n1\n P := n0.Next\n n1 := P.Next\n n0.Next = n1\n}\n
linked_list.js/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\n if n0.next == nil {\n return\n }\n // n0 -> P -> n1\n let P = n0.next\n let n1 = P?.next\n n0.next = n1\n P?.next = nil\n}\n
linked_list.ts/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\n if (!n0.next) return;\n // n0 -> P -> n1\n const P = n0.next;\n const n1 = P.next;\n n0.next = n1;\n}\n
linked_list.dart/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\n if (!n0.next) {\n return;\n }\n // n0 -> P -> n1\n const P = n0.next;\n const n1 = P.next;\n n0.next = n1;\n}\n
linked_list.rs/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n if (n0.next == null) return;\n // n0 -> P -> n1\n ListNode P = n0.next!;\n ListNode? n1 = P.next;\n n0.next = n1;\n}\n
linked_list.c/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n#[allow(non_snake_case)]\npub fn remove<T>(n0: &Rc<RefCell<ListNode<T>>>) {\n if n0.borrow().next.is_none() {return};\n // n0 -> P -> n1\n let P = n0.borrow_mut().next.take();\n if let Some(node) = P {\n let n1 = node.borrow_mut().next.take();\n n0.borrow_mut().next = n1;\n }\n}\n
linked_list.zig/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(ListNode *n0) {\n if (!n0->next)\n return;\n // n0 -> P -> n1\n ListNode *P = n0->next;\n ListNode *n1 = P->next;\n n0->next = n1;\n // \u91ca\u653e\u5185\u5b58\n free(P);\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#4","title":"4. \u00a0 \u8bbf\u95ee\u8282\u70b9","text":"// \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\n if (n0.?.next == null) return;\n // n0 -> P -> n1\n var P = n0.?.next;\n var n1 = P.?.next;\n n0.?.next = n1;\n}\n\u5728\u94fe\u8868\u4e2d\u8bbf\u95ee\u8282\u70b9\u7684\u6548\u7387\u8f83\u4f4e\u3002\u5982\u4e0a\u4e00\u8282\u6240\u8ff0\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u5143\u7d20\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u7a0b\u5e8f\u9700\u8981\u4ece\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u9010\u4e2a\u5411\u540e\u904d\u5386\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u8282\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8bbf\u95ee\u94fe\u8868\u7684\u7b2c \\(i\\) \u4e2a\u8282\u70b9\u9700\u8981\u5faa\u73af \\(i - 1\\) \u8f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py
linked_list.cppdef access(head: ListNode, index: int) -> ListNode | None:\n \"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\n for _ in range(index):\n if not head:\n return None\n head = head.next\n return head\n
linked_list.java/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n for (int i = 0; i < index; i++) {\n if (head == nullptr)\n return nullptr;\n head = head->next;\n }\n return head;\n}\n
linked_list.cs/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\n for (int i = 0; i < index; i++) {\n if (head == null)\n return null;\n head = head.next;\n }\n return head;\n}\n
linked_list.go/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? Access(ListNode? head, int index) {\n for (int i = 0; i < index; i++) {\n if (head == null)\n return null;\n head = head.next;\n }\n return head;\n}\n
linked_list.swift/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\n for i := 0; i < index; i++ {\n if head == nil {\n return nil\n }\n head = head.Next\n }\n return head\n}\n
linked_list.js/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -> ListNode? {\n var head: ListNode? = head\n for _ in 0 ..< index {\n if head == nil {\n return nil\n }\n head = head?.next\n }\n return head\n}\n
linked_list.ts/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\n for (let i = 0; i < index; i++) {\n if (!head) {\n return null;\n }\n head = head.next;\n }\n return head;\n}\n
linked_list.dart/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n for (let i = 0; i < index; i++) {\n if (!head) {\n return null;\n }\n head = head.next;\n }\n return head;\n}\n
linked_list.rs/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\n for (var i = 0; i < index; i++) {\n if (head == null) return null;\n head = head.next;\n }\n return head;\n}\n
linked_list.c/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\npub fn access<T>(head: Rc<RefCell<ListNode<T>>>, index: i32) -> Rc<RefCell<ListNode<T>>> {\n if index <= 0 {return head};\n if let Some(node) = &head.borrow_mut().next {\n return access(node.clone(), index - 1);\n }\n return head;\n}\n
linked_list.zig/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n for (int i = 0; i < index; i++) {\n if (head == NULL)\n return NULL;\n head = head->next;\n }\n return head;\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#5","title":"5. \u00a0 \u67e5\u627e\u8282\u70b9","text":"// \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\n var head = node;\n var i: i32 = 0;\n while (i < index) : (i += 1) {\n head = head.?.next;\n if (head == null) return null;\n }\n return head;\n}\n\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u5176\u4e2d\u503c\u4e3a
PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.pytarget\u7684\u8282\u70b9\uff0c\u8f93\u51fa\u8be5\u8282\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002\u6b64\u8fc7\u7a0b\u4e5f\u5c5e\u4e8e\u7ebf\u6027\u67e5\u627e\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
linked_list.cppdef find(head: ListNode, target: int) -> int:\n \"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n index = 0\n while head:\n if head.val == target:\n return index\n head = head.next\n index += 1\n return -1\n
linked_list.java/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n int index = 0;\n while (head != nullptr) {\n if (head->val == target)\n return index;\n head = head->next;\n index++;\n }\n return -1;\n}\n
linked_list.cs/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\n int index = 0;\n while (head != null) {\n if (head.val == target)\n return index;\n head = head.next;\n index++;\n }\n return -1;\n}\n
linked_list.go/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint Find(ListNode? head, int target) {\n int index = 0;\n while (head != null) {\n if (head.val == target)\n return index;\n head = head.next;\n index++;\n }\n return -1;\n}\n
linked_list.swift/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\n index := 0\n for head != nil {\n if head.Val == target {\n return index\n }\n head = head.Next\n index++\n }\n return -1\n}\n
linked_list.js/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -> Int {\n var head: ListNode? = head\n var index = 0\n while head != nil {\n if head?.val == target {\n return index\n }\n head = head?.next\n index += 1\n }\n return -1\n}\n
linked_list.ts/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\n let index = 0;\n while (head !== null) {\n if (head.val === target) {\n return index;\n }\n head = head.next;\n index += 1;\n }\n return -1;\n}\n
linked_list.dart/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\n let index = 0;\n while (head !== null) {\n if (head.val === target) {\n return index;\n }\n head = head.next;\n index += 1;\n }\n return -1;\n}\n
linked_list.rs/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\n int index = 0;\n while (head != null) {\n if (head.val == target) {\n return index;\n }\n head = head.next;\n index++;\n }\n return -1;\n}\n
linked_list.c/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\npub fn find<T: PartialEq>(head: Rc<RefCell<ListNode<T>>>, target: T, index: i32) -> i32 {\n if head.borrow().val == target {return index};\n if let Some(node) = &head.borrow_mut().next {\n return find(node.clone(), target, index + 1);\n }\n return -1;\n}\n
linked_list.zig/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n int index = 0;\n while (head) {\n if (head->val == target)\n return index;\n head = head->next;\n index++;\n }\n return -1;\n}\n
"},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","title":"4.2.2 \u00a0 \u6570\u7ec4 vs. \u94fe\u8868","text":"// \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\n var head = node;\n var index: i32 = 0;\n while (head != null) {\n if (head.?.val == target) return index;\n head = head.?.next;\n index += 1;\n }\n return -1;\n}\n\u8868 4-1 \u603b\u7ed3\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u7684\u5404\u9879\u7279\u70b9\u5e76\u5bf9\u6bd4\u4e86\u64cd\u4f5c\u6548\u7387\u3002\u7531\u4e8e\u5b83\u4eec\u91c7\u7528\u4e24\u79cd\u76f8\u53cd\u7684\u5b58\u50a8\u7b56\u7565\uff0c\u56e0\u6b64\u5404\u79cd\u6027\u8d28\u548c\u64cd\u4f5c\u6548\u7387\u4e5f\u5448\u73b0\u5bf9\u7acb\u7684\u7279\u70b9\u3002
\u8868 4-1 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u6548\u7387\u5bf9\u6bd4
\u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u5206\u6563\u5185\u5b58\u7a7a\u95f4 \u5bb9\u91cf\u6269\u5c55 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u53ef\u7075\u6d3b\u6269\u5c55 \u5185\u5b58\u6548\u7387 \u5143\u7d20\u5360\u7528\u5185\u5b58\u5c11\u3001\u4f46\u53ef\u80fd\u6d6a\u8d39\u7a7a\u95f4 \u5143\u7d20\u5360\u7528\u5185\u5b58\u591a \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3 \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"\u5982\u56fe 4-8 \u6240\u793a\uff0c\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u4e09\u79cd\u3002
- \u5355\u5411\u94fe\u8868\uff1a\u5373\u524d\u9762\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u8282\u70b9\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u8282\u70b9\u79f0\u4e3a\u5934\u8282\u70b9\uff0c\u5c06\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u79f0\u4e3a\u5c3e\u8282\u70b9\uff0c\u5c3e\u8282\u70b9\u6307\u5411\u7a7a \\(\\text{None}\\) \u3002
- \u73af\u5f62\u94fe\u8868\uff1a\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u8282\u70b9\u6307\u5411\u5934\u8282\u70b9\uff08\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u4efb\u610f\u8282\u70b9\u90fd\u53ef\u4ee5\u89c6\u4f5c\u5934\u8282\u70b9\u3002
- \u53cc\u5411\u94fe\u8868\uff1a\u4e0e\u5355\u5411\u94fe\u8868\u76f8\u6bd4\uff0c\u53cc\u5411\u94fe\u8868\u8bb0\u5f55\u4e86\u4e24\u4e2a\u65b9\u5411\u7684\u5f15\u7528\u3002\u53cc\u5411\u94fe\u8868\u7684\u8282\u70b9\u5b9a\u4e49\u540c\u65f6\u5305\u542b\u6307\u5411\u540e\u7ee7\u8282\u70b9\uff08\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\u548c\u524d\u9a71\u8282\u70b9\uff08\u4e0a\u4e00\u4e2a\u8282\u70b9\uff09\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u76f8\u8f83\u4e8e\u5355\u5411\u94fe\u8868\uff0c\u53cc\u5411\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u76f8\u5e94\u5730\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002
class ListNode:\n \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n def __init__(self, val: int):\n self.val: int = val # \u8282\u70b9\u503c\n self.next: ListNode | None = None # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n self.prev: ListNode | None = None # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n ListNode(int x) : val(x), next(nullptr), prev(nullptr) {} // \u6784\u9020\u51fd\u6570\n};\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n ListNode prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n ListNode(int x) { val = x; } // \u6784\u9020\u51fd\u6570\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) { // \u6784\u9020\u51fd\u6570\n int val = x; // \u8282\u70b9\u503c\n ListNode next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n ListNode prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\n Val int // \u8282\u70b9\u503c\n Next *DoublyListNode // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n Prev *DoublyListNode // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\n return &DoublyListNode{\n Val: val,\n Next: nil,\n Prev: nil,\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n var val: Int // \u8282\u70b9\u503c\n var next: ListNode? // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n var prev: ListNode? // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n\n init(x: Int) { // \u6784\u9020\u51fd\u6570\n val = x\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n constructor(val, next, prev) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = next === undefined ? null : next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n this.prev = prev === undefined ? null : prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n val: number;\n next: ListNode | null;\n prev: ListNode | null;\n constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = next === undefined ? null : next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n this.prev = prev === undefined ? null : prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n ListNode prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n ListNode(this.val, [this.next, this.prev]); // \u6784\u9020\u51fd\u6570\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\u578b */\n#[derive(Debug)]\nstruct ListNode {\n val: i32, // \u8282\u70b9\u503c\n next: Option<Rc<RefCell<ListNode>>>, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n prev: Option<Rc<RefCell<ListNode>>>, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nimpl ListNode {\n fn new(val: i32) -> Self {\n ListNode {\n val,\n next: None,\n prev: None,\n }\n }\n}\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n int val; // \u8282\u70b9\u503c\n struct ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n struct ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n ListNode *node;\n node = (ListNode *) malloc(sizeof(ListNode));\n node->val = val;\n node->next = NULL;\n node->prev = NULL;\n return node;\n}\n// \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n return struct {\n const Self = @This();\n\n val: T = 0, // \u8282\u70b9\u503c\n next: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n prev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n\n // \u6784\u9020\u51fd\u6570\n pub fn init(self: *Self, x: i32) void {\n self.val = x;\n self.next = null;\n self.prev = null;\n }\n };\n}\n\u56fe 4-8 \u00a0 \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b
"},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4 \u00a0 \u94fe\u8868\u5178\u578b\u5e94\u7528","text":"\u5355\u5411\u94fe\u8868\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u548c\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002
- \u6808\u4e0e\u961f\u5217\uff1a\u5f53\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u90fd\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\u65f6\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u540e\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u6808\uff1b\u5f53\u63d2\u5165\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\uff0c\u5220\u9664\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u53e6\u4e00\u7aef\u8fdb\u884c\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u5148\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u961f\u5217\u3002
- \u54c8\u5e0c\u8868\uff1a\u94fe\u5f0f\u5730\u5740\u662f\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u7684\u4e3b\u6d41\u65b9\u6848\u4e4b\u4e00\uff0c\u5728\u8be5\u65b9\u6848\u4e2d\uff0c\u6240\u6709\u51b2\u7a81\u7684\u5143\u7d20\u90fd\u4f1a\u88ab\u653e\u5230\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002
- \u56fe\uff1a\u90bb\u63a5\u8868\u662f\u8868\u793a\u56fe\u7684\u4e00\u79cd\u5e38\u7528\u65b9\u5f0f\uff0c\u5176\u4e2d\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u90fd\u4e0e\u4e00\u4e2a\u94fe\u8868\u76f8\u5173\u8054\uff0c\u94fe\u8868\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u4ee3\u8868\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u5176\u4ed6\u9876\u70b9\u3002
\u53cc\u5411\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5feb\u901f\u67e5\u627e\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u5143\u7d20\u7684\u573a\u666f\u3002
- \u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\uff1a\u6bd4\u5982\u5728\u7ea2\u9ed1\u6811\u3001B \u6811\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u8bbf\u95ee\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5728\u8282\u70b9\u4e2d\u4fdd\u5b58\u4e00\u4e2a\u6307\u5411\u7236\u8282\u70b9\u7684\u5f15\u7528\u6765\u5b9e\u73b0\uff0c\u7c7b\u4f3c\u4e8e\u53cc\u5411\u94fe\u8868\u3002
- \u6d4f\u89c8\u5668\u5386\u53f2\uff1a\u5728\u7f51\u9875\u6d4f\u89c8\u5668\u4e2d\uff0c\u5f53\u7528\u6237\u70b9\u51fb\u524d\u8fdb\u6216\u540e\u9000\u6309\u94ae\u65f6\uff0c\u6d4f\u89c8\u5668\u9700\u8981\u77e5\u9053\u7528\u6237\u8bbf\u95ee\u8fc7\u7684\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u7f51\u9875\u3002\u53cc\u5411\u94fe\u8868\u7684\u7279\u6027\u4f7f\u5f97\u8fd9\u79cd\u64cd\u4f5c\u53d8\u5f97\u7b80\u5355\u3002
- LRU \u7b97\u6cd5\uff1a\u5728\u7f13\u5b58\u6dd8\u6c70\uff08LRU\uff09\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5feb\u901f\u627e\u5230\u6700\u8fd1\u6700\u5c11\u4f7f\u7528\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u652f\u6301\u5feb\u901f\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u3002\u8fd9\u65f6\u5019\u4f7f\u7528\u53cc\u5411\u94fe\u8868\u5c31\u975e\u5e38\u5408\u9002\u3002
\u73af\u5f62\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5468\u671f\u6027\u64cd\u4f5c\u7684\u573a\u666f\uff0c\u6bd4\u5982\u64cd\u4f5c\u7cfb\u7edf\u7684\u8d44\u6e90\u8c03\u5ea6\u3002
- \u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\uff1a\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\u662f\u4e00\u79cd\u5e38\u89c1\u7684 CPU \u8c03\u5ea6\u7b97\u6cd5\uff0c\u5b83\u9700\u8981\u5bf9\u4e00\u7ec4\u8fdb\u7a0b\u8fdb\u884c\u5faa\u73af\u3002\u6bcf\u4e2a\u8fdb\u7a0b\u88ab\u8d4b\u4e88\u4e00\u4e2a\u65f6\u95f4\u7247\uff0c\u5f53\u65f6\u95f4\u7247\u7528\u5b8c\u65f6\uff0cCPU \u5c06\u5207\u6362\u5230\u4e0b\u4e00\u4e2a\u8fdb\u7a0b\u3002\u8fd9\u79cd\u5faa\u73af\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u73af\u5f62\u94fe\u8868\u6765\u5b9e\u73b0\u3002
- \u6570\u636e\u7f13\u51b2\u533a\uff1a\u5728\u67d0\u4e9b\u6570\u636e\u7f13\u51b2\u533a\u7684\u5b9e\u73b0\u4e2d\uff0c\u4e5f\u53ef\u80fd\u4f1a\u4f7f\u7528\u73af\u5f62\u94fe\u8868\u3002\u6bd4\u5982\u5728\u97f3\u9891\u3001\u89c6\u9891\u64ad\u653e\u5668\u4e2d\uff0c\u6570\u636e\u6d41\u53ef\u80fd\u4f1a\u88ab\u5206\u6210\u591a\u4e2a\u7f13\u51b2\u5757\u5e76\u653e\u5165\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\uff0c\u4ee5\u4fbf\u5b9e\u73b0\u65e0\u7f1d\u64ad\u653e\u3002
\u300c\u5217\u8868 list\u300d\u662f\u4e00\u4e2a\u62bd\u8c61\u7684\u6570\u636e\u7ed3\u6784\u6982\u5ff5\uff0c\u5b83\u8868\u793a\u5143\u7d20\u7684\u6709\u5e8f\u96c6\u5408\uff0c\u652f\u6301\u5143\u7d20\u8bbf\u95ee\u3001\u4fee\u6539\u3001\u6dfb\u52a0\u3001\u5220\u9664\u548c\u904d\u5386\u7b49\u64cd\u4f5c\uff0c\u65e0\u987b\u4f7f\u7528\u8005\u8003\u8651\u5bb9\u91cf\u9650\u5236\u7684\u95ee\u9898\u3002\u5217\u8868\u53ef\u4ee5\u57fa\u4e8e\u94fe\u8868\u6216\u6570\u7ec4\u5b9e\u73b0\u3002
- \u94fe\u8868\u5929\u7136\u53ef\u4ee5\u770b\u4f5c\u4e00\u4e2a\u5217\u8868\uff0c\u5176\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\u64cd\u4f5c\uff0c\u5e76\u4e14\u53ef\u4ee5\u7075\u6d3b\u52a8\u6001\u6269\u5bb9\u3002
- \u6570\u7ec4\u4e5f\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\uff0c\u4f46\u7531\u4e8e\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u53ea\u80fd\u770b\u4f5c\u4e00\u4e2a\u5177\u6709\u957f\u5ea6\u9650\u5236\u7684\u5217\u8868\u3002
\u5f53\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u5217\u8868\u65f6\uff0c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6027\u8d28\u4f1a\u5bfc\u81f4\u5217\u8868\u7684\u5b9e\u7528\u6027\u964d\u4f4e\u3002\u8fd9\u662f\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u65e0\u6cd5\u4e8b\u5148\u786e\u5b9a\u9700\u8981\u5b58\u50a8\u591a\u5c11\u6570\u636e\uff0c\u4ece\u800c\u96be\u4ee5\u9009\u62e9\u5408\u9002\u7684\u5217\u8868\u957f\u5ea6\u3002\u82e5\u957f\u5ea6\u8fc7\u5c0f\uff0c\u5219\u5f88\u53ef\u80fd\u65e0\u6cd5\u6ee1\u8db3\u4f7f\u7528\u9700\u6c42\uff1b\u82e5\u957f\u5ea6\u8fc7\u5927\uff0c\u5219\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u300c\u52a8\u6001\u6570\u7ec4 dynamic array\u300d\u6765\u5b9e\u73b0\u5217\u8868\u3002\u5b83\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u5404\u9879\u4f18\u70b9\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u8fdb\u884c\u52a8\u6001\u6269\u5bb9\u3002
\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u6807\u51c6\u5e93\u63d0\u4f9b\u7684\u5217\u8868\u662f\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982 Python \u4e2d\u7684
"},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1 \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u5217\u8868","text":"list\u3001Java \u4e2d\u7684ArrayList\u3001C++ \u4e2d\u7684vector\u548c C# \u4e2d\u7684List\u7b49\u3002\u5728\u63a5\u4e0b\u6765\u7684\u8ba8\u8bba\u4e2d\uff0c\u6211\u4eec\u5c06\u628a\u201c\u5217\u8868\u201d\u548c\u201c\u52a8\u6001\u6570\u7ec4\u201d\u89c6\u4e3a\u7b49\u540c\u7684\u6982\u5ff5\u3002\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u8fd9\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nnums1: list[int] = []\n# \u6709\u521d\u59cb\u503c\nnums: list[int] = [1, 3, 2, 5, 4]\n
list.java/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 nums\n// \u65e0\u521d\u59cb\u503c\nvector<int> nums1;\n// \u6709\u521d\u59cb\u503c\nvector<int> nums = { 1, 3, 2, 5, 4 };\n
list.cs/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<Integer> nums1 = new ArrayList<>();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList<Integer> nums = new ArrayList<>(Arrays.asList(numbers));\n
list_test.go/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> nums1 = [];\n// \u6709\u521d\u59cb\u503c\nint[] numbers = [1, 3, 2, 5, 4];\nList<int> nums = [.. numbers];\n
list.swift/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nnums1 := []int{}\n// \u6709\u521d\u59cb\u503c\nnums := []int{1, 3, 2, 5, 4}\n
list.js/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar nums = [1, 3, 2, 5, 4]\n
list.ts/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1 = [];\n// \u6709\u521d\u59cb\u503c\nconst nums = [1, 3, 2, 5, 4];\n
list.dart/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst nums: number[] = [1, 3, 2, 5, 4];\n
list.rs/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> nums1 = [];\n// \u6709\u521d\u59cb\u503c\nList<int> nums = [1, 3, 2, 5, 4];\n
list.c/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: Vec<i32> = Vec::new();\n// \u6709\u521d\u59cb\u503c\nlet nums: Vec<i32> = vec![1, 3, 2, 5, 4];\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"// \u521d\u59cb\u5316\u5217\u8868\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 });\n\u5217\u8868\u672c\u8d28\u4e0a\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u8bbf\u95ee\u5143\u7d20\nnum: int = nums[1] # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0 # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.java/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.cs/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums.get(1); // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums.set(1, 0); // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list_test.go/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.swift/* \u8bbf\u95ee\u5143\u7d20 */\nnum := nums[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.js/* \u8bbf\u95ee\u5143\u7d20 */\nlet num = nums[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.ts/* \u8bbf\u95ee\u5143\u7d20 */\nconst num = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.dart/* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.rs/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.c/* \u8bbf\u95ee\u5143\u7d20 */\nlet num: i32 = nums[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#3","title":"3. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20","text":"// \u8bbf\u95ee\u5143\u7d20\nvar num = nums.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n// \u66f4\u65b0\u5143\u7d20\nnums.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0 \n\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u76f8\u540c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u6e05\u7a7a\u5217\u8868\nnums.clear()\n\n# \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6) # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n# \u5220\u9664\u5143\u7d20\nnums.pop(3) # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.java/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(nums.begin() + 3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.erase(nums.begin() + 3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.cs/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list_test.go/* \u6e05\u7a7a\u5217\u8868 */\nnums.Clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.Insert(3, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.RemoveAt(3);\n
list.swift/* \u6e05\u7a7a\u5217\u8868 */\nnums = nil\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums = append(nums[:3], nums[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.js/* \u6e05\u7a7a\u5217\u8868 */\nnums.removeAll()\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.ts/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n
list.dart/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n
list.rs/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.removeAt(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.c/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#4","title":"4. \u00a0 \u904d\u5386\u5217\u8868","text":"// \u6e05\u7a7a\u5217\u8868\nnums.clearRetainingCapacity();\n\n// \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n\n// \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry nums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n// \u5220\u9664\u5143\u7d20\n_ = nums.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u6839\u636e\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u5404\u5143\u7d20\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in range(len(nums)):\n count += nums[i]\n\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\nfor num in nums:\n count += num\n
list.java/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < nums.size(); i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int num : nums) {\n count += num;\n}\n
list.cs/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < nums.size(); i++) {\n count += nums.get(i);\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\nfor (int num : nums) {\n count += num;\n}\n
list_test.go/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < nums.Count; i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int num in nums) {\n count += num;\n}\n
list.swift/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i < len(nums); i++ {\n count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _, num := range nums {\n count += num\n}\n
list.js/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor i in nums.indices {\n count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor num in nums {\n count += num\n}\n
list.ts/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < nums.length; i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n count += num;\n}\n
list.dart/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < nums.length; i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n count += num;\n}\n
list.rs/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (var i = 0; i < nums.length; i++) {\n count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (var num in nums) {\n count += num;\n}\n
list.c// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nlet mut _count = 0;\nfor i in 0..nums.len() {\n _count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\n_count = 0;\nfor num in &nums {\n _count += num;\n}\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#5","title":"5. \u00a0 \u62fc\u63a5\u5217\u8868","text":"// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < nums.items.len) : (i += 1) {\n count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (nums.items) |num| {\n count += num;\n}\n\u7ed9\u5b9a\u4e00\u4e2a\u65b0\u5217\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.pynums1\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u62fc\u63a5\u5230\u539f\u5217\u8868\u7684\u5c3e\u90e8\u3002
list.cpp# \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1 # \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.java/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector<int> nums1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n
list.cs/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<Integer> nums1 = new ArrayList<>(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list_test.go/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.swift/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...) // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.js/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.ts/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.dart/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.rs/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n
list.c/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1: Vec<i32> = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#6","title":"6. \u00a0 \u6392\u5e8f\u5217\u8868","text":"// \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u67e5\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u548c\u201c\u53cc\u6307\u9488\u201d\u7b97\u6cd5\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py
list.cpp# \u6392\u5e8f\u5217\u8868\nnums.sort() # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.java/* \u6392\u5e8f\u5217\u8868 */\nsort(nums.begin(), nums.end()); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.cs/* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(nums); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list_test.go/* \u6392\u5e8f\u5217\u8868 */\nnums.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.swift/* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(nums) // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.js/* \u6392\u5e8f\u5217\u8868 */\nnums.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.ts/* \u6392\u5e8f\u5217\u8868 */ \nnums.sort((a, b) => a - b); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.dart/* \u6392\u5e8f\u5217\u8868 */\nnums.sort((a, b) => a - b); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.rs/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.c/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
list.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
"},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2 \u00a0 \u5217\u8868\u5b9e\u73b0","text":"// \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u4e86\u5217\u8868\uff0c\u4f8b\u5982 Java\u3001C++\u3001Python \u7b49\u3002\u5b83\u4eec\u7684\u5b9e\u73b0\u6bd4\u8f83\u590d\u6742\uff0c\u5404\u4e2a\u53c2\u6570\u7684\u8bbe\u5b9a\u4e5f\u975e\u5e38\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u6269\u5bb9\u500d\u6570\u7b49\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u8fdb\u884c\u5b66\u4e60\u3002
\u4e3a\u4e86\u52a0\u6df1\u5bf9\u5217\u8868\u5de5\u4f5c\u539f\u7406\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5c1d\u8bd5\u5b9e\u73b0\u4e00\u4e2a\u7b80\u6613\u7248\u5217\u8868\uff0c\u5305\u62ec\u4ee5\u4e0b\u4e09\u4e2a\u91cd\u70b9\u8bbe\u8ba1\u3002
- \u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002
- \u6570\u91cf\u8bb0\u5f55\uff1a\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf
size\uff0c\u7528\u4e8e\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u548c\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002 - \u6269\u5bb9\u673a\u5236\uff1a\u82e5\u63d2\u5165\u5143\u7d20\u65f6\u5217\u8868\u5bb9\u91cf\u5df2\u6ee1\uff0c\u5219\u9700\u8981\u8fdb\u884c\u6269\u5bb9\u3002\u5148\u6839\u636e\u6269\u5bb9\u500d\u6570\u521b\u5efa\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u518d\u5c06\u5f53\u524d\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52a8\u81f3\u65b0\u6570\u7ec4\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002
my_list.cppclass MyList:\n \"\"\"\u5217\u8868\u7c7b\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._capacity: int = 10 # \u5217\u8868\u5bb9\u91cf\n self._arr: list[int] = [0] * self._capacity # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n self._size: int = 0 # \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n self._extend_ratio: int = 2 # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\n return self._size\n\n def capacity(self) -> int:\n \"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\n return self._capacity\n\n def get(self, index: int) -> int:\n \"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if index < 0 or index >= self._size:\n raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n return self._arr[index]\n\n def set(self, num: int, index: int):\n \"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\n if index < 0 or index >= self._size:\n raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n self._arr[index] = num\n\n def add(self, num: int):\n \"\"\"\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if self.size() == self.capacity():\n self.extend_capacity()\n self._arr[self._size] = num\n self._size += 1\n\n def insert(self, num: int, index: int):\n \"\"\"\u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\n if index < 0 or index >= self._size:\n raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if self._size == self.capacity():\n self.extend_capacity()\n # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for j in range(self._size - 1, index - 1, -1):\n self._arr[j + 1] = self._arr[j]\n self._arr[index] = num\n # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self._size += 1\n\n def remove(self, index: int) -> int:\n \"\"\"\u5220\u9664\u5143\u7d20\"\"\"\n if index < 0 or index >= self._size:\n raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n num = self._arr[index]\n # \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for j in range(index, self._size - 1):\n self._arr[j] = self._arr[j + 1]\n # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self._size -= 1\n # \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num\n\n def extend_capacity(self):\n \"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 __extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n self._capacity = len(self._arr)\n\n def to_array(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\n return self._arr[: self._size]\n
my_list.java/* \u5217\u8868\u7c7b */\nclass MyList {\n private:\n int *arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n int arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\n int arrSize = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n MyList() {\n arr = new int[arrCapacity];\n }\n\n /* \u6790\u6784\u65b9\u6cd5 */\n ~MyList() {\n delete[] arr;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n int size() {\n return arrSize;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n int capacity() {\n return arrCapacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n int get(int index) {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= size())\n throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n return arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n void set(int index, int num) {\n if (index < 0 || index >= size())\n throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n void add(int num) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size() == capacity())\n extendCapacity();\n arr[size()] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n void insert(int index, int num) {\n if (index < 0 || index >= size())\n throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size() == capacity())\n extendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int j = size() - 1; j >= index; j--) {\n arr[j + 1] = arr[j];\n }\n arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n int remove(int index) {\n if (index < 0 || index >= size())\n throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n int num = arr[index];\n // \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int j = index; j < size() - 1; j++) {\n arr[j] = arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n void extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n int newCapacity = capacity() * extendRatio;\n int *tmp = arr;\n arr = new int[newCapacity];\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n for (int i = 0; i < size(); i++) {\n arr[i] = tmp[i];\n }\n // \u91ca\u653e\u5185\u5b58\n delete[] tmp;\n arrCapacity = newCapacity;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\n vector<int> toVector() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n vector<int> vec(size());\n for (int i = 0; i < size(); i++) {\n vec[i] = arr[i];\n }\n return vec;\n }\n};\n
my_list.cs/* \u5217\u8868\u7c7b */\nclass MyList {\n private int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n private int capacity = 10; // \u5217\u8868\u5bb9\u91cf\n private int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n private int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public MyList() {\n arr = new int[capacity];\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n public int size() {\n return size;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n public int capacity() {\n return capacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n public int get(int index) {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= size)\n throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n return arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n public void set(int index, int num) {\n if (index < 0 || index >= size)\n throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n public void add(int num) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size == capacity())\n extendCapacity();\n arr[size] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n size++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n public void insert(int index, int num) {\n if (index < 0 || index >= size)\n throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size == capacity())\n extendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int j = size - 1; j >= index; j--) {\n arr[j + 1] = arr[j];\n }\n arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n size++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n public int remove(int index) {\n if (index < 0 || index >= size)\n throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n int num = arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int j = index; j < size - 1; j++) {\n arr[j] = arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n size--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n public void extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n arr = Arrays.copyOf(arr, capacity() * extendRatio);\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n capacity = arr.length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n public int[] toArray() {\n int size = size();\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] arr = new int[size];\n for (int i = 0; i < size; i++) {\n arr[i] = get(i);\n }\n return arr;\n }\n}\n
my_list.go/* \u5217\u8868\u7c7b */\nclass MyList {\n private int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n private int arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\n private int arrSize = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n private readonly int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public MyList() {\n arr = new int[arrCapacity];\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n public int Size() {\n return arrSize;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n public int Capacity() {\n return arrCapacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n public int Get(int index) {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= arrSize)\n throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n return arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n public void Set(int index, int num) {\n if (index < 0 || index >= arrSize)\n throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n public void Add(int num) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (arrSize == arrCapacity)\n ExtendCapacity();\n arr[arrSize] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n public void Insert(int index, int num) {\n if (index < 0 || index >= arrSize)\n throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (arrSize == arrCapacity)\n ExtendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (int j = arrSize - 1; j >= index; j--) {\n arr[j + 1] = arr[j];\n }\n arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n public int Remove(int index) {\n if (index < 0 || index >= arrSize)\n throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n int num = arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (int j = index; j < arrSize - 1; j++) {\n arr[j] = arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n arrSize--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n public void ExtendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a arrCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n Array.Resize(ref arr, arrCapacity * extendRatio);\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n arrCapacity = arr.Length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n public int[] ToArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] arr = new int[arrSize];\n for (int i = 0; i < arrSize; i++) {\n arr[i] = Get(i);\n }\n return arr;\n }\n}\n
my_list.swift/* \u5217\u8868\u7c7b */\ntype myList struct {\n arrCapacity int\n arr []int\n arrSize int\n extendRatio int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\n return &myList{\n arrCapacity: 10, // \u5217\u8868\u5bb9\u91cf\n arr: make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n arrSize: 0, // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n extendRatio: 2, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n }\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\n return l.arrSize\n}\n\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\n return l.arrCapacity\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if index < 0 || index >= l.arrSize {\n panic(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n return l.arr[index]\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\n if index < 0 || index >= l.arrSize {\n panic(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n l.arr[index] = num\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if l.arrSize == l.arrCapacity {\n l.extendCapacity()\n }\n l.arr[l.arrSize] = num\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n l.arrSize++\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\n if index < 0 || index >= l.arrSize {\n panic(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if l.arrSize == l.arrCapacity {\n l.extendCapacity()\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for j := l.arrSize - 1; j >= index; j-- {\n l.arr[j+1] = l.arr[j]\n }\n l.arr[index] = num\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n l.arrSize++\n}\n\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\n if index < 0 || index >= l.arrSize {\n panic(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n num := l.arr[index]\n // \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for j := index; j < l.arrSize-1; j++ {\n l.arr[j] = l.arr[j+1]\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n l.arrSize--\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n l.arrCapacity = len(l.arr)\n}\n\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n return l.arr[:l.arrSize]\n}\n
my_list.js/* \u5217\u8868\u7c7b */\nclass MyList {\n private var arr: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n private var _capacity = 10 // \u5217\u8868\u5bb9\u91cf\n private var _size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n private let extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init() {\n arr = Array(repeating: 0, count: _capacity)\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n func size() -> Int {\n _size\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n func capacity() -> Int {\n _capacity\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n func get(index: Int) -> Int {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\n if index < 0 || index >= _size {\n fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n return arr[index]\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n func set(index: Int, num: Int) {\n if index < 0 || index >= _size {\n fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n arr[index] = num\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n func add(num: Int) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if _size == _capacity {\n extendCapacity()\n }\n arr[_size] = num\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size += 1\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n func insert(index: Int, num: Int) {\n if index < 0 || index >= _size {\n fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if _size == _capacity {\n extendCapacity()\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for j in sequence(first: _size - 1, next: { $0 >= index + 1 ? $0 - 1 : nil }) {\n arr[j + 1] = arr[j]\n }\n arr[index] = num\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size += 1\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n @discardableResult\n func remove(index: Int) -> Int {\n if index < 0 || index >= _size {\n fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n }\n let num = arr[index]\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for j in index ..< (_size - 1) {\n arr[j] = arr[j + 1]\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size -= 1\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n func extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n arr = arr + Array(repeating: 0, count: _capacity * (extendRatio - 1))\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n _capacity = arr.count\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n func toArray() -> [Int] {\n var arr = Array(repeating: 0, count: _size)\n for i in 0 ..< _size {\n arr[i] = get(index: i)\n }\n return arr\n }\n}\n
my_list.ts/* \u5217\u8868\u7c7b */\nclass MyList {\n #arr = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n #capacity = 10; // \u5217\u8868\u5bb9\u91cf\n #size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n #extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.#arr = new Array(this.#capacity);\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n size() {\n return this.#size;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n capacity() {\n return this.#capacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n get(index) {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n return this.#arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n set(index, num) {\n if (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n this.#arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n add(num) {\n // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n if (this.#size === this.#capacity) {\n this.extendCapacity();\n }\n // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n this.#arr[this.#size] = num;\n this.#size++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n insert(index, num) {\n if (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (this.#size === this.#capacity) {\n this.extendCapacity();\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (let j = this.#size - 1; j >= index; j--) {\n this.#arr[j + 1] = this.#arr[j];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n this.#arr[index] = num;\n this.#size++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n remove(index) {\n if (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n let num = this.#arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (let j = index; j < this.#size - 1; j++) {\n this.#arr[j] = this.#arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n this.#size--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n this.#arr = this.#arr.concat(\n new Array(this.capacity() * (this.#extendRatio - 1))\n );\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n this.#capacity = this.#arr.length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n toArray() {\n let size = this.size();\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const arr = new Array(size);\n for (let i = 0; i < size; i++) {\n arr[i] = this.get(i);\n }\n return arr;\n }\n}\n
my_list.dart/* \u5217\u8868\u7c7b */\nclass MyList {\n private arr: Array<number>; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n private _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\n private _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n private extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.arr = new Array(this._capacity);\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n public size(): number {\n return this._size;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n public capacity(): number {\n return this._capacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n public get(index: number): number {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n return this.arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n public set(index: number, num: number): void {\n if (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n this.arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n public add(num: number): void {\n // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n if (this._size === this._capacity) this.extendCapacity();\n // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n this.arr[this._size] = num;\n this._size++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n public insert(index: number, num: number): void {\n if (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (this._size === this._capacity) {\n this.extendCapacity();\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (let j = this._size - 1; j >= index; j--) {\n this.arr[j + 1] = this.arr[j];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n this.arr[index] = num;\n this._size++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n public remove(index: number): number {\n if (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n let num = this.arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (let j = index; j < this._size - 1; j++) {\n this.arr[j] = this.arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n this._size--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n public extendCapacity(): void {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n this.arr = this.arr.concat(\n new Array(this.capacity() * (this.extendRatio - 1))\n );\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n this._capacity = this.arr.length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n public toArray(): number[] {\n let size = this.size();\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const arr = new Array(size);\n for (let i = 0; i < size; i++) {\n arr[i] = this.get(i);\n }\n return arr;\n }\n}\n
my_list.rs/* \u5217\u8868\u7c7b */\nclass MyList {\n late List<int> _arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n int _capacity = 10; // \u5217\u8868\u5bb9\u91cf\n int _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n int _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n /* \u6784\u9020\u65b9\u6cd5 */\n MyList() {\n _arr = List.filled(_capacity, 0);\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n int size() => _size;\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n int capacity() => _capacity;\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n int get(int index) {\n if (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n return _arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n void set(int index, int _num) {\n if (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n _arr[index] = _num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n void add(int _num) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (_size == _capacity) extendCapacity();\n _arr[_size] = _num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size++;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n void insert(int index, int _num) {\n if (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (_size == _capacity) extendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for (var j = _size - 1; j >= index; j--) {\n _arr[j + 1] = _arr[j];\n }\n _arr[index] = _num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size++;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n int remove(int index) {\n if (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n int _num = _arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for (var j = index; j < _size - 1; j++) {\n _arr[j] = _arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n _size--;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return _num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n void extendCapacity() {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n final _newNums = List.filled(_capacity * _extendRatio, 0);\n // \u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n List.copyRange(_newNums, 0, _arr);\n // \u66f4\u65b0 _arr \u7684\u5f15\u7528\n _arr = _newNums;\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n _capacity = _arr.length;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n List<int> toArray() {\n List<int> arr = [];\n for (var i = 0; i < _size; i++) {\n arr.add(get(i));\n }\n return arr;\n }\n}\n
my_list.c/* \u5217\u8868\u7c7b */\n#[allow(dead_code)]\nstruct MyList {\n arr: Vec<i32>, // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n capacity: usize, // \u5217\u8868\u5bb9\u91cf\n size: usize, // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n extend_ratio: usize, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n\n#[allow(unused,unused_comparisons)]\nimpl MyList {\n /* \u6784\u9020\u65b9\u6cd5 */\n pub fn new(capacity: usize) -> Self {\n let mut vec = Vec::new(); \n vec.resize(capacity, 0);\n Self {\n arr: vec,\n capacity,\n size: 0,\n extend_ratio: 2,\n }\n }\n\n /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n pub fn size(&self) -> usize {\n return self.size;\n }\n\n /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n pub fn capacity(&self) -> usize {\n return self.capacity;\n }\n\n /* \u8bbf\u95ee\u5143\u7d20 */\n pub fn get(&self, index: usize) -> i32 {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if index >= self.size {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n return self.arr[index];\n }\n\n /* \u66f4\u65b0\u5143\u7d20 */\n pub fn set(&mut self, index: usize, num: i32) {\n if index >= self.size {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n self.arr[index] = num;\n }\n\n /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n pub fn add(&mut self, num: i32) {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if self.size == self.capacity() {\n self.extend_capacity();\n }\n self.arr[self.size] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.size += 1;\n }\n\n /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n pub fn insert(&mut self, index: usize, num: i32) {\n if index >= self.size() {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if self.size == self.capacity() {\n self.extend_capacity();\n }\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n for j in (index..self.size).rev() {\n self.arr[j + 1] = self.arr[j];\n }\n self.arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.size += 1;\n }\n\n /* \u5220\u9664\u5143\u7d20 */\n pub fn remove(&mut self, index: usize) -> i32 {\n if index >= self.size() {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n let num = self.arr[index];\n // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n for j in (index..self.size - 1) {\n self.arr[j] = self.arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.size -= 1;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n /* \u5217\u8868\u6269\u5bb9 */\n pub fn extend_capacity(&mut self) {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n let new_capacity = self.capacity * self.extend_ratio;\n self.arr.resize(new_capacity, 0);\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n self.capacity = new_capacity;\n }\n\n /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n pub fn to_array(&mut self) -> Vec<i32> {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n let mut arr = Vec::new();\n for i in 0..self.size {\n arr.push(self.get(i));\n }\n arr\n }\n}\n
my_list.zig/* \u5217\u8868\u7c7b */\ntypedef struct {\n int *arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n int capacity; // \u5217\u8868\u5bb9\u91cf\n int size; // \u5217\u8868\u5927\u5c0f\n int extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n} MyList;\n\n/* \u6784\u9020\u51fd\u6570 */\nMyList *newMyList() {\n MyList *nums = malloc(sizeof(MyList));\n nums->capacity = 10;\n nums->arr = malloc(sizeof(int) * nums->capacity);\n nums->size = 0;\n nums->extendRatio = 2;\n return nums;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(MyList *nums) {\n free(nums->arr);\n free(nums);\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(MyList *nums) {\n return nums->size;\n}\n\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(MyList *nums) {\n return nums->capacity;\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(MyList *nums, int index) {\n assert(index >= 0 && index < nums->size);\n return nums->arr[index];\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\n assert(index >= 0 && index < nums->size);\n nums->arr[index] = num;\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\n if (size(nums) == capacity(nums)) {\n extendCapacity(nums); // \u6269\u5bb9\n }\n nums->arr[size(nums)] = num;\n nums->size++;\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\n assert(index >= 0 && index < size(nums));\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (size(nums) == capacity(nums)) {\n extendCapacity(nums); // \u6269\u5bb9\n }\n for (int i = size(nums); i > index; --i) {\n nums->arr[i] = nums->arr[i - 1];\n }\n nums->arr[index] = num;\n nums->size++;\n}\n\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeItem(MyList *nums, int index) {\n assert(index >= 0 && index < size(nums));\n int num = nums->arr[index];\n for (int i = index; i < size(nums) - 1; i++) {\n nums->arr[i] = nums->arr[i + 1];\n }\n nums->size--;\n return num;\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n // \u5148\u5206\u914d\u7a7a\u95f4\n int newCapacity = capacity(nums) * nums->extendRatio;\n int *extend = (int *)malloc(sizeof(int) * newCapacity);\n int *temp = nums->arr;\n\n // \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\n for (int i = 0; i < size(nums); i++)\n extend[i] = nums->arr[i];\n\n // \u91ca\u653e\u65e7\u6570\u636e\n free(temp);\n\n // \u66f4\u65b0\u65b0\u6570\u636e\n nums->arr = extend;\n nums->capacity = newCapacity;\n}\n\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(MyList *nums) {\n return nums->arr;\n}\n
"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *","text":"// \u5217\u8868\u7c7b\nfn MyList(comptime T: type) type {\n return struct {\n const Self = @This();\n\n arr: []T = undefined, // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n arrCapacity: usize = 10, // \u5217\u8868\u5bb9\u91cf\n numSize: usize = 0, // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n extendRatio: usize = 2, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n @memset(self.arr, @as(T, 0));\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n pub fn size(self: *Self) usize {\n return self.numSize;\n }\n\n // \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n pub fn capacity(self: *Self) usize {\n return self.arrCapacity;\n }\n\n // \u8bbf\u95ee\u5143\u7d20\n pub fn get(self: *Self, index: usize) T {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n return self.arr[index];\n } \n\n // \u66f4\u65b0\u5143\u7d20\n pub fn set(self: *Self, index: usize, num: T) void {\n // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n if (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n self.arr[index] = num;\n } \n\n // \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\n pub fn add(self: *Self, num: T) !void {\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (self.size() == self.capacity()) try self.extendCapacity();\n self.arr[self.size()] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.numSize += 1;\n } \n\n // \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\n pub fn insert(self: *Self, index: usize, num: T) !void {\n if (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n if (self.size() == self.capacity()) try self.extendCapacity();\n // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n var j = self.size() - 1;\n while (j >= index) : (j -= 1) {\n self.arr[j + 1] = self.arr[j];\n }\n self.arr[index] = num;\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.numSize += 1;\n }\n\n // \u5220\u9664\u5143\u7d20\n pub fn remove(self: *Self, index: usize) T {\n if (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n var num = self.arr[index];\n // \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n var j = index;\n while (j < self.size() - 1) : (j += 1) {\n self.arr[j] = self.arr[j + 1];\n }\n // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n self.numSize -= 1;\n // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n return num;\n }\n\n // \u5217\u8868\u6269\u5bb9\n pub fn extendCapacity(self: *Self) !void {\n // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n var newCapacity = self.capacity() * self.extendRatio;\n var extend = try self.mem_allocator.alloc(T, newCapacity);\n @memset(extend, @as(T, 0));\n // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n std.mem.copy(T, extend, self.arr);\n self.arr = extend;\n // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n self.arrCapacity = newCapacity;\n }\n\n // \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n pub fn toArray(self: *Self) ![]T {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n var arr = try self.mem_allocator.alloc(T, self.size());\n @memset(arr, @as(T, 0));\n for (arr, 0..) |*num, i| {\n num.* = self.get(i);\n }\n return arr;\n }\n };\n}\n\u5728\u672c\u7ae0\u7684\u524d\u4e24\u8282\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u8fd9\u4e24\u79cd\u57fa\u7840\u4e14\u91cd\u8981\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5b83\u4eec\u5206\u522b\u4ee3\u8868\u4e86\u201c\u8fde\u7eed\u5b58\u50a8\u201d\u548c\u201c\u5206\u6563\u5b58\u50a8\u201d\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u3002
\u5b9e\u9645\u4e0a\uff0c\u7269\u7406\u7ed3\u6784\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u51b3\u5b9a\u4e86\u7a0b\u5e8f\u5bf9\u5185\u5b58\u548c\u7f13\u5b58\u7684\u4f7f\u7528\u6548\u7387\uff0c\u8fdb\u800c\u5f71\u54cd\u7b97\u6cd5\u7a0b\u5e8f\u7684\u6574\u4f53\u6027\u80fd\u3002
"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","title":"4.4.1 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u8bbe\u5907","text":"\u8ba1\u7b97\u673a\u4e2d\u5305\u62ec\u4e09\u79cd\u7c7b\u578b\u7684\u5b58\u50a8\u8bbe\u5907\uff1a\u300c\u786c\u76d8 hard disk\u300d\u3001\u300c\u5185\u5b58 random-access memory, RAM\u300d\u3001\u300c\u7f13\u5b58 cache memory\u300d\u3002\u8868 4-2 \u5c55\u793a\u4e86\u5b83\u4eec\u5728\u8ba1\u7b97\u673a\u7cfb\u7edf\u4e2d\u7684\u4e0d\u540c\u89d2\u8272\u548c\u6027\u80fd\u7279\u70b9\u3002
\u8868 4-2 \u00a0 \u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u8bbe\u5907
\u786c\u76d8 \u5185\u5b58 \u7f13\u5b58 \u7528\u9014 \u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5305\u62ec\u64cd\u4f5c\u7cfb\u7edf\u3001\u7a0b\u5e8f\u3001\u6587\u4ef6\u7b49 \u4e34\u65f6\u5b58\u50a8\u5f53\u524d\u8fd0\u884c\u7684\u7a0b\u5e8f\u548c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e \u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u51cf\u5c11 CPU \u8bbf\u95ee\u5185\u5b58\u7684\u6b21\u6570 \u6613\u5931\u6027 \u65ad\u7535\u540e\u6570\u636e\u4e0d\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u5bb9\u91cf \u8f83\u5927\uff0cTB \u7ea7\u522b \u8f83\u5c0f\uff0cGB \u7ea7\u522b \u975e\u5e38\u5c0f\uff0cMB \u7ea7\u522b \u901f\u5ea6 \u8f83\u6162\uff0c\u51e0\u767e\u5230\u51e0\u5343 MB/s \u8f83\u5feb\uff0c\u51e0\u5341 GB/s \u975e\u5e38\u5feb\uff0c\u51e0\u5341\u5230\u51e0\u767e GB/s \u4ef7\u683c \u8f83\u4fbf\u5b9c\uff0c\u51e0\u6bdb\u5230\u51e0\u5143 / GB \u8f83\u8d35\uff0c\u51e0\u5341\u5230\u51e0\u767e\u5143 / GB \u975e\u5e38\u8d35\uff0c\u968f CPU \u6253\u5305\u8ba1\u4ef7\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf\u60f3\u8c61\u4e3a\u56fe 4-9 \u6240\u793a\u7684\u91d1\u5b57\u5854\u7ed3\u6784\u3002\u8d8a\u9760\u8fd1\u91d1\u5b57\u5854\u9876\u7aef\u7684\u5b58\u50a8\u8bbe\u5907\u7684\u901f\u5ea6\u8d8a\u5feb\u3001\u5bb9\u91cf\u8d8a\u5c0f\u3001\u6210\u672c\u8d8a\u9ad8\u3002\u8fd9\u79cd\u591a\u5c42\u7ea7\u7684\u8bbe\u8ba1\u5e76\u975e\u5076\u7136\uff0c\u800c\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u548c\u5de5\u7a0b\u5e08\u4eec\u7ecf\u8fc7\u6df1\u601d\u719f\u8651\u7684\u7ed3\u679c\u3002
- \u786c\u76d8\u96be\u4ee5\u88ab\u5185\u5b58\u53d6\u4ee3\u3002\u9996\u5148\uff0c\u5185\u5b58\u4e2d\u7684\u6570\u636e\u5728\u65ad\u7535\u540e\u4f1a\u4e22\u5931\uff0c\u56e0\u6b64\u5b83\u4e0d\u9002\u5408\u957f\u671f\u5b58\u50a8\u6570\u636e\uff1b\u5176\u6b21\uff0c\u5185\u5b58\u7684\u6210\u672c\u662f\u786c\u76d8\u7684\u51e0\u5341\u500d\uff0c\u8fd9\u4f7f\u5f97\u5b83\u96be\u4ee5\u5728\u6d88\u8d39\u8005\u5e02\u573a\u666e\u53ca\u3002
- \u7f13\u5b58\u7684\u5927\u5bb9\u91cf\u548c\u9ad8\u901f\u5ea6\u96be\u4ee5\u517c\u5f97\u3002\u968f\u7740 L1\u3001L2\u3001L3 \u7f13\u5b58\u7684\u5bb9\u91cf\u9010\u6b65\u589e\u5927\uff0c\u5176\u7269\u7406\u5c3a\u5bf8\u4f1a\u53d8\u5927\uff0c\u4e0e CPU \u6838\u5fc3\u4e4b\u95f4\u7684\u7269\u7406\u8ddd\u79bb\u4f1a\u53d8\u8fdc\uff0c\u4ece\u800c\u5bfc\u81f4\u6570\u636e\u4f20\u8f93\u65f6\u95f4\u589e\u52a0\uff0c\u5143\u7d20\u8bbf\u95ee\u5ef6\u8fdf\u53d8\u9ad8\u3002\u5728\u5f53\u524d\u6280\u672f\u4e0b\uff0c\u591a\u5c42\u7ea7\u7684\u7f13\u5b58\u7ed3\u6784\u662f\u5bb9\u91cf\u3001\u901f\u5ea6\u548c\u6210\u672c\u4e4b\u95f4\u7684\u6700\u4f73\u5e73\u8861\u70b9\u3002
\u56fe 4-9 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf
Note
\u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u5c42\u6b21\u7ed3\u6784\u4f53\u73b0\u4e86\u901f\u5ea6\u3001\u5bb9\u91cf\u548c\u6210\u672c\u4e09\u8005\u4e4b\u95f4\u7684\u7cbe\u5999\u5e73\u8861\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u79cd\u6743\u8861\u666e\u904d\u5b58\u5728\u4e8e\u6240\u6709\u5de5\u4e1a\u9886\u57df\uff0c\u5b83\u8981\u6c42\u6211\u4eec\u5728\u4e0d\u540c\u7684\u4f18\u52bf\u548c\u9650\u5236\u4e4b\u95f4\u627e\u5230\u6700\u4f73\u5e73\u8861\u70b9\u3002
\u603b\u7684\u6765\u8bf4\uff0c\u786c\u76d8\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u5927\u91cf\u6570\u636e\uff0c\u5185\u5b58\u7528\u4e8e\u4e34\u65f6\u5b58\u50a8\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\uff0c\u800c\u7f13\u5b58\u5219\u7528\u4e8e\u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u4ee5\u63d0\u9ad8\u7a0b\u5e8f\u8fd0\u884c\u6548\u7387\u3002\u4e09\u8005\u5171\u540c\u534f\u4f5c\uff0c\u786e\u4fdd\u8ba1\u7b97\u673a\u7cfb\u7edf\u9ad8\u6548\u8fd0\u884c\u3002
\u5982\u56fe 4-10 \u6240\u793a\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4f1a\u4ece\u786c\u76d8\u4e2d\u88ab\u8bfb\u53d6\u5230\u5185\u5b58\u4e2d\uff0c\u4f9b CPU \u8ba1\u7b97\u4f7f\u7528\u3002\u7f13\u5b58\u53ef\u4ee5\u770b\u4f5c CPU \u7684\u4e00\u90e8\u5206\uff0c\u5b83\u901a\u8fc7\u667a\u80fd\u5730\u4ece\u5185\u5b58\u52a0\u8f7d\u6570\u636e\uff0c\u7ed9 CPU \u63d0\u4f9b\u9ad8\u901f\u7684\u6570\u636e\u8bfb\u53d6\uff0c\u4ece\u800c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\uff0c\u51cf\u5c11\u5bf9\u8f83\u6162\u7684\u5185\u5b58\u7684\u4f9d\u8d56\u3002
\u56fe 4-10 \u00a0 \u786c\u76d8\u3001\u5185\u5b58\u548c\u7f13\u5b58\u4e4b\u95f4\u7684\u6570\u636e\u6d41\u901a
"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","title":"4.4.2 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u5185\u5b58\u6548\u7387","text":"\u5728\u5185\u5b58\u7a7a\u95f4\u5229\u7528\u65b9\u9762\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5404\u81ea\u5177\u6709\u4f18\u52bf\u548c\u5c40\u9650\u6027\u3002
\u4e00\u65b9\u9762\uff0c\u5185\u5b58\u662f\u6709\u9650\u7684\uff0c\u4e14\u540c\u4e00\u5757\u5185\u5b58\u4e0d\u80fd\u88ab\u591a\u4e2a\u7a0b\u5e8f\u5171\u4eab\uff0c\u56e0\u6b64\u6211\u4eec\u5e0c\u671b\u6570\u636e\u7ed3\u6784\u80fd\u591f\u5c3d\u53ef\u80fd\u9ad8\u6548\u5730\u5229\u7528\u7a7a\u95f4\u3002\u6570\u7ec4\u7684\u5143\u7d20\u7d27\u5bc6\u6392\u5217\uff0c\u4e0d\u9700\u8981\u989d\u5916\u7684\u7a7a\u95f4\u6765\u5b58\u50a8\u94fe\u8868\u8282\u70b9\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u56e0\u6b64\u7a7a\u95f4\u6548\u7387\u66f4\u9ad8\u3002\u7136\u800c\uff0c\u6570\u7ec4\u9700\u8981\u4e00\u6b21\u6027\u5206\u914d\u8db3\u591f\u7684\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u8fd9\u53ef\u80fd\u5bfc\u81f4\u5185\u5b58\u6d6a\u8d39\uff0c\u6570\u7ec4\u6269\u5bb9\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6210\u672c\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u94fe\u8868\u4ee5\u201c\u8282\u70b9\u201d\u4e3a\u5355\u4f4d\u8fdb\u884c\u52a8\u6001\u5185\u5b58\u5206\u914d\u548c\u56de\u6536\uff0c\u63d0\u4f9b\u4e86\u66f4\u5927\u7684\u7075\u6d3b\u6027\u3002
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"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","title":"4.4.3 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u7f13\u5b58\u6548\u7387","text":"\u7f13\u5b58\u867d\u7136\u5728\u7a7a\u95f4\u5bb9\u91cf\u4e0a\u8fdc\u5c0f\u4e8e\u5185\u5b58\uff0c\u4f46\u5b83\u6bd4\u5185\u5b58\u5feb\u5f97\u591a\uff0c\u5728\u7a0b\u5e8f\u6267\u884c\u901f\u5ea6\u4e0a\u8d77\u7740\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\u3002\u7531\u4e8e\u7f13\u5b58\u7684\u5bb9\u91cf\u6709\u9650\uff0c\u53ea\u80fd\u5b58\u50a8\u4e00\u5c0f\u90e8\u5206\u9891\u7e41\u8bbf\u95ee\u7684\u6570\u636e\uff0c\u56e0\u6b64\u5f53 CPU \u5c1d\u8bd5\u8bbf\u95ee\u7684\u6570\u636e\u4e0d\u5728\u7f13\u5b58\u4e2d\u65f6\uff0c\u5c31\u4f1a\u53d1\u751f\u300c\u7f13\u5b58\u672a\u547d\u4e2d cache miss\u300d\uff0c\u6b64\u65f6 CPU \u4e0d\u5f97\u4e0d\u4ece\u901f\u5ea6\u8f83\u6162\u7684\u5185\u5b58\u4e2d\u52a0\u8f7d\u6240\u9700\u6570\u636e\u3002
\u663e\u7136\uff0c\u201c\u7f13\u5b58\u672a\u547d\u4e2d\u201d\u8d8a\u5c11\uff0cCPU \u8bfb\u5199\u6570\u636e\u7684\u6548\u7387\u5c31\u8d8a\u9ad8\uff0c\u7a0b\u5e8f\u6027\u80fd\u4e5f\u5c31\u8d8a\u597d\u3002\u6211\u4eec\u5c06 CPU \u4ece\u7f13\u5b58\u4e2d\u6210\u529f\u83b7\u53d6\u6570\u636e\u7684\u6bd4\u4f8b\u79f0\u4e3a\u300c\u7f13\u5b58\u547d\u4e2d\u7387 cache hit rate\u300d\uff0c\u8fd9\u4e2a\u6307\u6807\u901a\u5e38\u7528\u6765\u8861\u91cf\u7f13\u5b58\u6548\u7387\u3002
\u4e3a\u4e86\u5c3d\u53ef\u80fd\u8fbe\u5230\u66f4\u9ad8\u7684\u6548\u7387\uff0c\u7f13\u5b58\u4f1a\u91c7\u53d6\u4ee5\u4e0b\u6570\u636e\u52a0\u8f7d\u673a\u5236\u3002
- \u7f13\u5b58\u884c\uff1a\u7f13\u5b58\u4e0d\u662f\u5355\u4e2a\u5b57\u8282\u5730\u5b58\u50a8\u4e0e\u52a0\u8f7d\u6570\u636e\uff0c\u800c\u662f\u4ee5\u7f13\u5b58\u884c\u4e3a\u5355\u4f4d\u3002\u76f8\u6bd4\u4e8e\u5355\u4e2a\u5b57\u8282\u7684\u4f20\u8f93\uff0c\u7f13\u5b58\u884c\u7684\u4f20\u8f93\u5f62\u5f0f\u66f4\u52a0\u9ad8\u6548\u3002
- \u9884\u53d6\u673a\u5236\uff1a\u5904\u7406\u5668\u4f1a\u5c1d\u8bd5\u9884\u6d4b\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\uff08\u4f8b\u5982\u987a\u5e8f\u8bbf\u95ee\u3001\u56fa\u5b9a\u6b65\u957f\u8df3\u8dc3\u8bbf\u95ee\u7b49\uff09\uff0c\u5e76\u6839\u636e\u7279\u5b9a\u6a21\u5f0f\u5c06\u6570\u636e\u52a0\u8f7d\u81f3\u7f13\u5b58\u4e4b\u4e2d\uff0c\u4ece\u800c\u63d0\u5347\u547d\u4e2d\u7387\u3002
- \u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u9644\u8fd1\u7684\u6570\u636e\u53ef\u80fd\u8fd1\u671f\u4e5f\u4f1a\u88ab\u8bbf\u95ee\u3002\u56e0\u6b64\uff0c\u7f13\u5b58\u5728\u52a0\u8f7d\u67d0\u4e00\u6570\u636e\u65f6\uff0c\u4e5f\u4f1a\u52a0\u8f7d\u5176\u9644\u8fd1\u7684\u6570\u636e\uff0c\u4ee5\u63d0\u9ad8\u547d\u4e2d\u7387\u3002
- \u65f6\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u5728\u4e0d\u4e45\u7684\u5c06\u6765\u5f88\u53ef\u80fd\u518d\u6b21\u88ab\u8bbf\u95ee\u3002\u7f13\u5b58\u5229\u7528\u8fd9\u4e00\u539f\u7406\uff0c\u901a\u8fc7\u4fdd\u7559\u6700\u8fd1\u8bbf\u95ee\u8fc7\u7684\u6570\u636e\u6765\u63d0\u9ad8\u547d\u4e2d\u7387\u3002
\u5b9e\u9645\u4e0a\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5bf9\u7f13\u5b58\u7684\u5229\u7528\u6548\u7387\u662f\u4e0d\u540c\u7684\uff0c\u4e3b\u8981\u4f53\u73b0\u5728\u4ee5\u4e0b\u51e0\u4e2a\u65b9\u9762\u3002
- \u5360\u7528\u7a7a\u95f4\uff1a\u94fe\u8868\u5143\u7d20\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u7a7a\u95f4\u66f4\u591a\uff0c\u5bfc\u81f4\u7f13\u5b58\u4e2d\u5bb9\u7eb3\u7684\u6709\u6548\u6570\u636e\u91cf\u66f4\u5c11\u3002
- \u7f13\u5b58\u884c\uff1a\u94fe\u8868\u6570\u636e\u5206\u6563\u5728\u5185\u5b58\u5404\u5904\uff0c\u800c\u7f13\u5b58\u662f\u201c\u6309\u884c\u52a0\u8f7d\u201d\u7684\uff0c\u56e0\u6b64\u52a0\u8f7d\u5230\u65e0\u6548\u6570\u636e\u7684\u6bd4\u4f8b\u66f4\u9ad8\u3002
- \u9884\u53d6\u673a\u5236\uff1a\u6570\u7ec4\u6bd4\u94fe\u8868\u7684\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\u66f4\u5177\u201c\u53ef\u9884\u6d4b\u6027\u201d\uff0c\u5373\u7cfb\u7edf\u66f4\u5bb9\u6613\u731c\u51fa\u5373\u5c06\u88ab\u52a0\u8f7d\u7684\u6570\u636e\u3002
- \u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u6570\u7ec4\u88ab\u5b58\u50a8\u5728\u96c6\u4e2d\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u56e0\u6b64\u88ab\u52a0\u8f7d\u6570\u636e\u9644\u8fd1\u7684\u6570\u636e\u66f4\u6709\u53ef\u80fd\u5373\u5c06\u88ab\u8bbf\u95ee\u3002
\u603b\u4f53\u800c\u8a00\uff0c\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u5728\u64cd\u4f5c\u6548\u7387\u4e0a\u901a\u5e38\u4f18\u4e8e\u94fe\u8868\u3002\u8fd9\u4f7f\u5f97\u5728\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u5f80\u5f80\u66f4\u53d7\u6b22\u8fce\u3002
\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9ad8\u7f13\u5b58\u6548\u7387\u5e76\u4e0d\u610f\u5473\u7740\u6570\u7ec4\u5728\u6240\u6709\u60c5\u51b5\u4e0b\u90fd\u4f18\u4e8e\u94fe\u8868\u3002\u5b9e\u9645\u5e94\u7528\u4e2d\u9009\u62e9\u54ea\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u6765\u51b3\u5b9a\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5b9e\u73b0\u201c\u6808\u201d\u6570\u636e\u7ed3\u6784\uff08\u4e0b\u4e00\u7ae0\u4f1a\u8be6\u7ec6\u4ecb\u7ecd\uff09\uff0c\u4f46\u5b83\u4eec\u9002\u7528\u4e8e\u4e0d\u540c\u573a\u666f\u3002
- \u5728\u505a\u7b97\u6cd5\u9898\u65f6\uff0c\u6211\u4eec\u4f1a\u503e\u5411\u4e8e\u9009\u62e9\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\uff0c\u56e0\u4e3a\u5b83\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u64cd\u4f5c\u6548\u7387\u548c\u968f\u673a\u8bbf\u95ee\u7684\u80fd\u529b\uff0c\u4ee3\u4ef7\u4ec5\u662f\u9700\u8981\u9884\u5148\u4e3a\u6570\u7ec4\u5206\u914d\u4e00\u5b9a\u7684\u5185\u5b58\u7a7a\u95f4\u3002
- \u5982\u679c\u6570\u636e\u91cf\u975e\u5e38\u5927\u3001\u52a8\u6001\u6027\u5f88\u9ad8\u3001\u6808\u7684\u9884\u671f\u5927\u5c0f\u96be\u4ee5\u4f30\u8ba1\uff0c\u90a3\u4e48\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u66f4\u52a0\u5408\u9002\u3002\u94fe\u8868\u80fd\u591f\u5c06\u5927\u91cf\u6570\u636e\u5206\u6563\u5b58\u50a8\u4e8e\u5185\u5b58\u7684\u4e0d\u540c\u90e8\u5206\uff0c\u5e76\u4e14\u907f\u514d\u4e86\u6570\u7ec4\u6269\u5bb9\u4ea7\u751f\u7684\u989d\u5916\u5f00\u9500\u3002
- \u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5206\u522b\u4ee3\u8868\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u4e24\u79cd\u5b58\u50a8\u65b9\u5f0f\uff1a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u4e24\u8005\u7684\u7279\u70b9\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u6027\u3002
- \u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5360\u7528\u5185\u5b58\u8f83\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002
- \u94fe\u8868\u901a\u8fc7\u66f4\u6539\u5f15\u7528\uff08\u6307\u9488\uff09\u5b9e\u73b0\u9ad8\u6548\u7684\u8282\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\uff1b\u4f46\u8282\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u8f83\u591a\u3002
- \u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u5355\u5411\u94fe\u8868\u3001\u73af\u5f62\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\uff0c\u5b83\u4eec\u5206\u522b\u5177\u6709\u5404\u81ea\u7684\u5e94\u7528\u573a\u666f\u3002
- \u5217\u8868\u662f\u4e00\u79cd\u652f\u6301\u589e\u5220\u67e5\u6539\u7684\u5143\u7d20\u6709\u5e8f\u96c6\u5408\uff0c\u901a\u5e38\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\uff0c\u5176\u4fdd\u7559\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u540c\u65f6\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\u3002
- \u5217\u8868\u7684\u51fa\u73b0\u5927\u5e45\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
- \u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u6570\u7ec4\u53ef\u63d0\u4f9b\u66f4\u9ad8\u7684\u5185\u5b58\u7a7a\u95f4\u6548\u7387\uff0c\u800c\u94fe\u8868\u5219\u5728\u5185\u5b58\u4f7f\u7528\u4e0a\u66f4\u52a0\u7075\u6d3b\u3002
- \u7f13\u5b58\u901a\u8fc7\u7f13\u5b58\u884c\u3001\u9884\u53d6\u673a\u5236\u4ee5\u53ca\u7a7a\u95f4\u5c40\u90e8\u6027\u548c\u65f6\u95f4\u5c40\u90e8\u6027\u7b49\u6570\u636e\u52a0\u8f7d\u673a\u5236\uff0c\u4e3a CPU \u63d0\u4f9b\u5feb\u901f\u6570\u636e\u8bbf\u95ee\uff0c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\u3002
- \u7531\u4e8e\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u901a\u5e38\u6bd4\u94fe\u8868\u66f4\u9ad8\u6548\u3002\u5728\u9009\u62e9\u6570\u636e\u7ed3\u6784\u65f6\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u548c\u573a\u666f\u505a\u51fa\u6070\u5f53\u9009\u62e9\u3002
\u6570\u7ec4\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5b58\u50a8\u5728\u5806\u4e0a\uff0c\u5bf9\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u5426\u6709\u5f71\u54cd\uff1f
\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5806\u4e0a\u7684\u6570\u7ec4\u90fd\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u6570\u636e\u64cd\u4f5c\u6548\u7387\u57fa\u672c\u4e00\u81f4\u3002\u7136\u800c\uff0c\u6808\u548c\u5806\u5177\u6709\u5404\u81ea\u7684\u7279\u70b9\uff0c\u4ece\u800c\u5bfc\u81f4\u4ee5\u4e0b\u4e0d\u540c\u70b9\u3002
- \u5206\u914d\u548c\u91ca\u653e\u6548\u7387\uff1a\u6808\u662f\u4e00\u5757\u8f83\u5c0f\u7684\u5185\u5b58\uff0c\u5206\u914d\u7531\u7f16\u8bd1\u5668\u81ea\u52a8\u5b8c\u6210\uff1b\u800c\u5806\u5185\u5b58\u76f8\u5bf9\u66f4\u5927\uff0c\u53ef\u4ee5\u5728\u4ee3\u7801\u4e2d\u52a8\u6001\u5206\u914d\uff0c\u66f4\u5bb9\u6613\u788e\u7247\u5316\u3002\u56e0\u6b64\uff0c\u5806\u4e0a\u7684\u5206\u914d\u548c\u91ca\u653e\u64cd\u4f5c\u901a\u5e38\u6bd4\u6808\u4e0a\u7684\u6162\u3002
- \u5927\u5c0f\u9650\u5236\uff1a\u6808\u5185\u5b58\u76f8\u5bf9\u8f83\u5c0f\uff0c\u5806\u7684\u5927\u5c0f\u4e00\u822c\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\u3002\u56e0\u6b64\u5806\u66f4\u52a0\u9002\u5408\u5b58\u50a8\u5927\u578b\u6570\u7ec4\u3002
- \u7075\u6d3b\u6027\uff1a\u6808\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u9700\u8981\u5728\u7f16\u8bd1\u65f6\u786e\u5b9a\uff0c\u800c\u5806\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u53ef\u4ee5\u5728\u8fd0\u884c\u65f6\u52a8\u6001\u786e\u5b9a\u3002
\u4e3a\u4ec0\u4e48\u6570\u7ec4\u8981\u6c42\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u5374\u6ca1\u6709\u5f3a\u8c03\u540c\u7c7b\u578b\u5462\uff1f
\u94fe\u8868\u7531\u8282\u70b9\u7ec4\u6210\uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u5f15\u7528\uff08\u6307\u9488\uff09\u8fde\u63a5\uff0c\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5b58\u50a8\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f8b\u5982 int\u3001double\u3001string\u3001object \u7b49\u3002
\u76f8\u5bf9\u5730\uff0c\u6570\u7ec4\u5143\u7d20\u5219\u5fc5\u987b\u662f\u76f8\u540c\u7c7b\u578b\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u8ba1\u7b97\u504f\u79fb\u91cf\u6765\u83b7\u53d6\u5bf9\u5e94\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u540c\u65f6\u5305\u542b int \u548c long \u4e24\u79cd\u7c7b\u578b\uff0c\u5355\u4e2a\u5143\u7d20\u5206\u522b\u5360\u7528 4 bytes \u548c 8 bytes \uff0c\u6b64\u65f6\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5305\u542b\u4e86\u4e24\u79cd\u957f\u5ea6\u7684\u5143\u7d20\u3002
# \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\n\u5220\u9664\u8282\u70b9\u540e\uff0c\u662f\u5426\u9700\u8981\u628a
P.next\u8bbe\u4e3a \\(\\text{None}\\) \u5462\uff1f\u4e0d\u4fee\u6539
P.next\u4e5f\u53ef\u4ee5\u3002\u4ece\u8be5\u94fe\u8868\u7684\u89d2\u5ea6\u770b\uff0c\u4ece\u5934\u8282\u70b9\u904d\u5386\u5230\u5c3e\u8282\u70b9\u5df2\u7ecf\u4e0d\u4f1a\u9047\u5230P\u4e86\u3002\u8fd9\u610f\u5473\u7740\u8282\u70b9P\u5df2\u7ecf\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u4e86\uff0c\u6b64\u65f6\u8282\u70b9P\u6307\u5411\u54ea\u91cc\u90fd\u4e0d\u4f1a\u5bf9\u8be5\u94fe\u8868\u4ea7\u751f\u5f71\u54cd\u3002\u4ece\u5783\u573e\u56de\u6536\u7684\u89d2\u5ea6\u770b\uff0c\u5bf9\u4e8e Java\u3001Python\u3001Go \u7b49\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\u7684\u8bed\u8a00\u6765\u8bf4\uff0c\u8282\u70b9
P\u662f\u5426\u88ab\u56de\u6536\u53d6\u51b3\u4e8e\u662f\u5426\u4ecd\u5b58\u5728\u6307\u5411\u5b83\u7684\u5f15\u7528\uff0c\u800c\u4e0d\u662fP.next\u7684\u503c\u3002\u5728 C \u548c C++ \u7b49\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u624b\u52a8\u91ca\u653e\u8282\u70b9\u5185\u5b58\u3002\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u5220\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u67e5\u627e\u5143\u7d20\uff0c\u90a3\u4e3a\u4ec0\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f
\u5982\u679c\u662f\u5148\u67e5\u627e\u5143\u7d20\u3001\u518d\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u786e\u5b9e\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u94fe\u8868\u7684 \\(O(1)\\) \u589e\u5220\u7684\u4f18\u52bf\u53ef\u4ee5\u5728\u5176\u4ed6\u5e94\u7528\u4e0a\u5f97\u5230\u4f53\u73b0\u3002\u4f8b\u5982\uff0c\u53cc\u5411\u961f\u5217\u9002\u5408\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u7ef4\u62a4\u4e00\u4e2a\u6307\u9488\u53d8\u91cf\u59cb\u7ec8\u6307\u5411\u5934\u8282\u70b9\u3001\u5c3e\u8282\u70b9\uff0c\u6bcf\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002
\u56fe\u201c\u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f\u201d\u4e2d\uff0c\u6d45\u84dd\u8272\u7684\u5b58\u50a8\u8282\u70b9\u6307\u9488\u662f\u5360\u7528\u4e00\u5757\u5185\u5b58\u5730\u5740\u5417\uff1f\u8fd8\u662f\u548c\u8282\u70b9\u503c\u5404\u5360\u4e00\u534a\u5462\uff1f
\u8be5\u793a\u610f\u56fe\u53ea\u662f\u5b9a\u6027\u8868\u793a\uff0c\u5b9a\u91cf\u8868\u793a\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002
- \u4e0d\u540c\u7c7b\u578b\u7684\u8282\u70b9\u503c\u5360\u7528\u7684\u7a7a\u95f4\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 int\u3001long\u3001double \u548c\u5b9e\u4f8b\u5bf9\u8c61\u7b49\u3002
- \u6307\u9488\u53d8\u91cf\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u6839\u636e\u6240\u4f7f\u7528\u7684\u64cd\u4f5c\u7cfb\u7edf\u53ca\u7f16\u8bd1\u73af\u5883\u800c\u5b9a\uff0c\u5927\u591a\u4e3a 8 \u5b57\u8282\u6216 4 \u5b57\u8282\u3002
\u5728\u5217\u8868\u672b\u5c3e\u6dfb\u52a0\u5143\u7d20\u662f\u5426\u65f6\u65f6\u523b\u523b\u90fd\u4e3a \\(O(1)\\) \uff1f
\u5982\u679c\u6dfb\u52a0\u5143\u7d20\u65f6\u8d85\u51fa\u5217\u8868\u957f\u5ea6\uff0c\u5219\u9700\u8981\u5148\u6269\u5bb9\u5217\u8868\u518d\u6dfb\u52a0\u3002\u7cfb\u7edf\u4f1a\u7533\u8bf7\u4e00\u5757\u65b0\u7684\u5185\u5b58\uff0c\u5e76\u5c06\u539f\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u642c\u8fd0\u8fc7\u53bb\uff0c\u8fd9\u65f6\u5019\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4f1a\u662f \\(O(n)\\) \u3002
\u201c\u5217\u8868\u7684\u51fa\u73b0\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u201d\uff0c\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u662f\u6307\u989d\u5916\u589e\u52a0\u7684\u53d8\u91cf\u5982\u5bb9\u91cf\u3001\u957f\u5ea6\u3001\u6269\u5bb9\u500d\u6570\u6240\u5360\u7684\u5185\u5b58\u5417\uff1f
\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u4e3b\u8981\u6709\u4e24\u65b9\u9762\u542b\u4e49\uff1a\u4e00\u65b9\u9762\uff0c\u5217\u8868\u90fd\u4f1a\u8bbe\u5b9a\u4e00\u4e2a\u521d\u59cb\u957f\u5ea6\uff0c\u6211\u4eec\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u8fd9\u4e48\u591a\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u4e3a\u4e86\u9632\u6b62\u9891\u7e41\u6269\u5bb9\uff0c\u6269\u5bb9\u4e00\u822c\u4f1a\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u8fd9\u6837\u4e00\u6765\uff0c\u4e5f\u4f1a\u51fa\u73b0\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u4eec\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6ee1\u5b83\u4eec\u3002
\u5728 Python \u4e2d\u521d\u59cb\u5316
n = [1, 2, 3]\u540e\uff0c\u8fd9 3 \u4e2a\u5143\u7d20\u7684\u5730\u5740\u662f\u76f8\u8fde\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316m = [2, 1, 3]\u4f1a\u53d1\u73b0\u5b83\u4eec\u6bcf\u4e2a\u5143\u7d20\u7684 id \u5e76\u4e0d\u662f\u8fde\u7eed\u7684\uff0c\u800c\u662f\u5206\u522b\u8ddfn\u4e2d\u7684\u76f8\u540c\u3002\u8fd9\u4e9b\u5143\u7d20\u7684\u5730\u5740\u4e0d\u8fde\u7eed\uff0c\u90a3\u4e48m\u8fd8\u662f\u6570\u7ec4\u5417\uff1f\u5047\u5982\u628a\u5217\u8868\u5143\u7d20\u6362\u6210\u94fe\u8868\u8282\u70b9
n = [n1, n2, n3, n4, n5]\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u8fd9 5 \u4e2a\u8282\u70b9\u5bf9\u8c61\u4e5f\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u3002\u7136\u800c\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5217\u8868\u7d22\u5f15\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u8282\u70b9\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u8282\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5b58\u50a8\u7684\u662f\u8282\u70b9\u7684\u5f15\u7528\uff0c\u800c\u975e\u8282\u70b9\u672c\u8eab\u3002\u4e0e\u8bb8\u591a\u8bed\u8a00\u4e0d\u540c\uff0cPython \u4e2d\u7684\u6570\u5b57\u4e5f\u88ab\u5305\u88c5\u4e3a\u5bf9\u8c61\uff0c\u5217\u8868\u4e2d\u5b58\u50a8\u7684\u4e0d\u662f\u6570\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5bf9\u6570\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u4e24\u4e2a\u6570\u7ec4\u4e2d\u7684\u76f8\u540c\u6570\u5b57\u62e5\u6709\u540c\u4e00\u4e2a id \uff0c\u5e76\u4e14\u8fd9\u4e9b\u6570\u5b57\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002
C++ STL \u91cc\u9762\u7684
std::list\u5df2\u7ecf\u5b9e\u73b0\u4e86\u53cc\u5411\u94fe\u8868\uff0c\u4f46\u597d\u50cf\u4e00\u4e9b\u7b97\u6cd5\u4e66\u4e0a\u4e0d\u600e\u4e48\u76f4\u63a5\u4f7f\u7528\u5b83\uff0c\u662f\u4e0d\u662f\u56e0\u4e3a\u6709\u4ec0\u4e48\u5c40\u9650\u6027\u5462\uff1f\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u5f80\u5f80\u66f4\u9752\u7750\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u7b97\u6cd5\uff0c\u800c\u53ea\u5728\u5fc5\u8981\u65f6\u624d\u4f7f\u7528\u94fe\u8868\uff0c\u4e3b\u8981\u6709\u4e24\u4e2a\u539f\u56e0\u3002
- \u7a7a\u95f4\u5f00\u9500\uff1a\u7531\u4e8e\u6bcf\u4e2a\u5143\u7d20\u9700\u8981\u4e24\u4e2a\u989d\u5916\u7684\u6307\u9488\uff08\u4e00\u4e2a\u7528\u4e8e\u524d\u4e00\u4e2a\u5143\u7d20\uff0c\u4e00\u4e2a\u7528\u4e8e\u540e\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u6240\u4ee5
std::list\u901a\u5e38\u6bd4std::vector\u66f4\u5360\u7528\u7a7a\u95f4\u3002 - \u7f13\u5b58\u4e0d\u53cb\u597d\uff1a\u7531\u4e8e\u6570\u636e\u4e0d\u662f\u8fde\u7eed\u5b58\u653e\u7684\uff0c\u56e0\u6b64
std::list\u5bf9\u7f13\u5b58\u7684\u5229\u7528\u7387\u8f83\u4f4e\u3002\u4e00\u822c\u60c5\u51b5\u4e0b\uff0cstd::vector\u7684\u6027\u80fd\u4f1a\u66f4\u597d\u3002
\u53e6\u4e00\u65b9\u9762\uff0c\u5fc5\u8981\u4f7f\u7528\u94fe\u8868\u7684\u60c5\u51b5\u4e3b\u8981\u662f\u4e8c\u53c9\u6811\u548c\u56fe\u3002\u6808\u548c\u961f\u5217\u5f80\u5f80\u4f1a\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684
stack\u548cqueue\uff0c\u800c\u975e\u94fe\u8868\u3002\u521d\u59cb\u5316\u5217\u8868
res = [0] * self.size()\u64cd\u4f5c\uff0c\u4f1a\u5bfc\u81f4res\u7684\u6bcf\u4e2a\u5143\u7d20\u5f15\u7528\u76f8\u540c\u7684\u5730\u5740\u5417\uff1f\u4e0d\u4f1a\u3002\u4f46\u4e8c\u7ef4\u6570\u7ec4\u4f1a\u6709\u8fd9\u4e2a\u95ee\u9898\uff0c\u4f8b\u5982\u521d\u59cb\u5316\u4e8c\u7ef4\u5217\u8868
res = [[0] * self.size()]\uff0c\u5219\u591a\u6b21\u5f15\u7528\u4e86\u540c\u4e00\u4e2a\u5217\u8868[0]\u3002\u5728\u5220\u9664\u8282\u70b9\u4e2d\uff0c\u9700\u8981\u65ad\u5f00\u8be5\u8282\u70b9\u4e0e\u5176\u540e\u7ee7\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u6307\u5411\u5417\uff1f
\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\uff08\u505a\u9898\uff09\u7684\u89d2\u5ea6\u770b\uff0c\u4e0d\u65ad\u5f00\u6ca1\u6709\u5173\u7cfb\uff0c\u53ea\u8981\u4fdd\u8bc1\u7a0b\u5e8f\u7684\u903b\u8f91\u662f\u6b63\u786e\u7684\u5c31\u884c\u3002\u4ece\u6807\u51c6\u5e93\u7684\u89d2\u5ea6\u770b\uff0c\u65ad\u5f00\u66f4\u52a0\u5b89\u5168\u3001\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3002\u5982\u679c\u4e0d\u65ad\u5f00\uff0c\u5047\u8bbe\u88ab\u5220\u9664\u8282\u70b9\u672a\u88ab\u6b63\u5e38\u56de\u6536\uff0c\u90a3\u4e48\u5b83\u4f1a\u5f71\u54cd\u540e\u7ee7\u8282\u70b9\u7684\u5185\u5b58\u56de\u6536\u3002
"},{"location":"chapter_backtracking/","title":"\u7b2c 13 \u7ae0 \u00a0 \u56de\u6eaf","text":"Abstract
\u6211\u4eec\u5982\u540c\u8ff7\u5bab\u4e2d\u7684\u63a2\u7d22\u8005\uff0c\u5728\u524d\u8fdb\u7684\u9053\u8def\u4e0a\u53ef\u80fd\u4f1a\u9047\u5230\u56f0\u96be\u3002
\u56de\u6eaf\u7684\u529b\u91cf\u8ba9\u6211\u4eec\u80fd\u591f\u91cd\u65b0\u5f00\u59cb\uff0c\u4e0d\u65ad\u5c1d\u8bd5\uff0c\u6700\u7ec8\u627e\u5230\u901a\u5f80\u5149\u660e\u7684\u51fa\u53e3\u3002
"},{"location":"chapter_backtracking/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5
- 13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898
- 13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898
- 13.4 \u00a0 N \u7687\u540e\u95ee\u9898
- 13.5 \u00a0 \u5c0f\u7ed3
\u300c\u56de\u6eaf\u7b97\u6cd5 backtracking algorithm\u300d\u662f\u4e00\u79cd\u901a\u8fc7\u7a77\u4e3e\u6765\u89e3\u51b3\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4ece\u4e00\u4e2a\u521d\u59cb\u72b6\u6001\u51fa\u53d1\uff0c\u66b4\u529b\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5f53\u9047\u5230\u6b63\u786e\u7684\u89e3\u5219\u5c06\u5176\u8bb0\u5f55\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5c1d\u8bd5\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u90fd\u65e0\u6cd5\u627e\u5230\u89e3\u4e3a\u6b62\u3002
\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u6765\u904d\u5386\u89e3\u7a7a\u95f4\u3002\u5728\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5229\u7528\u524d\u5e8f\u904d\u5386\u6784\u9020\u4e00\u4e2a\u56de\u6eaf\u95ee\u9898\uff0c\u9010\u6b65\u4e86\u89e3\u56de\u6eaf\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002
\u4f8b\u9898\u4e00
\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\uff0c\u641c\u7d22\u5e76\u8bb0\u5f55\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u8282\u70b9\u5217\u8868\u3002
\u5bf9\u4e8e\u6b64\u9898\uff0c\u6211\u4eec\u524d\u5e8f\u904d\u5386\u8fd9\u68f5\u6811\uff0c\u5e76\u5224\u65ad\u5f53\u524d\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \uff0c\u82e5\u662f\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u7684\u503c\u52a0\u5165\u7ed3\u679c\u5217\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_i_compact.pyres\u4e4b\u4e2d\u3002\u76f8\u5173\u8fc7\u7a0b\u5b9e\u73b0\u5982\u56fe 13-1 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff1a
preorder_traversal_i_compact.cppdef pre_order(root: TreeNode):\n \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00\"\"\"\n if root is None:\n return\n if root.val == 7:\n # \u8bb0\u5f55\u89e3\n res.append(root)\n pre_order(root.left)\n pre_order(root.right)\n
preorder_traversal_i_compact.java/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n if (root == nullptr) {\n return;\n }\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n res.push_back(root);\n }\n preOrder(root->left);\n preOrder(root->right);\n}\n
preorder_traversal_i_compact.cs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\n if (root == null) {\n return;\n }\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(root);\n }\n preOrder(root.left);\n preOrder(root.right);\n}\n
preorder_traversal_i_compact.go/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid PreOrder(TreeNode? root) {\n if (root == null) {\n return;\n }\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.Add(root);\n }\n PreOrder(root.left);\n PreOrder(root.right);\n}\n
preorder_traversal_i_compact.swift/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n if root == nil {\n return\n }\n if (root.Val).(int) == 7 {\n // \u8bb0\u5f55\u89e3\n *res = append(*res, root)\n }\n preOrderI(root.Left, res)\n preOrderI(root.Right, res)\n}\n
preorder_traversal_i_compact.js/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n if root.val == 7 {\n // \u8bb0\u5f55\u89e3\n res.append(root)\n }\n preOrder(root: root.left)\n preOrder(root: root.right)\n}\n
preorder_traversal_i_compact.ts/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root, res) {\n if (root === null) {\n return;\n }\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push(root);\n }\n preOrder(root.left, res);\n preOrder(root.right, res);\n}\n
preorder_traversal_i_compact.dart/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n if (root === null) {\n return;\n }\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push(root);\n }\n preOrder(root.left, res);\n preOrder(root.right, res);\n}\n
preorder_traversal_i_compact.rs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode? root, List<TreeNode> res) {\n if (root == null) {\n return;\n }\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(root);\n }\n preOrder(root.left, res);\n preOrder(root.right, res);\n}\n
preorder_traversal_i_compact.c/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfn pre_order(res: &mut Vec<Rc<RefCell<TreeNode>>>, root: Option<Rc<RefCell<TreeNode>>>) {\n if root.is_none() {\n return;\n }\n if let Some(node) = root {\n if node.borrow().val == 7 {\n // \u8bb0\u5f55\u89e3\n res.push(node.clone());\n }\n pre_order(res, node.borrow().left.clone());\n pre_order(res, node.borrow().right.clone());\n }\n}\n
preorder_traversal_i_compact.zig/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n if (root == NULL) {\n return;\n }\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n res[resSize++] = root;\n }\n preOrder(root->left);\n preOrder(root->right);\n}\n[class]{}-[func]{preOrder}\n\u56fe 13-1 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u4e2d\u641c\u7d22\u8282\u70b9
"},{"location":"chapter_backtracking/backtracking_algorithm/#1311","title":"13.1.1 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000","text":"\u4e4b\u6240\u4ee5\u79f0\u4e4b\u4e3a\u56de\u6eaf\u7b97\u6cd5\uff0c\u662f\u56e0\u4e3a\u8be5\u7b97\u6cd5\u5728\u641c\u7d22\u89e3\u7a7a\u95f4\u65f6\u4f1a\u91c7\u7528\u201c\u5c1d\u8bd5\u201d\u4e0e\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u5f53\u7b97\u6cd5\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u9047\u5230\u67d0\u4e2a\u72b6\u6001\u65e0\u6cd5\u7ee7\u7eed\u524d\u8fdb\u6216\u65e0\u6cd5\u5f97\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u65f6\uff0c\u5b83\u4f1a\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u5c1d\u8bd5\u5176\u4ed6\u53ef\u80fd\u7684\u9009\u62e9\u3002
\u5bf9\u4e8e\u4f8b\u9898\u4e00\uff0c\u8bbf\u95ee\u6bcf\u4e2a\u8282\u70b9\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5c1d\u8bd5\u201d\uff0c\u800c\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u8fd4\u56de\u7236\u8282\u70b9\u7684
return\u5219\u8868\u793a\u201c\u56de\u9000\u201d\u3002\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u56de\u9000\u5e76\u4e0d\u4ec5\u4ec5\u5305\u62ec\u51fd\u6570\u8fd4\u56de\u3002\u4e3a\u89e3\u91ca\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u5bf9\u4f8b\u9898\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002
\u4f8b\u9898\u4e8c
\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\u3002
\u5728\u4f8b\u9898\u4e00\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u5217\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_ii_compact.pypath\u8bb0\u5f55\u8bbf\u95ee\u8fc7\u7684\u8282\u70b9\u8def\u5f84\u3002\u5f53\u8bbf\u95ee\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u590d\u5236path\u5e76\u6dfb\u52a0\u8fdb\u7ed3\u679c\u5217\u8868res\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0cres\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
preorder_traversal_ii_compact.cppdef pre_order(root: TreeNode):\n \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c\"\"\"\n if root is None:\n return\n # \u5c1d\u8bd5\n path.append(root)\n if root.val == 7:\n # \u8bb0\u5f55\u89e3\n res.append(list(path))\n pre_order(root.left)\n pre_order(root.right)\n # \u56de\u9000\n path.pop()\n
preorder_traversal_ii_compact.java/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n if (root == nullptr) {\n return;\n }\n // \u5c1d\u8bd5\n path.push_back(root);\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n res.push_back(path);\n }\n preOrder(root->left);\n preOrder(root->right);\n // \u56de\u9000\n path.pop_back();\n}\n
preorder_traversal_ii_compact.cs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\n if (root == null) {\n return;\n }\n // \u5c1d\u8bd5\n path.add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(new ArrayList<>(path));\n }\n preOrder(root.left);\n preOrder(root.right);\n // \u56de\u9000\n path.remove(path.size() - 1);\n}\n
preorder_traversal_ii_compact.go/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid PreOrder(TreeNode? root) {\n if (root == null) {\n return;\n }\n // \u5c1d\u8bd5\n path.Add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.Add(new List<TreeNode>(path));\n }\n PreOrder(root.left);\n PreOrder(root.right);\n // \u56de\u9000\n path.RemoveAt(path.Count - 1);\n}\n
preorder_traversal_ii_compact.swift/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n if root == nil {\n return\n }\n // \u5c1d\u8bd5\n *path = append(*path, root)\n if root.Val.(int) == 7 {\n // \u8bb0\u5f55\u89e3\n *res = append(*res, append([]*TreeNode{}, *path...))\n }\n preOrderII(root.Left, res, path)\n preOrderII(root.Right, res, path)\n // \u56de\u9000\n *path = (*path)[:len(*path)-1]\n}\n
preorder_traversal_ii_compact.js/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n // \u5c1d\u8bd5\n path.append(root)\n if root.val == 7 {\n // \u8bb0\u5f55\u89e3\n res.append(path)\n }\n preOrder(root: root.left)\n preOrder(root: root.right)\n // \u56de\u9000\n path.removeLast()\n}\n
preorder_traversal_ii_compact.ts/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(root, path, res) {\n if (root === null) {\n return;\n }\n // \u5c1d\u8bd5\n path.push(root);\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push([...path]);\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.pop();\n}\n
preorder_traversal_ii_compact.dart/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(\n root: TreeNode | null,\n path: TreeNode[],\n res: TreeNode[][]\n): void {\n if (root === null) {\n return;\n }\n // \u5c1d\u8bd5\n path.push(root);\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push([...path]);\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.pop();\n}\n
preorder_traversal_ii_compact.rs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(\n TreeNode? root,\n List<TreeNode> path,\n List<List<TreeNode>> res,\n) {\n if (root == null) {\n return;\n }\n\n // \u5c1d\u8bd5\n path.add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(List.from(path));\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.removeLast();\n}\n
preorder_traversal_ii_compact.c/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfn pre_order(res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>, path: &mut Vec<Rc<RefCell<TreeNode>>>, root: Option<Rc<RefCell<TreeNode>>>) {\n if root.is_none() {\n return;\n }\n if let Some(node) = root {\n // \u5c1d\u8bd5\n path.push(node.clone());\n if node.borrow().val == 7 {\n // \u8bb0\u5f55\u89e3\n res.push(path.clone());\n }\n pre_order(res, path, node.borrow().left.clone());\n pre_order(res, path, node.borrow().right.clone());\n // \u56de\u9000\n path.remove(path.len() - 1);\n }\n}\n
preorder_traversal_ii_compact.zig/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n if (root == NULL) {\n return;\n }\n // \u5c1d\u8bd5\n path[pathSize++] = root;\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n for (int i = 0; i < pathSize; ++i) {\n res[resSize][i] = path[i];\n }\n resSize++;\n }\n preOrder(root->left);\n preOrder(root->right);\n // \u56de\u9000\n pathSize--;\n}\n[class]{}-[func]{preOrder}\n\u5728\u6bcf\u6b21\u201c\u5c1d\u8bd5\u201d\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u5f53\u524d\u8282\u70b9\u6dfb\u52a0\u8fdb
path\u6765\u8bb0\u5f55\u8def\u5f84\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u4eec\u9700\u8981\u5c06\u8be5\u8282\u70b9\u4ecepath\u4e2d\u5f39\u51fa\uff0c\u4ee5\u6062\u590d\u672c\u6b21\u5c1d\u8bd5\u4e4b\u524d\u7684\u72b6\u6001\u3002\u89c2\u5bdf\u56fe 13-2 \u6240\u793a\u7684\u8fc7\u7a0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5c1d\u8bd5\u548c\u56de\u9000\u7406\u89e3\u4e3a\u201c\u524d\u8fdb\u201d\u4e0e\u201c\u64a4\u9500\u201d\uff0c\u4e24\u4e2a\u64cd\u4f5c\u4e92\u4e3a\u9006\u5411\u3002
<1><2><3><4><5><6><7><8><9><10><11>\u56fe 13-2 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000
"},{"location":"chapter_backtracking/backtracking_algorithm/#1312","title":"13.1.2 \u00a0 \u526a\u679d","text":"\u590d\u6742\u7684\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u4e00\u4e2a\u6216\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u7ea6\u675f\u6761\u4ef6\u901a\u5e38\u53ef\u7528\u4e8e\u201c\u526a\u679d\u201d\u3002
\u4f8b\u9898\u4e09
\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\uff0c\u5e76\u8981\u6c42\u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u3002
\u4e3a\u4e86\u6ee1\u8db3\u4ee5\u4e0a\u7ea6\u675f\u6761\u4ef6\uff0c\u6211\u4eec\u9700\u8981\u6dfb\u52a0\u526a\u679d\u64cd\u4f5c\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u82e5\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\uff0c\u5219\u63d0\u524d\u8fd4\u56de\uff0c\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_iii_compact.py
preorder_traversal_iii_compact.cppdef pre_order(root: TreeNode):\n \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09\"\"\"\n # \u526a\u679d\n if root is None or root.val == 3:\n return\n # \u5c1d\u8bd5\n path.append(root)\n if root.val == 7:\n # \u8bb0\u5f55\u89e3\n res.append(list(path))\n pre_order(root.left)\n pre_order(root.right)\n # \u56de\u9000\n path.pop()\n
preorder_traversal_iii_compact.java/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n // \u526a\u679d\n if (root == nullptr || root->val == 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.push_back(root);\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n res.push_back(path);\n }\n preOrder(root->left);\n preOrder(root->right);\n // \u56de\u9000\n path.pop_back();\n}\n
preorder_traversal_iii_compact.cs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n // \u526a\u679d\n if (root == null || root.val == 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(new ArrayList<>(path));\n }\n preOrder(root.left);\n preOrder(root.right);\n // \u56de\u9000\n path.remove(path.size() - 1);\n}\n
preorder_traversal_iii_compact.go/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid PreOrder(TreeNode? root) {\n // \u526a\u679d\n if (root == null || root.val == 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.Add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.Add(new List<TreeNode>(path));\n }\n PreOrder(root.left);\n PreOrder(root.right);\n // \u56de\u9000\n path.RemoveAt(path.Count - 1);\n}\n
preorder_traversal_iii_compact.swift/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n // \u526a\u679d\n if root == nil || root.Val == 3 {\n return\n }\n // \u5c1d\u8bd5\n *path = append(*path, root)\n if root.Val.(int) == 7 {\n // \u8bb0\u5f55\u89e3\n *res = append(*res, *path)\n }\n preOrderIII(root.Left, res, path)\n preOrderIII(root.Right, res, path)\n // \u56de\u9000\n *path = (*path)[:len(*path)-1]\n}\n
preorder_traversal_iii_compact.js/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrder(root: TreeNode?) {\n // \u526a\u679d\n guard let root = root, root.val != 3 else {\n return\n }\n // \u5c1d\u8bd5\n path.append(root)\n if root.val == 7 {\n // \u8bb0\u5f55\u89e3\n res.append(path)\n }\n preOrder(root: root.left)\n preOrder(root: root.right)\n // \u56de\u9000\n path.removeLast()\n}\n
preorder_traversal_iii_compact.ts/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(root, path, res) {\n // \u526a\u679d\n if (root === null || root.val === 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.push(root);\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push([...path]);\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.pop();\n}\n
preorder_traversal_iii_compact.dart/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(\n root: TreeNode | null,\n path: TreeNode[],\n res: TreeNode[][]\n): void {\n // \u526a\u679d\n if (root === null || root.val === 3) {\n return;\n }\n // \u5c1d\u8bd5\n path.push(root);\n if (root.val === 7) {\n // \u8bb0\u5f55\u89e3\n res.push([...path]);\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.pop();\n}\n
preorder_traversal_iii_compact.rs/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(\n TreeNode? root,\n List<TreeNode> path,\n List<List<TreeNode>> res,\n) {\n if (root == null || root.val == 3) {\n return;\n }\n\n // \u5c1d\u8bd5\n path.add(root);\n if (root.val == 7) {\n // \u8bb0\u5f55\u89e3\n res.add(List.from(path));\n }\n preOrder(root.left, path, res);\n preOrder(root.right, path, res);\n // \u56de\u9000\n path.removeLast();\n}\n
preorder_traversal_iii_compact.c/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfn pre_order(res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>, path: &mut Vec<Rc<RefCell<TreeNode>>>, root: Option<Rc<RefCell<TreeNode>>>) {\n // \u526a\u679d\n if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n return;\n }\n if let Some(node) = root {\n // \u5c1d\u8bd5\n path.push(node.clone());\n if node.borrow().val == 7 {\n // \u8bb0\u5f55\u89e3\n res.push(path.clone());\n }\n pre_order(res, path, node.borrow().left.clone());\n pre_order(res, path, node.borrow().right.clone());\n // \u56de\u9000\n path.remove(path.len() - 1);\n }\n}\n
preorder_traversal_iii_compact.zig/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n // \u526a\u679d\n if (root == NULL || root->val == 3) {\n return;\n }\n // \u5c1d\u8bd5\n path[pathSize++] = root;\n if (root->val == 7) {\n // \u8bb0\u5f55\u89e3\n for (int i = 0; i < pathSize; i++) {\n res[resSize][i] = path[i];\n }\n resSize++;\n }\n preOrder(root->left);\n preOrder(root->right);\n // \u56de\u9000\n pathSize--;\n}\n[class]{}-[func]{preOrder}\n\u201c\u526a\u679d\u201d\u662f\u4e00\u4e2a\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8bcd\u3002\u5982\u56fe 13-3 \u6240\u793a\uff0c\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u641c\u7d22\u5206\u652f\uff0c\u907f\u514d\u8bb8\u591a\u65e0\u610f\u4e49\u7684\u5c1d\u8bd5\uff0c\u4ece\u800c\u63d0\u9ad8\u4e86\u641c\u7d22\u6548\u7387\u3002
\u56fe 13-3 \u00a0 \u6839\u636e\u7ea6\u675f\u6761\u4ef6\u526a\u679d
"},{"location":"chapter_backtracking/backtracking_algorithm/#1313","title":"13.1.3 \u00a0 \u6846\u67b6\u4ee3\u7801","text":"\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c1d\u8bd5\u5c06\u56de\u6eaf\u7684\u201c\u5c1d\u8bd5\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u4f53\u6846\u67b6\u63d0\u70bc\u51fa\u6765\uff0c\u63d0\u5347\u4ee3\u7801\u7684\u901a\u7528\u6027\u3002
\u5728\u4ee5\u4e0b\u6846\u67b6\u4ee3\u7801\u4e2d\uff0c
PythonC++JavaC#GoSwiftJSTSDartRustCZigstate\u8868\u793a\u95ee\u9898\u7684\u5f53\u524d\u72b6\u6001\uff0cchoices\u8868\u793a\u5f53\u524d\u72b6\u6001\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9009\u62e9\uff1adef backtrack(state: State, choices: list[choice], res: list[state]):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\"\"\"\n # \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if is_solution(state):\n # \u8bb0\u5f55\u89e3\n record_solution(state, res)\n # \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices:\n # \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if is_valid(state, choice):\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n make_choice(state, choice)\n backtrack(state, choices, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undo_choice(state, choice)\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector<Choice *> &choices, vector<State *> &res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (Choice choice : choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice> choices, List<State> res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (Choice choice : choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid Backtrack(State state, List<Choice> choices, List<State> res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (IsSolution(state)) {\n // \u8bb0\u5f55\u89e3\n RecordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n foreach (Choice choice in choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (IsValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n MakeChoice(state, choice);\n Backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n UndoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if isSolution(state) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res)\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for _, choice := range choices {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if isValid(state, choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice)\n backtrack(state, choices, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice)\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if isSolution(state: state) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state: state, res: &res)\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if isValid(state: state, choice: choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state: &state, choice: choice)\n backtrack(state: &state, choices: choices, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state: &state, choice: choice)\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (let choice of choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (let choice of choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice>, List<State> res) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (Choice choice in choices) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfn backtrack(state: &mut State, choices: &Vec<Choice>, res: &mut Vec<State>) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if is_solution(state) {\n // \u8bb0\u5f55\u89e3\n record_solution(state, res);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if is_valid(state, choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n make_choice(state, choice);\n backtrack(state, choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undo_choice(state, choice);\n }\n }\n}\n/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res, numRes);\n // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < numChoices; i++) {\n // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, &choices[i])) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, &choices[i]);\n backtrack(state, choices, numChoices, res, numRes);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, &choices[i]);\n }\n }\n}\n\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u57fa\u4e8e\u6846\u67b6\u4ee3\u7801\u6765\u89e3\u51b3\u4f8b\u9898\u4e09\u3002\u72b6\u6001
PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_iii_template.pystate\u4e3a\u8282\u70b9\u904d\u5386\u8def\u5f84\uff0c\u9009\u62e9choices\u4e3a\u5f53\u524d\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\uff0c\u7ed3\u679cres\u662f\u8def\u5f84\u5217\u8868\uff1a
preorder_traversal_iii_template.cppdef is_solution(state: list[TreeNode]) -> bool:\n \"\"\"\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3\"\"\"\n return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n \"\"\"\u8bb0\u5f55\u89e3\"\"\"\n res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -> bool:\n \"\"\"\u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5\"\"\"\n return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n \"\"\"\u66f4\u65b0\u72b6\u6001\"\"\"\n state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n \"\"\"\u6062\u590d\u72b6\u6001\"\"\"\n state.pop()\n\ndef backtrack(\n state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09\"\"\"\n # \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if is_solution(state):\n # \u8bb0\u5f55\u89e3\n record_solution(state, res)\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices:\n # \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if is_valid(state, choice):\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n make_choice(state, choice)\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, [choice.left, choice.right], res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undo_choice(state, choice)\n
preorder_traversal_iii_template.java/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(vector<TreeNode *> &state) {\n return !state.empty() && state.back()->val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(vector<TreeNode *> &state, vector<vector<TreeNode *>> &res) {\n res.push_back(state);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector<TreeNode *> &state, TreeNode *choice) {\n return choice != nullptr && choice->val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(vector<TreeNode *> &state, TreeNode *choice) {\n state.push_back(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(vector<TreeNode *> &state, TreeNode *choice) {\n state.pop_back();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(vector<TreeNode *> &state, vector<TreeNode *> &choices, vector<vector<TreeNode *>> &res) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (TreeNode *choice : choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n vector<TreeNode *> nextChoices{choice->left, choice->right};\n backtrack(state, nextChoices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n
preorder_traversal_iii_template.cs/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nboolean isSolution(List<TreeNode> state) {\n return !state.isEmpty() && state.get(state.size() - 1).val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\n res.add(new ArrayList<>(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List<TreeNode> state, TreeNode choice) {\n return choice != null && choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List<TreeNode> state, TreeNode choice) {\n state.add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List<TreeNode> state, TreeNode choice) {\n state.remove(state.size() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (TreeNode choice : choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, Arrays.asList(choice.left, choice.right), res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n
preorder_traversal_iii_template.go/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool IsSolution(List<TreeNode> state) {\n return state.Count != 0 && state[^1].val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid RecordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\n res.Add(new List<TreeNode>(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool IsValid(List<TreeNode> state, TreeNode choice) {\n return choice != null && choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid MakeChoice(List<TreeNode> state, TreeNode choice) {\n state.Add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid UndoChoice(List<TreeNode> state, TreeNode choice) {\n state.RemoveAt(state.Count - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid Backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (IsSolution(state)) {\n // \u8bb0\u5f55\u89e3\n RecordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n foreach (TreeNode choice in choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (IsValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n MakeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, [choice.left!, choice.right!], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n UndoChoice(state, choice);\n }\n }\n}\n
preorder_traversal_iii_template.swift/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\n return len(*state) != 0 && (*state)[len(*state)-1].Val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n *res = append(*res, *state)\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n return choice != nil && choice.Val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n *state = append(*state, choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n *state = (*state)[:len(*state)-1]\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if isSolution(state) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res)\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for _, choice := range *choices {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if isValid(state, choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n temp := make([]*TreeNode, 0)\n temp = append(temp, choice.Left, choice.Right)\n backtrackIII(state, &temp, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice)\n }\n }\n}\n
preorder_traversal_iii_template.js/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state: [TreeNode]) -> Bool {\n !state.isEmpty && state.last!.val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n res.append(state)\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -> Bool {\n choice != nil && choice!.val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n state.append(choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n state.removeLast()\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if isSolution(state: state) {\n recordSolution(state: state, res: &res)\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if isValid(state: state, choice: choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state: &state, choice: choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, choices: [choice.left, choice.right].compactMap { $0 }, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state: &state, choice: choice)\n }\n }\n}\n
preorder_traversal_iii_template.ts/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state) {\n return state && state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state, res) {\n res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\n return choice !== null && choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state, choice) {\n state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state) {\n state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(state, choices, res) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (const choice of choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, [choice.left, choice.right], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state);\n }\n }\n}\n
preorder_traversal_iii_template.dart/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\n return state && state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n return choice !== null && choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state: TreeNode[]): void {\n state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(\n state: TreeNode[],\n choices: TreeNode[],\n res: TreeNode[][]\n): void {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (const choice of choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, [choice.left, choice.right], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state);\n }\n }\n}\n
preorder_traversal_iii_template.rs/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(List<TreeNode> state) {\n return state.isNotEmpty && state.last.val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\n res.add(List.from(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(List<TreeNode> state, TreeNode? choice) {\n return choice != null && choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List<TreeNode> state, TreeNode? choice) {\n state.add(choice!);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List<TreeNode> state, TreeNode? choice) {\n state.removeLast();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(\n List<TreeNode> state,\n List<TreeNode?> choices,\n List<List<TreeNode>> res,\n) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution(state)) {\n // \u8bb0\u5f55\u89e3\n recordSolution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (TreeNode? choice in choices) {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(state, choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(state, choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, [choice!.left, choice.right], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice(state, choice);\n }\n }\n}\n
preorder_traversal_iii_template.c/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfn is_solution(state: &mut Vec<Rc<RefCell<TreeNode>>>) -> bool {\n return !state.is_empty() && state.get(state.len() - 1).unwrap().borrow().val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfn record_solution(state: &mut Vec<Rc<RefCell<TreeNode>>>, res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>) {\n res.push(state.clone());\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfn is_valid(_: &mut Vec<Rc<RefCell<TreeNode>>>, choice: Rc<RefCell<TreeNode>>) -> bool {\n return choice.borrow().val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfn make_choice(state: &mut Vec<Rc<RefCell<TreeNode>>>, choice: Rc<RefCell<TreeNode>>) {\n state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfn undo_choice(state: &mut Vec<Rc<RefCell<TreeNode>>>, _: Rc<RefCell<TreeNode>>) {\n state.remove(state.len() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfn backtrack(state: &mut Vec<Rc<RefCell<TreeNode>>>, choices: &mut Vec<Rc<RefCell<TreeNode>>>, res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if is_solution(state) {\n // \u8bb0\u5f55\u89e3\n record_solution(state, res);\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if is_valid(state, choice.clone()) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n make_choice(state, choice.clone());\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, &mut vec![choice.borrow().left.clone().unwrap(), choice.borrow().right.clone().unwrap()], res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undo_choice(state, choice.clone());\n }\n }\n}\n
preorder_traversal_iii_template.zig/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(void) {\n return pathSize > 0 && path[pathSize - 1]->val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(void) {\n for (int i = 0; i < pathSize; i++) {\n res[resSize][i] = path[i];\n }\n resSize++;\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(TreeNode *choice) {\n return choice != NULL && choice->val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(TreeNode *choice) {\n path[pathSize++] = choice;\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(void) {\n pathSize--;\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(TreeNode *choices[2]) {\n // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n if (isSolution()) {\n // \u8bb0\u5f55\u89e3\n recordSolution();\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < 2; i++) {\n TreeNode *choice = choices[i];\n // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n if (isValid(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n makeChoice(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n TreeNode *nextChoices[2] = {choice->left, choice->right};\n backtrack(nextChoices);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n undoChoice();\n }\n }\n}\n[class]{}-[func]{isSolution}\n\n[class]{}-[func]{recordSolution}\n\n[class]{}-[func]{isValid}\n\n[class]{}-[func]{makeChoice}\n\n[class]{}-[func]{undoChoice}\n\n[class]{}-[func]{backtrack}\n\u6839\u636e\u9898\u610f\uff0c\u6211\u4eec\u5728\u627e\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u540e\u5e94\u8be5\u7ee7\u7eed\u641c\u7d22\uff0c\u56e0\u6b64\u9700\u8981\u5c06\u8bb0\u5f55\u89e3\u4e4b\u540e\u7684
return\u8bed\u53e5\u5220\u9664\u3002\u56fe 13-4 \u5bf9\u6bd4\u4e86\u4fdd\u7559\u6216\u5220\u9664return\u8bed\u53e5\u7684\u641c\u7d22\u8fc7\u7a0b\u3002\u56fe 13-4 \u00a0 \u4fdd\u7559\u4e0e\u5220\u9664 return \u7684\u641c\u7d22\u8fc7\u7a0b\u5bf9\u6bd4
\u76f8\u6bd4\u57fa\u4e8e\u524d\u5e8f\u904d\u5386\u7684\u4ee3\u7801\u5b9e\u73b0\uff0c\u57fa\u4e8e\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\u7684\u4ee3\u7801\u5b9e\u73b0\u867d\u7136\u663e\u5f97\u5570\u5506\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u56de\u6eaf\u95ee\u9898\u53ef\u4ee5\u5728\u8be5\u6846\u67b6\u4e0b\u89e3\u51b3\u3002\u6211\u4eec\u53ea\u9700\u6839\u636e\u5177\u4f53\u95ee\u9898\u6765\u5b9a\u4e49
"},{"location":"chapter_backtracking/backtracking_algorithm/#1314","title":"13.1.4 \u00a0 \u5e38\u7528\u672f\u8bed","text":"state\u548cchoices\uff0c\u5e76\u5b9e\u73b0\u6846\u67b6\u4e2d\u7684\u5404\u4e2a\u65b9\u6cd5\u5373\u53ef\u3002\u4e3a\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u7b97\u6cd5\u95ee\u9898\uff0c\u6211\u4eec\u603b\u7ed3\u4e00\u4e0b\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5e38\u7528\u672f\u8bed\u7684\u542b\u4e49\uff0c\u5e76\u5bf9\u7167\u4f8b\u9898\u4e09\u7ed9\u51fa\u5bf9\u5e94\u793a\u4f8b\uff0c\u5982\u8868 13-1 \u6240\u793a\u3002
\u8868 13-1 \u00a0 \u5e38\u89c1\u7684\u56de\u6eaf\u7b97\u6cd5\u672f\u8bed
\u540d\u8bcd \u5b9a\u4e49 \u4f8b\u9898\u4e09 \u89e3\uff08solution\uff09 \u89e3\u662f\u6ee1\u8db3\u95ee\u9898\u7279\u5b9a\u6761\u4ef6\u7684\u7b54\u6848\uff0c\u53ef\u80fd\u6709\u4e00\u4e2a\u6216\u591a\u4e2a \u6839\u8282\u70b9\u5230\u8282\u70b9 \\(7\\) \u7684\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u6240\u6709\u8def\u5f84 \u7ea6\u675f\u6761\u4ef6\uff08constraint\uff09 \u7ea6\u675f\u6761\u4ef6\u662f\u95ee\u9898\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u6761\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u526a\u679d \u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u8282\u70b9 \\(3\\) \u72b6\u6001\uff08state\uff09 \u72b6\u6001\u8868\u793a\u95ee\u9898\u5728\u67d0\u4e00\u65f6\u523b\u7684\u60c5\u51b5\uff0c\u5305\u62ec\u5df2\u7ecf\u505a\u51fa\u7684\u9009\u62e9 \u5f53\u524d\u5df2\u8bbf\u95ee\u7684\u8282\u70b9\u8def\u5f84\uff0c\u5373path\u8282\u70b9\u5217\u8868 \u5c1d\u8bd5\uff08attempt\uff09 \u5c1d\u8bd5\u662f\u6839\u636e\u53ef\u7528\u9009\u62e9\u6765\u63a2\u7d22\u89e3\u7a7a\u95f4\u7684\u8fc7\u7a0b\uff0c\u5305\u62ec\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\uff0c\u68c0\u67e5\u662f\u5426\u4e3a\u89e3 \u9012\u5f52\u8bbf\u95ee\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\uff0c\u5c06\u8282\u70b9\u6dfb\u52a0\u8fdbpath\uff0c\u5224\u65ad\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \u56de\u9000\uff08backtracking\uff09 \u56de\u9000\u6307\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u72b6\u6001\u65f6\uff0c\u64a4\u9500\u524d\u9762\u505a\u51fa\u7684\u9009\u62e9\uff0c\u56de\u5230\u4e0a\u4e00\u4e2a\u72b6\u6001 \u5f53\u8d8a\u8fc7\u53f6\u8282\u70b9\u3001\u7ed3\u675f\u8282\u70b9\u8bbf\u95ee\u3001\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\u7ec8\u6b62\u641c\u7d22\uff0c\u51fd\u6570\u8fd4\u56de \u526a\u679d\uff08pruning\uff09 \u526a\u679d\u662f\u6839\u636e\u95ee\u9898\u7279\u6027\u548c\u7ea6\u675f\u6761\u4ef6\u907f\u514d\u65e0\u610f\u4e49\u7684\u641c\u7d22\u8def\u5f84\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u7d22\u6548\u7387 \u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22Tip
\u95ee\u9898\u3001\u89e3\u3001\u72b6\u6001\u7b49\u6982\u5ff5\u662f\u901a\u7528\u7684\uff0c\u5728\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u4e2d\u90fd\u6709\u6d89\u53ca\u3002
"},{"location":"chapter_backtracking/backtracking_algorithm/#1315","title":"13.1.5 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u7b97\u6cd5\uff0c\u5b83\u5c1d\u8bd5\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\u76f4\u5230\u627e\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u4f18\u70b9\u5728\u4e8e\u80fd\u591f\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002
\u7136\u800c\uff0c\u5728\u5904\u7406\u5927\u89c4\u6a21\u6216\u8005\u590d\u6742\u95ee\u9898\u65f6\uff0c\u56de\u6eaf\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u53ef\u80fd\u96be\u4ee5\u63a5\u53d7\u3002
- \u65f6\u95f4\uff1a\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u9700\u8981\u904d\u5386\u72b6\u6001\u7a7a\u95f4\u7684\u6240\u6709\u53ef\u80fd\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u6307\u6570\u9636\u6216\u9636\u4e58\u9636\u3002
- \u7a7a\u95f4\uff1a\u5728\u9012\u5f52\u8c03\u7528\u4e2d\u9700\u8981\u4fdd\u5b58\u5f53\u524d\u7684\u72b6\u6001\uff08\u4f8b\u5982\u8def\u5f84\u3001\u7528\u4e8e\u526a\u679d\u7684\u8f85\u52a9\u53d8\u91cf\u7b49\uff09\uff0c\u5f53\u6df1\u5ea6\u5f88\u5927\u65f6\uff0c\u7a7a\u95f4\u9700\u6c42\u53ef\u80fd\u4f1a\u53d8\u5f97\u5f88\u5927\u3002
\u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u7684\u6700\u4f73\u89e3\u51b3\u65b9\u6848\u3002\u5bf9\u4e8e\u8fd9\u4e9b\u95ee\u9898\uff0c\u7531\u4e8e\u65e0\u6cd5\u9884\u6d4b\u54ea\u4e9b\u9009\u62e9\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u8fdb\u884c\u904d\u5386\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5173\u952e\u662f\u5982\u4f55\u4f18\u5316\u6548\u7387\uff0c\u5e38\u89c1\u7684\u6548\u7387\u4f18\u5316\u65b9\u6cd5\u6709\u4e24\u79cd\u3002
- \u526a\u679d\uff1a\u907f\u514d\u641c\u7d22\u90a3\u4e9b\u80af\u5b9a\u4e0d\u4f1a\u4ea7\u751f\u89e3\u7684\u8def\u5f84\uff0c\u4ece\u800c\u8282\u7701\u65f6\u95f4\u548c\u7a7a\u95f4\u3002
- \u542f\u53d1\u5f0f\u641c\u7d22\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8ba1\u503c\uff0c\u4ece\u800c\u4f18\u5148\u641c\u7d22\u6700\u6709\u53ef\u80fd\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\u3002
\u56de\u6eaf\u7b97\u6cd5\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u641c\u7d22\u95ee\u9898\u3001\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u548c\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u3002
\u641c\u7d22\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u89e3\u51b3\u65b9\u6848\u3002
- \u5168\u6392\u5217\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7ec4\u5408\u3002
- \u5b50\u96c6\u548c\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\u548c\u4e00\u4e2a\u76ee\u6807\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u4e3a\u76ee\u6807\u548c\u7684\u5b50\u96c6\u3002
- \u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5706\u76d8\uff0c\u8981\u6c42\u5c06\u6240\u6709\u5706\u76d8\u4ece\u4e00\u6839\u67f1\u5b50\u79fb\u52a8\u5230\u53e6\u4e00\u6839\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\uff0c\u4e14\u4e0d\u80fd\u5c06\u5927\u5706\u76d8\u653e\u5728\u5c0f\u5706\u76d8\u4e0a\u3002
\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u6240\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u89e3\u3002
- \\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e \\(n\\) \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u4e92\u4e0d\u653b\u51fb\u3002
- \u6570\u72ec\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7f51\u683c\u4e2d\u586b\u5165\u6570\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u4e2a \\(3 \\times 3\\) \u5b50\u7f51\u683c\u4e2d\u7684\u6570\u5b57\u4e0d\u91cd\u590d\u3002
- \u56fe\u7740\u8272\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u7528\u6700\u5c11\u7684\u989c\u8272\u7ed9\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u7740\u8272\uff0c\u4f7f\u5f97\u76f8\u90bb\u9876\u70b9\u989c\u8272\u4e0d\u540c\u3002
\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u4e00\u4e2a\u7ec4\u5408\u7a7a\u95f4\u4e2d\u627e\u5230\u6ee1\u8db3\u67d0\u4e9b\u6761\u4ef6\u7684\u6700\u4f18\u89e3\u3002
- 0-1 \u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u80cc\u5305\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u4ef7\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5185\uff0c\u9009\u62e9\u7269\u54c1\u4f7f\u5f97\u603b\u4ef7\u503c\u6700\u5927\u3002
- \u65c5\u884c\u5546\u95ee\u9898\uff1a\u5728\u4e00\u4e2a\u56fe\u4e2d\uff0c\u4ece\u4e00\u4e2a\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u6240\u6709\u5176\u4ed6\u70b9\u6070\u597d\u4e00\u6b21\u540e\u8fd4\u56de\u8d77\u70b9\uff0c\u6c42\u6700\u77ed\u8def\u5f84\u3002
- \u6700\u5927\u56e2\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u56fe\uff0c\u5373\u5b50\u56fe\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u90fd\u6709\u8fb9\u76f8\u8fde\u3002
\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff0c\u56de\u6eaf\u4e0d\u662f\u6700\u4f18\u89e3\u51b3\u65b9\u6848\u3002
- 0-1 \u80cc\u5305\u95ee\u9898\u901a\u5e38\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\uff0c\u4ee5\u8fbe\u5230\u66f4\u9ad8\u7684\u65f6\u95f4\u6548\u7387\u3002
- \u65c5\u884c\u5546\u662f\u4e00\u4e2a\u8457\u540d\u7684 NP-Hard \u95ee\u9898\uff0c\u5e38\u7528\u89e3\u6cd5\u6709\u9057\u4f20\u7b97\u6cd5\u548c\u8681\u7fa4\u7b97\u6cd5\u7b49\u3002
- \u6700\u5927\u56e2\u95ee\u9898\u662f\u56fe\u8bba\u4e2d\u7684\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\uff0c\u53ef\u7528\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u542f\u53d1\u5f0f\u7b97\u6cd5\u6765\u89e3\u51b3\u3002
Question
\u6839\u636e\u56fd\u9645\u8c61\u68cb\u7684\u89c4\u5219\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u51fb\u4e0e\u540c\u5904\u4e00\u884c\u3001\u4e00\u5217\u6216\u4e00\u6761\u659c\u7ebf\u4e0a\u7684\u68cb\u5b50\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u7687\u540e\u548c\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5bfb\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u95f4\u65e0\u6cd5\u76f8\u4e92\u653b\u51fb\u7684\u6446\u653e\u65b9\u6848\u3002
\u5982\u56fe 13-15 \u6240\u793a\uff0c\u5f53 \\(n = 4\\) \u65f6\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u4e24\u4e2a\u89e3\u3002\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\u5171\u6709 \\(n^2\\) \u4e2a\u683c\u5b50\uff0c\u7ed9\u51fa\u4e86\u6240\u6709\u7684\u9009\u62e9
choices\u3002\u5728\u9010\u4e2a\u653e\u7f6e\u7687\u540e\u7684\u8fc7\u7a0b\u4e2d\uff0c\u68cb\u76d8\u72b6\u6001\u5728\u4e0d\u65ad\u5730\u53d8\u5316\uff0c\u6bcf\u4e2a\u65f6\u523b\u7684\u68cb\u76d8\u5c31\u662f\u72b6\u6001state\u3002\u56fe 13-15 \u00a0 4 \u7687\u540e\u95ee\u9898\u7684\u89e3
\u56fe 13-16 \u5c55\u793a\u4e86\u672c\u9898\u7684\u4e09\u4e2a\u7ea6\u675f\u6761\u4ef6\uff1a\u591a\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u3001\u540c\u4e00\u5bf9\u89d2\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u89d2\u7ebf\u5206\u4e3a\u4e3b\u5bf9\u89d2\u7ebf
\\\u548c\u6b21\u5bf9\u89d2\u7ebf/\u4e24\u79cd\u3002\u56fe 13-16 \u00a0 n \u7687\u540e\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6
"},{"location":"chapter_backtracking/n_queens_problem/#1","title":"1. \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565","text":"\u7687\u540e\u7684\u6570\u91cf\u548c\u68cb\u76d8\u7684\u884c\u6570\u90fd\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u6211\u4eec\u5bb9\u6613\u5f97\u5230\u4e00\u4e2a\u63a8\u8bba\uff1a\u68cb\u76d8\u6bcf\u884c\u90fd\u5141\u8bb8\u4e14\u53ea\u5141\u8bb8\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002
\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\uff0c\u76f4\u81f3\u6700\u540e\u4e00\u884c\u7ed3\u675f\u3002
\u56fe 13-17 \u6240\u793a\u4e3a \\(4\\) \u7687\u540e\u95ee\u9898\u7684\u9010\u884c\u653e\u7f6e\u8fc7\u7a0b\u3002\u53d7\u753b\u5e45\u9650\u5236\uff0c\u56fe 13-17 \u4ec5\u5c55\u5f00\u4e86\u7b2c\u4e00\u884c\u7684\u5176\u4e2d\u4e00\u4e2a\u641c\u7d22\u5206\u652f\uff0c\u5e76\u4e14\u5c06\u4e0d\u6ee1\u8db3\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u65b9\u6848\u90fd\u8fdb\u884c\u4e86\u526a\u679d\u3002
\u56fe 13-17 \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565
\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9010\u884c\u653e\u7f6e\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73b0\u591a\u4e2a\u7687\u540e\u7684\u6240\u6709\u641c\u7d22\u5206\u652f\u3002
"},{"location":"chapter_backtracking/n_queens_problem/#2","title":"2. \u00a0 \u5217\u4e0e\u5bf9\u89d2\u7ebf\u526a\u679d","text":"\u4e3a\u4e86\u6ee1\u8db3\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5e03\u5c14\u578b\u6570\u7ec4
cols\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u51b3\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u4eec\u901a\u8fc7cols\u5c06\u5df2\u6709\u7687\u540e\u7684\u5217\u8fdb\u884c\u526a\u679d\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u52a8\u6001\u66f4\u65b0cols\u7684\u72b6\u6001\u3002\u90a3\u4e48\uff0c\u5982\u4f55\u5904\u7406\u5bf9\u89d2\u7ebf\u7ea6\u675f\u5462\uff1f\u8bbe\u68cb\u76d8\u4e2d\u67d0\u4e2a\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\((row, col)\\) \uff0c\u9009\u5b9a\u77e9\u9635\u4e2d\u7684\u67d0\u6761\u4e3b\u5bf9\u89d2\u7ebf\uff0c\u6211\u4eec\u53d1\u73b0\u8be5\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u51cf\u5217\u7d22\u5f15\u90fd\u76f8\u7b49\uff0c\u5373\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684 \\(row - col\\) \u4e3a\u6052\u5b9a\u503c\u3002
\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u4e24\u4e2a\u683c\u5b50\u6ee1\u8db3 \\(row_1 - col_1 = row_2 - col_2\\) \uff0c\u5219\u5b83\u4eec\u4e00\u5b9a\u5904\u5728\u540c\u4e00\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u3002\u5229\u7528\u8be5\u89c4\u5f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u56fe 13-18 \u6240\u793a\u7684\u6570\u7ec4
diags1\u8bb0\u5f55\u6bcf\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002\u540c\u7406\uff0c\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 \\(row + col\\) \u662f\u6052\u5b9a\u503c\u3002\u6211\u4eec\u540c\u6837\u4e5f\u53ef\u4ee5\u501f\u52a9\u6570\u7ec4
diags2\u6765\u5904\u7406\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u56fe 13-18 \u00a0 \u5904\u7406\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f
"},{"location":"chapter_backtracking/n_queens_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u8bf7\u6ce8\u610f\uff0c\\(n\\) \u7ef4\u65b9\u9635\u4e2d \\(row - col\\) \u7684\u8303\u56f4\u662f \\([-n + 1, n - 1]\\) \uff0c\\(row + col\\) \u7684\u8303\u56f4\u662f \\([0, 2n - 2]\\) \uff0c\u6240\u4ee5\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\u7684\u6570\u91cf\u90fd\u4e3a \\(2n - 1\\) \uff0c\u5373\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig n_queens.pydiags1\u548cdiags2\u7684\u957f\u5ea6\u90fd\u4e3a \\(2n - 1\\) \u3002
n_queens.cppdef backtrack(\n row: int,\n n: int,\n state: list[list[str]],\n res: list[list[list[str]]],\n cols: list[bool],\n diags1: list[bool],\n diags2: list[bool],\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e\"\"\"\n # \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n:\n res.append([list(row) for row in state])\n return\n # \u904d\u5386\u6240\u6709\u5217\n for col in range(n):\n # \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n diag1 = row - col + n - 1\n diag2 = row + col\n # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n # \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\"\n cols[col] = diags1[diag1] = diags2[diag2] = True\n # \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2)\n # \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\"\n cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -> list[list[list[str]]]:\n \"\"\"\u6c42\u89e3 N \u7687\u540e\"\"\"\n # \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n state = [[\"#\" for _ in range(n)] for _ in range(n)]\n cols = [False] * n # \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n diags1 = [False] * (2 * n - 1) # \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n diags2 = [False] * (2 * n - 1) # \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n res = []\n backtrack(0, n, state, res, cols, diags1, diags2)\n\n return res\n
n_queens.java/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, vector<vector<string>> &state, vector<vector<vector<string>>> &res, vector<bool> &cols,\n vector<bool> &diags1, vector<bool> &diags2) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\";\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\";\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nvector<vector<vector<string>>> nQueens(int n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n vector<vector<string>> state(n, vector<string>(n, \"#\"));\n vector<bool> cols(n, false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n vector<bool> diags1(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n vector<bool> diags2(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n vector<vector<vector<string>>> res;\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n\n return res;\n}\n
n_queens.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, List<List<String>> state, List<List<List<String>>> res,\n boolean[] cols, boolean[] diags1, boolean[] diags2) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n List<List<String>> copyState = new ArrayList<>();\n for (List<String> sRow : state) {\n copyState.add(new ArrayList<>(sRow));\n }\n res.add(copyState);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state.get(row).set(col, \"Q\");\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state.get(row).set(col, \"#\");\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<String>>> nQueens(int n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n List<List<String>> state = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n List<String> row = new ArrayList<>();\n for (int j = 0; j < n; j++) {\n row.add(\"#\");\n }\n state.add(row);\n }\n boolean[] cols = new boolean[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n boolean[] diags1 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n boolean[] diags2 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n List<List<List<String>>> res = new ArrayList<>();\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n\n return res;\n}\n
n_queens.go/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid Backtrack(int row, int n, List<List<string>> state, List<List<List<string>>> res,\n bool[] cols, bool[] diags1, bool[] diags2) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n List<List<string>> copyState = [];\n foreach (List<string> sRow in state) {\n copyState.Add(new List<string>(sRow));\n }\n res.Add(copyState);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\";\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\";\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<string>>> NQueens(int n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n List<List<string>> state = [];\n for (int i = 0; i < n; i++) {\n List<string> row = [];\n for (int j = 0; j < n; j++) {\n row.Add(\"#\");\n }\n state.Add(row);\n }\n bool[] cols = new bool[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n bool[] diags1 = new bool[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n bool[] diags2 = new bool[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n List<List<List<string>>> res = [];\n\n Backtrack(0, n, state, res, cols, diags1, diags2);\n\n return res;\n}\n
n_queens.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n {\n newState := make([][]string, len(*state))\n for i, _ := range newState {\n newState[i] = make([]string, len((*state)[0]))\n copy(newState[i], (*state)[i])\n\n }\n *res = append(*res, newState)\n }\n // \u904d\u5386\u6240\u6709\u5217\n for col := 0; col < n; col++ {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n diag1 := row - col + n - 1\n diag2 := row + col\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if !(*cols)[col] && !(*diags1)[diag1] && !(*diags2)[diag2] {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n (*state)[row][col] = \"Q\"\n (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row+1, n, state, res, cols, diags1, diags2)\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n (*state)[row][col] = \"#\"\n (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n }\n }\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n {\n newState := make([][]string, len(*state))\n for i, _ := range newState {\n newState[i] = make([]string, len((*state)[0]))\n copy(newState[i], (*state)[i])\n\n }\n *res = append(*res, newState)\n }\n // \u904d\u5386\u6240\u6709\u5217\n for col := 0; col < n; col++ {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n diag1 := row - col + n - 1\n diag2 := row + col\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if !(*cols)[col] && !(*diags1)[diag1] && !(*diags2)[diag2] {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n (*state)[row][col] = \"Q\"\n (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row+1, n, state, res, cols, diags1, diags2)\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n (*state)[row][col] = \"#\"\n (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n }\n }\n}\n\nfunc nQueens(n int) [][][]string {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n state := make([][]string, n)\n for i := 0; i < n; i++ {\n row := make([]string, n)\n for i := 0; i < n; i++ {\n row[i] = \"#\"\n }\n state[i] = row\n }\n // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n cols := make([]bool, n)\n diags1 := make([]bool, 2*n-1)\n diags2 := make([]bool, 2*n-1)\n res := make([][][]string, 0)\n backtrack(0, n, &state, &res, &cols, &diags1, &diags2)\n return res\n}\n
n_queens.js/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u5217\n for col in 0 ..< n {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n let diag1 = row - col + n - 1\n let diag2 = row + col\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if !cols[col] && !diags1[diag1] && !diags2[diag2] {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\"\n cols[col] = true\n diags1[diag1] = true\n diags2[diag2] = true\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row: row + 1, n: n, state: &state, res: &res, cols: &cols, diags1: &diags1, diags2: &diags2)\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\"\n cols[col] = false\n diags1[diag1] = false\n diags2[diag2] = false\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfunc nQueens(n: Int) -> [[[String]]] {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n var cols = Array(repeating: false, count: n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n var diags1 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n var diags2 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n var res: [[[String]]] = []\n\n backtrack(row: 0, n: n, state: &state, res: &res, cols: &cols, diags1: &diags1, diags2: &diags2)\n\n return res\n}\n
n_queens.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row === n) {\n res.push(state.map((row) => row.slice()));\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (let col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n const diag1 = row - col + n - 1;\n const diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = 'Q';\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = '#';\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n const state = Array.from({ length: n }, () => Array(n).fill('#'));\n const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n const res = [];\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n return res;\n}\n
n_queens.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(\n row: number,\n n: number,\n state: string[][],\n res: string[][][],\n cols: boolean[],\n diags1: boolean[],\n diags2: boolean[]\n): void {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row === n) {\n res.push(state.map((row) => row.slice()));\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (let col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n const diag1 = row - col + n - 1;\n const diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = 'Q';\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = '#';\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n const state = Array.from({ length: n }, () => Array(n).fill('#'));\n const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n const res: string[][][] = [];\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n return res;\n}\n
n_queens.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(\n int row,\n int n,\n List<List<String>> state,\n List<List<List<String>>> res,\n List<bool> cols,\n List<bool> diags1,\n List<bool> diags2,\n) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n List<List<String>> copyState = [];\n for (List<String> sRow in state) {\n copyState.add(List.from(sRow));\n }\n res.add(copyState);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = \"Q\";\n cols[col] = true;\n diags1[diag1] = true;\n diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = \"#\";\n cols[col] = false;\n diags1[diag1] = false;\n diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<String>>> nQueens(int n) {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n List<List<String>> state = List.generate(n, (index) => List.filled(n, \"#\"));\n List<bool> cols = List.filled(n, false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n List<bool> diags1 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n List<bool> diags2 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n List<List<List<String>>> res = [];\n\n backtrack(0, n, state, res, cols, diags1, diags2);\n\n return res;\n}\n
n_queens.c/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfn backtrack(row: usize, n: usize, state: &mut Vec<Vec<String>>, res: &mut Vec<Vec<Vec<String>>>,\n cols: &mut [bool], diags1: &mut [bool], diags2: &mut [bool]) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if row == n {\n let mut copy_state: Vec<Vec<String>> = Vec::new();\n for s_row in state.clone() {\n copy_state.push(s_row);\n }\n res.push(copy_state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for col in 0..n {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n let diag1 = row + n - 1 - col;\n let diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if !cols[col] && !diags1[diag1] && !diags2[diag2] {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state.get_mut(row).unwrap()[col] = \"Q\".into();\n (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state.get_mut(row).unwrap()[col] = \"#\".into();\n (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfn n_queens(n: usize) -> Vec<Vec<Vec<String>>> {\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n let mut state: Vec<Vec<String>> = Vec::new();\n for _ in 0..n {\n let mut row: Vec<String> = Vec::new();\n for _ in 0..n {\n row.push(\"#\".into());\n }\n state.push(row);\n }\n let mut cols = vec![false; n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n let mut diags1 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n let mut diags2 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n let mut res: Vec<Vec<Vec<String>>> = Vec::new();\n\n backtrack(0, n, &mut state, &mut res, &mut cols, &mut diags1, &mut diags2);\n\n res\n}\n
n_queens.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (row == n) {\n res[*resSize] = (char **)malloc(sizeof(char *) * n);\n for (int i = 0; i < n; ++i) {\n res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n strcpy(res[*resSize][i], state[i]);\n }\n (*resSize)++;\n return;\n }\n // \u904d\u5386\u6240\u6709\u5217\n for (int col = 0; col < n; col++) {\n // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n int diag1 = row - col + n - 1;\n int diag2 = row + col;\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n state[row][col] = 'Q';\n cols[col] = diags1[diag1] = diags2[diag2] = true;\n // \u653e\u7f6e\u4e0b\u4e00\u884c\n backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n state[row][col] = '#';\n cols[col] = diags1[diag1] = diags2[diag2] = false;\n }\n }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nchar ***nQueens(int n, int *returnSize) {\n char state[MAX_SIZE][MAX_SIZE];\n // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n for (int i = 0; i < n; ++i) {\n for (int j = 0; j < n; ++j) {\n state[i][j] = '#';\n }\n state[i][n] = '\\0';\n }\n bool cols[MAX_SIZE] = {false}; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n bool diags1[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n bool diags2[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n\n char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n *returnSize = 0;\n backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n return res;\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{nQueens}\n\u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u8651\u5217\u7ea6\u675f\uff0c\u5219\u4ece\u7b2c\u4e00\u884c\u5230\u6700\u540e\u4e00\u884c\u5206\u522b\u6709 \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u4e2a\u9009\u62e9\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!)\\) \u3002\u5b9e\u9645\u4e0a\uff0c\u6839\u636e\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u526a\u679d\u4e5f\u80fd\u591f\u5927\u5e45\u7f29\u5c0f\u641c\u7d22\u7a7a\u95f4\uff0c\u56e0\u800c\u641c\u7d22\u6548\u7387\u5f80\u5f80\u4f18\u4e8e\u4ee5\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u3002
\u6570\u7ec4
"},{"location":"chapter_backtracking/permutations_problem/","title":"13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898","text":"state\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff0c\u6570\u7ec4cols\u3001diags1\u548cdiags2\u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5168\u6392\u5217\u95ee\u9898\u662f\u56de\u6eaf\u7b97\u6cd5\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u7684\u5b9a\u4e49\u662f\u5728\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff08\u5982\u4e00\u4e2a\u6570\u7ec4\u6216\u5b57\u7b26\u4e32\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u627e\u51fa\u5176\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002
\u8868 13-2 \u5217\u4e3e\u4e86\u51e0\u4e2a\u793a\u4f8b\u6570\u636e\uff0c\u5305\u62ec\u8f93\u5165\u6570\u7ec4\u548c\u5bf9\u5e94\u7684\u6240\u6709\u6392\u5217\u3002
\u8868 13-2 \u00a0 \u5168\u6392\u5217\u793a\u4f8b
\u8f93\u5165\u6570\u7ec4 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1321","title":"13.2.1 \u00a0 \u65e0\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"Question
\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u5176\u4e2d\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002
\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\u3002\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u4eec\u5148\u9009\u62e9 \\(1\\) \uff0c\u518d\u9009\u62e9 \\(3\\) \uff0c\u6700\u540e\u9009\u62e9 \\(2\\) \uff0c\u5219\u83b7\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u9500\u4e00\u4e2a\u9009\u62e9\uff0c\u4e4b\u540e\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002
\u4ece\u56de\u6eaf\u4ee3\u7801\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9009\u96c6\u5408
choices\u662f\u8f93\u5165\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72b6\u6001state\u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9009\u62e9\u7684\u5143\u7d20\u3002\u8bf7\u6ce8\u610f\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u5141\u8bb8\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u56e0\u6b64state\u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u5e94\u8be5\u662f\u552f\u4e00\u7684\u3002\u5982\u56fe 13-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u641c\u7d22\u8fc7\u7a0b\u5c55\u5f00\u6210\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u5f53\u524d\u72b6\u6001
state\u3002\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u7ecf\u8fc7\u4e09\u8f6e\u9009\u62e9\u540e\u5230\u8fbe\u53f6\u8282\u70b9\uff0c\u6bcf\u4e2a\u53f6\u8282\u70b9\u90fd\u5bf9\u5e94\u4e00\u4e2a\u6392\u5217\u3002\u56fe 13-5 \u00a0 \u5168\u6392\u5217\u7684\u9012\u5f52\u6811
"},{"location":"chapter_backtracking/permutations_problem/#1","title":"1. \u00a0 \u91cd\u590d\u9009\u62e9\u526a\u679d","text":"\u4e3a\u4e86\u5b9e\u73b0\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u6211\u4eec\u8003\u8651\u5f15\u5165\u4e00\u4e2a\u5e03\u5c14\u578b\u6570\u7ec4
selected\uff0c\u5176\u4e2dselected[i]\u8868\u793achoices[i]\u662f\u5426\u5df2\u88ab\u9009\u62e9\uff0c\u5e76\u57fa\u4e8e\u5b83\u5b9e\u73b0\u4ee5\u4e0b\u526a\u679d\u64cd\u4f5c\u3002- \u5728\u505a\u51fa\u9009\u62e9
choice[i]\u540e\uff0c\u6211\u4eec\u5c31\u5c06selected[i]\u8d4b\u503c\u4e3a \\(\\text{True}\\) \uff0c\u4ee3\u8868\u5b83\u5df2\u88ab\u9009\u62e9\u3002 - \u904d\u5386\u9009\u62e9\u5217\u8868
choices\u65f6\uff0c\u8df3\u8fc7\u6240\u6709\u5df2\u88ab\u9009\u62e9\u7684\u8282\u70b9\uff0c\u5373\u526a\u679d\u3002
\u5982\u56fe 13-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u7b2c\u4e00\u8f6e\u9009\u62e9 1 \uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9 3 \uff0c\u7b2c\u4e09\u8f6e\u9009\u62e9 2 \uff0c\u5219\u9700\u8981\u5728\u7b2c\u4e8c\u8f6e\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f6e\u526a\u6389\u5143\u7d20 1 \u548c\u5143\u7d20 3 \u7684\u5206\u652f\u3002
\u56fe 13-6 \u00a0 \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b
\u89c2\u5bdf\u56fe 13-6 \u53d1\u73b0\uff0c\u8be5\u526a\u679d\u64cd\u4f5c\u5c06\u641c\u7d22\u7a7a\u95f4\u5927\u5c0f\u4ece \\(O(n^n)\\) \u51cf\u5c0f\u81f3 \\(O(n!)\\) \u3002
"},{"location":"chapter_backtracking/permutations_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u60f3\u6e05\u695a\u4ee5\u4e0a\u4fe1\u606f\u4e4b\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u4ee3\u7801\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u4e3a\u4e86\u7f29\u77ed\u6574\u4f53\u4ee3\u7801\uff0c\u6211\u4eec\u4e0d\u5355\u72ec\u5b9e\u73b0\u6846\u67b6\u4ee3\u7801\u4e2d\u7684\u5404\u4e2a\u51fd\u6570\uff0c\u800c\u662f\u5c06\u5b83\u4eec\u5c55\u5f00\u5728
PythonC++JavaC#GoSwiftJSTSDartRustCZig permutations_i.pybacktrack()\u51fd\u6570\u4e2d\uff1a
permutations_i.cppdef backtrack(\n state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if len(state) == len(choices):\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for i, choice in enumerate(choices):\n # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if not selected[i]:\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = True\n state.append(choice)\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = False\n state.pop()\n\ndef permutations_i(nums: list[int]) -> list[list[int]]:\n \"\"\"\u5168\u6392\u5217 I\"\"\"\n res = []\n backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n return res\n
permutations_i.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.size() == choices.size()) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.size(); i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.push_back(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop_back();\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nvector<vector<int>> permutationsI(vector<int> nums) {\n vector<int> state;\n vector<bool> selected(nums.size(), false);\n vector<vector<int>> res;\n backtrack(state, nums, selected, res);\n return res;\n}\n
permutations_i.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List<Integer> state, int[] choices, boolean[] selected, List<List<Integer>> res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.size() == choices.length) {\n res.add(new ArrayList<Integer>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.remove(state.size() - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nList<List<Integer>> permutationsI(int[] nums) {\n List<List<Integer>> res = new ArrayList<List<Integer>>();\n backtrack(new ArrayList<Integer>(), nums, new boolean[nums.length], res);\n return res;\n}\n
permutations_i.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid Backtrack(List<int> state, int[] choices, bool[] selected, List<List<int>> res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.Count == choices.Length) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.Length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.Add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.RemoveAt(state.Count - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nList<List<int>> PermutationsI(int[] nums) {\n List<List<int>> res = [];\n Backtrack([], nums, new bool[nums.Length], res);\n return res;\n}\n
permutations_i.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if len(*state) == len(*choices) {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i := 0; i < len(*choices); i++ {\n choice := (*choices)[i]\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if !(*selected)[i] {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n (*selected)[i] = true\n *state = append(*state, choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackI(state, choices, selected, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n (*selected)[i] = false\n *state = (*state)[:len(*state)-1]\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\n res := make([][]int, 0)\n state := make([]int, 0)\n selected := make([]bool, len(nums))\n backtrackI(&state, &nums, &selected, &res)\n return res\n}\n
permutations_i.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if state.count == choices.count {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (i, choice) in choices.enumerated() {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if !selected[i] {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true\n state.append(choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, choices: choices, selected: &selected, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false\n state.removeLast()\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -> [[Int]] {\n var state: [Int] = []\n var selected = Array(repeating: false, count: nums.count)\n var res: [[Int]] = []\n backtrack(state: &state, choices: nums, selected: &selected, res: &res)\n return res\n}\n
permutations_i.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length === choices.length) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n choices.forEach((choice, i) => {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop();\n }\n });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\n const res = [];\n backtrack([], nums, Array(nums.length).fill(false), res);\n return res;\n}\n
permutations_i.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\n state: number[],\n choices: number[],\n selected: boolean[],\n res: number[][]\n): void {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length === choices.length) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n choices.forEach((choice, i) => {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop();\n }\n });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\n const res: number[][] = [];\n backtrack([], nums, Array(nums.length).fill(false), res);\n return res;\n}\n
permutations_i.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(\n List<int> state,\n List<int> choices,\n List<bool> selected,\n List<List<int>> res,\n) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length == choices.length) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.removeLast();\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nList<List<int>> permutationsI(List<int> nums) {\n List<List<int>> res = [];\n backtrack([], nums, List.filled(nums.length, false), res);\n return res;\n}\n
permutations_i.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfn backtrack(mut state: Vec<i32>, choices: &[i32], selected: &mut [bool], res: &mut Vec<Vec<i32>>) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if state.len() == choices.len() {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i in 0..choices.len() {\n let choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if !selected[i] {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.remove(state.len() - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nfn permutations_i(nums: &mut [i32]) -> Vec<Vec<i32>> {\n let mut res = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n backtrack(Vec::new(), nums, &mut vec![false; nums.len()], &mut res);\n res\n}\n
permutations_i.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (stateSize == choicesSize) {\n res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n for (int i = 0; i < choicesSize; i++) {\n res[*resSize][i] = state[i];\n }\n (*resSize)++;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choicesSize; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n if (!selected[i]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n selected[i] = true;\n state[stateSize] = choice;\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n }\n }\n}\n\n/* \u5168\u6392\u5217 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n int *state = (int *)malloc(numsSize * sizeof(int));\n bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n for (int i = 0; i < numsSize; i++) {\n selected[i] = false;\n }\n int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n *returnSize = 0;\n\n backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n free(state);\n free(selected);\n\n return res;\n}\n
"},{"location":"chapter_backtracking/permutations_problem/#1322","title":"13.2.2 \u00a0 \u8003\u8651\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsI}\nQuestion
\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u590d\u7684\u6392\u5217\u3002
\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 1, 2]\\) \u3002\u4e3a\u4e86\u65b9\u4fbf\u533a\u5206\u4e24\u4e2a\u91cd\u590d\u5143\u7d20 \\(1\\) \uff0c\u6211\u4eec\u5c06\u7b2c\u4e8c\u4e2a \\(1\\) \u8bb0\u4e3a \\(\\hat{1}\\) \u3002
\u5982\u56fe 13-7 \u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u662f\u91cd\u590d\u7684\u3002
\u56fe 13-7 \u00a0 \u91cd\u590d\u6392\u5217
\u90a3\u4e48\u5982\u4f55\u53bb\u9664\u91cd\u590d\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u76f4\u63a5\u5bf9\u6392\u5217\u7ed3\u679c\u8fdb\u884c\u53bb\u91cd\u3002\u7136\u800c\u8fd9\u6837\u505a\u4e0d\u591f\u4f18\u96c5\uff0c\u56e0\u4e3a\u751f\u6210\u91cd\u590d\u6392\u5217\u7684\u641c\u7d22\u5206\u652f\u6ca1\u6709\u5fc5\u8981\uff0c\u5e94\u5f53\u63d0\u524d\u8bc6\u522b\u5e76\u526a\u679d\uff0c\u8fd9\u6837\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002
"},{"location":"chapter_backtracking/permutations_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"\u89c2\u5bdf\u56fe 13-8 \uff0c\u5728\u7b2c\u4e00\u8f6e\u4e2d\uff0c\u9009\u62e9 \\(1\\) \u6216\u9009\u62e9 \\(\\hat{1}\\) \u662f\u7b49\u4ef7\u7684\uff0c\u5728\u8fd9\u4e24\u4e2a\u9009\u62e9\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u590d\u7684\u3002\u56e0\u6b64\u5e94\u8be5\u628a \\(\\hat{1}\\) \u526a\u679d\u3002
\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(2\\) \u4e4b\u540e\uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u5e94\u5c06\u7b2c\u4e8c\u8f6e\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002
\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u5728\u67d0\u4e00\u8f6e\u9009\u62e9\u4e2d\uff0c\u4fdd\u8bc1\u591a\u4e2a\u76f8\u7b49\u7684\u5143\u7d20\u4ec5\u88ab\u9009\u62e9\u4e00\u6b21\u3002
\u56fe 13-8 \u00a0 \u91cd\u590d\u6392\u5217\u526a\u679d
"},{"location":"chapter_backtracking/permutations_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5728\u4e0a\u4e00\u9898\u7684\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u8003\u8651\u5728\u6bcf\u4e00\u8f6e\u9009\u62e9\u4e2d\u5f00\u542f\u4e00\u4e2a\u54c8\u5e0c\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig permutations_ii.pyduplicated\uff0c\u7528\u4e8e\u8bb0\u5f55\u8be5\u8f6e\u4e2d\u5df2\u7ecf\u5c1d\u8bd5\u8fc7\u7684\u5143\u7d20\uff0c\u5e76\u5c06\u91cd\u590d\u5143\u7d20\u526a\u679d\uff1a
permutations_ii.cppdef backtrack(\n state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if len(state) == len(choices):\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n duplicated = set[int]()\n for i, choice in enumerate(choices):\n # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if not selected[i] and choice not in duplicated:\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice) # \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = True\n state.append(choice)\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = False\n state.pop()\n\ndef permutations_ii(nums: list[int]) -> list[list[int]]:\n \"\"\"\u5168\u6392\u5217 II\"\"\"\n res = []\n backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n return res\n
permutations_ii.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.size() == choices.size()) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n unordered_set<int> duplicated;\n for (int i = 0; i < choices.size(); i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && duplicated.find(choice) == duplicated.end()) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.emplace(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.push_back(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop_back();\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nvector<vector<int>> permutationsII(vector<int> nums) {\n vector<int> state;\n vector<bool> selected(nums.size(), false);\n vector<vector<int>> res;\n backtrack(state, nums, selected, res);\n return res;\n}\n
permutations_ii.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List<Integer> state, int[] choices, boolean[] selected, List<List<Integer>> res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.size() == choices.length) {\n res.add(new ArrayList<Integer>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n Set<Integer> duplicated = new HashSet<Integer>();\n for (int i = 0; i < choices.length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.contains(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.remove(state.size() - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nList<List<Integer>> permutationsII(int[] nums) {\n List<List<Integer>> res = new ArrayList<List<Integer>>();\n backtrack(new ArrayList<Integer>(), nums, new boolean[nums.length], res);\n return res;\n}\n
permutations_ii.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid Backtrack(List<int> state, int[] choices, bool[] selected, List<List<int>> res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.Count == choices.Length) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n HashSet<int> duplicated = [];\n for (int i = 0; i < choices.Length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.Contains(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.Add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.Add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.RemoveAt(state.Count - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nList<List<int>> PermutationsII(int[] nums) {\n List<List<int>> res = [];\n Backtrack([], nums, new bool[nums.Length], res);\n return res;\n}\n
permutations_ii.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if len(*state) == len(*choices) {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n duplicated := make(map[int]struct{}, 0)\n for i := 0; i < len(*choices); i++ {\n choice := (*choices)[i]\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if _, ok := duplicated[choice]; !ok && !(*selected)[i] {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n duplicated[choice] = struct{}{}\n (*selected)[i] = true\n *state = append(*state, choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackI(state, choices, selected, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n (*selected)[i] = false\n *state = (*state)[:len(*state)-1]\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\n res := make([][]int, 0)\n state := make([]int, 0)\n selected := make([]bool, len(nums))\n backtrackII(&state, &nums, &selected, &res)\n return res\n}\n
permutations_ii.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if state.count == choices.count {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n var duplicated: Set<Int> = []\n for (i, choice) in choices.enumerated() {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if !selected[i], !duplicated.contains(choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.insert(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true\n state.append(choice)\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, choices: choices, selected: &selected, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false\n state.removeLast()\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -> [[Int]] {\n var state: [Int] = []\n var selected = Array(repeating: false, count: nums.count)\n var res: [[Int]] = []\n backtrack(state: &state, choices: nums, selected: &selected, res: &res)\n return res\n}\n
permutations_ii.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length === choices.length) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n const duplicated = new Set();\n choices.forEach((choice, i) => {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.has(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop();\n }\n });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\n const res = [];\n backtrack([], nums, Array(nums.length).fill(false), res);\n return res;\n}\n
permutations_ii.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\n state: number[],\n choices: number[],\n selected: boolean[],\n res: number[][]\n): void {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length === choices.length) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n const duplicated = new Set();\n choices.forEach((choice, i) => {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.has(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.pop();\n }\n });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\n const res: number[][] = [];\n backtrack([], nums, Array(nums.length).fill(false), res);\n return res;\n}\n
permutations_ii.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(\n List<int> state,\n List<int> choices,\n List<bool> selected,\n List<List<int>> res,\n) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (state.length == choices.length) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n Set<int> duplicated = {};\n for (int i = 0; i < choices.length; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated.contains(choice)) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.add(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.removeLast();\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nList<List<int>> permutationsII(List<int> nums) {\n List<List<int>> res = [];\n backtrack([], nums, List.filled(nums.length, false), res);\n return res;\n}\n
permutations_ii.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfn backtrack(mut state: Vec<i32>, choices: &[i32], selected: &mut [bool], res: &mut Vec<Vec<i32>>) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if state.len() == choices.len() {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n let mut duplicated = HashSet::<i32>::new();\n for i in 0..choices.len() {\n let choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if !selected[i] && !duplicated.contains(&choice) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated.insert(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state.push(choice);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), choices, selected, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n state.remove(state.len() - 1);\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nfn permutations_ii(nums: &mut [i32]) -> Vec<Vec<i32>> {\n let mut res = Vec::new();\n backtrack(Vec::new(), nums, &mut vec![false; nums.len()], &mut res);\n res\n}\n
permutations_ii.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n if (stateSize == choicesSize) {\n res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n for (int i = 0; i < choicesSize; i++) {\n res[*resSize][i] = state[i];\n }\n (*resSize)++;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n bool duplicated[MAX_SIZE] = {false};\n for (int i = 0; i < choicesSize; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n if (!selected[i] && !duplicated[choice]) {\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n duplicated[choice] = true; // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n selected[i] = true;\n state[stateSize] = choice;\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n selected[i] = false;\n }\n }\n}\n\n/* \u5168\u6392\u5217 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n int *state = (int *)malloc(numsSize * sizeof(int));\n bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n for (int i = 0; i < numsSize; i++) {\n selected[i] = false;\n }\n int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n *returnSize = 0;\n\n backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n free(state);\n free(selected);\n\n return res;\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsII}\n\u5047\u8bbe\u5143\u7d20\u4e24\u4e24\u4e4b\u95f4\u4e92\u4e0d\u76f8\u540c\uff0c\u5219 \\(n\\) \u4e2a\u5143\u7d20\u5171\u6709 \\(n!\\) \u79cd\u6392\u5217\uff08\u9636\u4e58\uff09\uff1b\u5728\u8bb0\u5f55\u7ed3\u679c\u65f6\uff0c\u9700\u8981\u590d\u5236\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5217\u8868\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!n)\\) \u3002
\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002
"},{"location":"chapter_backtracking/permutations_problem/#3","title":"3. \u00a0 \u4e24\u79cd\u526a\u679d\u5bf9\u6bd4","text":"selected\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u540c\u4e00\u65f6\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u4e2aduplicated\uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u867d\u7136
selected\u548cduplicated\u90fd\u7528\u4e8e\u526a\u679d\uff0c\u4f46\u4e24\u8005\u7684\u76ee\u6807\u4e0d\u540c\u3002- \u91cd\u590d\u9009\u62e9\u526a\u679d\uff1a\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\u4e2d\u53ea\u6709\u4e00\u4e2a
selected\u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5f53\u524d\u72b6\u6001\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u5176\u4f5c\u7528\u662f\u9632\u6b62choices\u4e2d\u7684\u4efb\u4e00\u5143\u7d20\u5728state\u4e2d\u91cd\u590d\u51fa\u73b0\u3002 - \u76f8\u7b49\u5143\u7d20\u526a\u679d\uff1a\u6bcf\u8f6e\u9009\u62e9\uff08\u6bcf\u4e2a\u8c03\u7528\u7684
backtrack\u51fd\u6570\uff09\u90fd\u5305\u542b\u4e00\u4e2aduplicated\u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5728\u672c\u8f6e\u904d\u5386\uff08for\u5faa\u73af\uff09\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9009\u62e9\u8fc7\uff0c\u5176\u4f5c\u7528\u662f\u4fdd\u8bc1\u76f8\u7b49\u7684\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002
\u56fe 13-9 \u5c55\u793a\u4e86\u4e24\u4e2a\u526a\u679d\u6761\u4ef6\u7684\u751f\u6548\u8303\u56f4\u3002\u6ce8\u610f\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u9009\u62e9\uff0c\u4ece\u6839\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\u6784\u6210\u4e00\u4e2a\u6392\u5217\u3002
\u56fe 13-9 \u00a0 \u4e24\u79cd\u526a\u679d\u6761\u4ef6\u7684\u4f5c\u7528\u8303\u56f4
"},{"location":"chapter_backtracking/subset_sum_problem/","title":"13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898","text":""},{"location":"chapter_backtracking/subset_sum_problem/#1331","title":"13.3.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4
nums\u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570target\uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8etarget\u3002\u7ed9\u5b9a\u6570\u7ec4\u65e0\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ef\u4ee5\u88ab\u9009\u53d6\u591a\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6807\u6574\u6570 \\(9\\) \uff0c\u89e3\u4e3a \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002
- \u8f93\u5165\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u91cd\u590d\u9009\u53d6\u3002
- \u5b50\u96c6\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u6bd4\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u4e2a\u5b50\u96c6\u3002
\u7c7b\u4f3c\u4e8e\u5168\u6392\u5217\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\uff0c\u5e76\u5728\u9009\u62e9\u8fc7\u7a0b\u4e2d\u5b9e\u65f6\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u5f53\u5143\u7d20\u548c\u7b49\u4e8e
target\u65f6\uff0c\u5c31\u5c06\u5b50\u96c6\u8bb0\u5f55\u81f3\u7ed3\u679c\u5217\u8868\u3002\u800c\u4e0e\u5168\u6392\u5217\u95ee\u9898\u4e0d\u540c\u7684\u662f\uff0c\u672c\u9898\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u9009\u53d6\uff0c\u56e0\u6b64\u65e0\u987b\u501f\u52a9
PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_i_naive.pyselected\u5e03\u5c14\u5217\u8868\u6765\u8bb0\u5f55\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u53ef\u4ee5\u5bf9\u5168\u6392\u5217\u4ee3\u7801\u8fdb\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u9898\u4ee3\u7801\uff1a
subset_sum_i_naive.cppdef backtrack(\n state: list[int],\n target: int,\n total: int,\n choices: list[int],\n res: list[list[int]],\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if total == target:\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for i in range(len(choices)):\n # \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if total + choices[i] > target:\n continue\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.append(choices[i])\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -> list[list[int]]:\n \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09\"\"\"\n state = [] # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n total = 0 # \u5b50\u96c6\u548c\n res = [] # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res)\n return res\n
subset_sum_i_naive.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector<int> &state, int target, int total, vector<int> &choices, vector<vector<int>> &res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (size_t i = 0; i < choices.size(); i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.push_back(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop_back();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvector<vector<int>> subsetSumINaive(vector<int> &nums, int target) {\n vector<int> state; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n int total = 0; // \u5b50\u96c6\u548c\n vector<vector<int>> res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<Integer> state, int target, int total, int[] choices, List<List<Integer>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n res.add(new ArrayList<>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.remove(state.size() - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<Integer>> subsetSumINaive(int[] nums, int target) {\n List<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n int total = 0; // \u5b50\u96c6\u548c\n List<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List<int> state, int target, int total, int[] choices, List<List<int>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.Length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.Add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.RemoveAt(state.Count - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<int>> SubsetSumINaive(int[] nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n int total = 0; // \u5b50\u96c6\u548c\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n Backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == total {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i := 0; i < len(*choices); i++ {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if total+(*choices)[i] > target {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n *state = append(*state, (*choices)[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n *state = (*state)[:len(*state)-1]\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n total := 0 // \u5b50\u96c6\u548c\n res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrackSubsetSumINaive(total, target, &state, &nums, &res)\n return res\n}\n
subset_sum_i_naive.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if total == target {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i in stride(from: 0, to: choices.count, by: 1) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if total + choices[i] > target {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.append(choices[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, target: target, total: total + choices[i], choices: choices, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast()\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums: [Int], target: Int) -> [[Int]] {\n var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n let total = 0 // \u5b50\u96c6\u548c\n var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state: &state, target: target, total: total, choices: nums, res: &res)\n return res\n}\n
subset_sum_i_naive.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, total, choices, res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total === target) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (let i = 0; i < choices.length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums, target) {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n const total = 0; // \u5b50\u96c6\u548c\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n state: number[],\n target: number,\n total: number,\n choices: number[],\n res: number[][]\n): void {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total === target) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (let i = 0; i < choices.length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n const total = 0; // \u5b50\u96c6\u548c\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n List<int> state,\n int target,\n int total,\n List<int> choices,\n List<List<int>> res,\n) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choices.length; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<int>> subsetSumINaive(List<int> nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n int total = 0; // \u5143\u7d20\u548c\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, res);\n return res;\n}\n
subset_sum_i_naive.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(mut state: Vec<i32>, target: i32, total: i32, choices: &[i32], res: &mut Vec<Vec<i32>>) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if total == target {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for i in 0..choices.len() {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if total + choices[i] > target {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), target, total + choices[i], choices, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfn subset_sum_i_naive(nums: &[i32], target: i32) -> Vec<Vec<i32>> {\n let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n let total = 0; // \u5b50\u96c6\u548c\n let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, total, nums, &mut res);\n res\n}\n
subset_sum_i_naive.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (total == target) {\n for (int i = 0; i < stateSize; i++) {\n res[resSize][i] = state[i];\n }\n resColSizes[resSize++] = stateSize;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < choicesSize; i++) {\n // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n if (total + choices[i] > target) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n state[stateSize++] = choices[i];\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(target, total + choices[i], choices, choicesSize);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n stateSize--;\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n resSize = 0; // \u521d\u59cb\u5316\u89e3\u7684\u6570\u91cf\u4e3a0\n backtrack(target, 0, nums, numsSize);\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumINaive}\n\u5411\u4ee5\u4e0a\u4ee3\u7801\u8f93\u5165\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u8f93\u51fa\u7ed3\u679c\u4e3a \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u867d\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u4e3a \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u590d\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002
\u8fd9\u662f\u56e0\u4e3a\u641c\u7d22\u8fc7\u7a0b\u662f\u533a\u5206\u9009\u62e9\u987a\u5e8f\u7684\uff0c\u7136\u800c\u5b50\u96c6\u4e0d\u533a\u5206\u9009\u62e9\u987a\u5e8f\u3002\u5982\u56fe 13-10 \u6240\u793a\uff0c\u5148\u9009 \\(4\\) \u540e\u9009 \\(5\\) \u4e0e\u5148\u9009 \\(5\\) \u540e\u9009 \\(4\\) \u662f\u4e0d\u540c\u7684\u5206\u652f\uff0c\u4f46\u5bf9\u5e94\u540c\u4e00\u4e2a\u5b50\u96c6\u3002
\u56fe 13-10 \u00a0 \u5b50\u96c6\u641c\u7d22\u4e0e\u8d8a\u754c\u526a\u679d
\u4e3a\u4e86\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u601d\u8def\u662f\u5bf9\u7ed3\u679c\u5217\u8868\u8fdb\u884c\u53bb\u91cd\u3002\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u6709\u4e24\u65b9\u9762\u539f\u56e0\u3002
- \u5f53\u6570\u7ec4\u5143\u7d20\u8f83\u591a\uff0c\u5c24\u5176\u662f\u5f53
target\u8f83\u5927\u65f6\uff0c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u4ea7\u751f\u5927\u91cf\u7684\u91cd\u590d\u5b50\u96c6\u3002 - \u6bd4\u8f83\u5b50\u96c6\uff08\u6570\u7ec4\uff09\u7684\u5f02\u540c\u975e\u5e38\u8017\u65f6\uff0c\u9700\u8981\u5148\u6392\u5e8f\u6570\u7ec4\uff0c\u518d\u6bd4\u8f83\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684\u5f02\u540c\u3002
\u6211\u4eec\u8003\u8651\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u901a\u8fc7\u526a\u679d\u8fdb\u884c\u53bb\u91cd\u3002\u89c2\u5bdf\u56fe 13-11 \uff0c\u91cd\u590d\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u987a\u5e8f\u9009\u62e9\u6570\u7ec4\u5143\u7d20\u65f6\u4ea7\u751f\u7684\uff0c\u4f8b\u5982\u4ee5\u4e0b\u60c5\u51b5\u3002
- \u5f53\u7b2c\u4e00\u8f6e\u548c\u7b2c\u4e8c\u8f6e\u5206\u522b\u9009\u62e9 \\(3\\) \u548c \\(4\\) \u65f6\uff0c\u4f1a\u751f\u6210\u5305\u542b\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8bb0\u4e3a \\([3, 4, \\dots]\\) \u3002
- \u4e4b\u540e\uff0c\u5f53\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(4\\) \u65f6\uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \uff0c\u56e0\u4e3a\u8be5\u9009\u62e9\u4ea7\u751f\u7684\u5b50\u96c6 \\([4, 3, \\dots]\\) \u548c\u7b2c
1.\u6b65\u4e2d\u751f\u6210\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002
\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e00\u5c42\u7684\u9009\u62e9\u90fd\u662f\u4ece\u5de6\u5230\u53f3\u88ab\u9010\u4e2a\u5c1d\u8bd5\u7684\uff0c\u56e0\u6b64\u8d8a\u9760\u53f3\u7684\u5206\u652f\u88ab\u526a\u6389\u7684\u8d8a\u591a\u3002
- \u524d\u4e24\u8f6e\u9009\u62e9 \\(3\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([3, 5, \\dots]\\) \u3002
- \u524d\u4e24\u8f6e\u9009\u62e9 \\(4\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([4, 5, \\dots]\\) \u3002
- \u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(5\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u4e3a\u5b50\u96c6 \\([5, 3, \\dots]\\) \u548c \\([5, 4, \\dots]\\) \u4e0e\u7b2c
1.\u6b65\u548c\u7b2c2.\u6b65\u4e2d\u63cf\u8ff0\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002
\u56fe 13-11 \u00a0 \u4e0d\u540c\u9009\u62e9\u987a\u5e8f\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6
\u603b\u7ed3\u6765\u770b\uff0c\u7ed9\u5b9a\u8f93\u5165\u6570\u7ec4 \\([x_1, x_2, \\dots, x_n]\\) \uff0c\u8bbe\u641c\u7d22\u8fc7\u7a0b\u4e2d\u7684\u9009\u62e9\u5e8f\u5217\u4e3a \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\) \uff0c\u5219\u8be5\u9009\u62e9\u5e8f\u5217\u9700\u8981\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4e0d\u6ee1\u8db3\u8be5\u6761\u4ef6\u7684\u9009\u62e9\u5e8f\u5217\u90fd\u4f1a\u9020\u6210\u91cd\u590d\uff0c\u5e94\u5f53\u526a\u679d\u3002
"},{"location":"chapter_backtracking/subset_sum_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u4e3a\u5b9e\u73b0\u8be5\u526a\u679d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u53d8\u91cf
start\uff0c\u7528\u4e8e\u6307\u793a\u904d\u5386\u8d77\u59cb\u70b9\u3002\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i\\) \u5f00\u59cb\u904d\u5386\u3002\u8fd9\u6837\u505a\u5c31\u53ef\u4ee5\u8ba9\u9009\u62e9\u5e8f\u5217\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4ece\u800c\u4fdd\u8bc1\u5b50\u96c6\u552f\u4e00\u3002\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u8fd8\u5bf9\u4ee3\u7801\u8fdb\u884c\u4e86\u4ee5\u4e0b\u4e24\u9879\u4f18\u5316\u3002
- \u5728\u5f00\u542f\u641c\u7d22\u524d\uff0c\u5148\u5c06\u6570\u7ec4
nums\u6392\u5e8f\u3002\u5728\u904d\u5386\u6240\u6709\u9009\u62e9\u65f6\uff0c\u5f53\u5b50\u96c6\u548c\u8d85\u8fc7target\u65f6\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\uff0c\u56e0\u4e3a\u540e\u8fb9\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7target\u3002 - \u7701\u53bb\u5143\u7d20\u548c\u53d8\u91cf
total\uff0c\u901a\u8fc7\u5728target\u4e0a\u6267\u884c\u51cf\u6cd5\u6765\u7edf\u8ba1\u5143\u7d20\u548c\uff0c\u5f53target\u7b49\u4e8e \\(0\\) \u65f6\u8bb0\u5f55\u89e3\u3002
subset_sum_i.cppdef backtrack(\n state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0:\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for i in range(start, len(choices)):\n # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0:\n break\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.append(choices[i])\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -> list[list[int]]:\n \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\n state = [] # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort() # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n start = 0 # \u904d\u5386\u8d77\u59cb\u70b9\n res = [] # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res)\n return res\n
subset_sum_i.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choices.size(); i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push_back(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop_back();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector<vector<int>> subsetSumI(vector<int> &nums, int target) {\n vector<int> state; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n vector<vector<int>> res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<Integer> state, int target, int[] choices, int start, List<List<Integer>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.add(new ArrayList<>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.remove(state.size() - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<Integer>> subsetSumI(int[] nums, int target) {\n List<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List<int> state, int target, int[] choices, int start, List<List<int>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choices.Length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.Add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.RemoveAt(state.Count - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<int>> SubsetSumI(int[] nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n Backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for i := start; i < len(*choices); i++ {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target-(*choices)[i] < 0 {\n break\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n *state = append(*state, (*choices)[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n *state = (*state)[:len(*state)-1]\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\n state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n sort.Ints(nums) // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n start := 0 // \u904d\u5386\u8d77\u59cb\u70b9\n res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrackSubsetSumI(start, target, &state, &nums, &res)\n return res\n}\n
subset_sum_i.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for i in stride(from: start, to: choices.count, by: 1) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0 {\n break\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.append(choices[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, target: target - choices[i], choices: choices, start: i, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast()\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums: [Int], target: Int) -> [[Int]] {\n var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state: &state, target: target, choices: nums, start: start, res: &res)\n return res\n}\n
subset_sum_i.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, choices, start, res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target === 0) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (let i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums, target) {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort((a, b) => a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n state: number[],\n target: number,\n choices: number[],\n start: number,\n res: number[][]\n): void {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target === 0) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (let i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort((a, b) => a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n List<int> state,\n int target,\n List<int> choices,\n int start,\n List<List<int>> res,\n) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<int>> subsetSumI(List<int> nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_i.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(mut state: Vec<i32>, target: i32, choices: &[i32], start: usize, res: &mut Vec<Vec<i32>>) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for i in start..choices.len() {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0 {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfn subset_sum_i(nums: &mut [i32], target: i32) -> Vec<Vec<i32>> {\n let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, &mut res);\n res\n}\n
subset_sum_i.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n for (int i = 0; i < stateSize; ++i) {\n res[resSize][i] = state[i];\n }\n resColSizes[resSize++] = stateSize;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n for (int i = start; i < choicesSize; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state[stateSize] = choices[i];\n stateSize++;\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(target - choices[i], choices, choicesSize, i);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n stateSize--;\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n qsort(nums, numsSize, sizeof(int), cmp); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n backtrack(target, nums, numsSize, start);\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumI}\n\u56fe 13-12 \u6240\u793a\u4e3a\u5c06\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u8f93\u5165\u4ee5\u4e0a\u4ee3\u7801\u540e\u7684\u6574\u4f53\u56de\u6eaf\u8fc7\u7a0b\u3002
\u56fe 13-12 \u00a0 \u5b50\u96c6\u548c I \u56de\u6eaf\u8fc7\u7a0b
"},{"location":"chapter_backtracking/subset_sum_problem/#1332","title":"13.3.2 \u00a0 \u8003\u8651\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4
nums\u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570target\uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8etarget\u3002\u7ed9\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u53ef\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002\u76f8\u6bd4\u4e8e\u4e0a\u9898\uff0c\u672c\u9898\u7684\u8f93\u5165\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd9\u5f15\u5165\u4e86\u65b0\u7684\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u7ed9\u5b9a\u6570\u7ec4 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u5219\u73b0\u6709\u4ee3\u7801\u7684\u8f93\u51fa\u7ed3\u679c\u4e3a \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u51fa\u73b0\u4e86\u91cd\u590d\u5b50\u96c6\u3002
\u9020\u6210\u8fd9\u79cd\u91cd\u590d\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f6e\u4e2d\u88ab\u591a\u6b21\u9009\u62e9\u3002\u5728\u56fe 13-13 \u4e2d\uff0c\u7b2c\u4e00\u8f6e\u5171\u6709\u4e09\u4e2a\u9009\u62e9\uff0c\u5176\u4e2d\u4e24\u4e2a\u90fd\u4e3a \\(4\\) \uff0c\u4f1a\u4ea7\u751f\u4e24\u4e2a\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\uff0c\u4ece\u800c\u8f93\u51fa\u91cd\u590d\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f6e\u7684\u4e24\u4e2a \\(4\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002
\u56fe 13-13 \u00a0 \u76f8\u7b49\u5143\u7d20\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6
"},{"location":"chapter_backtracking/subset_sum_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5b9e\u73b0\u65b9\u5f0f\u6bd4\u8f83\u5de7\u5999\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u90bb\u7684\u3002\u8fd9\u610f\u5473\u7740\u5728\u67d0\u8f6e\u9009\u62e9\u4e2d\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u4e0e\u5176\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u5219\u8bf4\u660e\u5b83\u5df2\u7ecf\u88ab\u9009\u62e9\u8fc7\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u8fc7\u5f53\u524d\u5143\u7d20\u3002
\u4e0e\u6b64\u540c\u65f6\uff0c\u672c\u9898\u89c4\u5b9a\u6bcf\u4e2a\u6570\u7ec4\u5143\u7d20\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5e78\u8fd0\u7684\u662f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5229\u7528\u53d8\u91cf
"},{"location":"chapter_backtracking/subset_sum_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_ii.pystart\u6765\u6ee1\u8db3\u8be5\u7ea6\u675f\uff1a\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i + 1\\) \u5f00\u59cb\u5411\u540e\u904d\u5386\u3002\u8fd9\u6837\u65e2\u80fd\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u590d\u9009\u62e9\u5143\u7d20\u3002
subset_sum_ii.cppdef backtrack(\n state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0:\n res.append(list(state))\n return\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n # \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for i in range(start, len(choices)):\n # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0:\n break\n # \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if i > start and choices[i] == choices[i - 1]:\n continue\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.append(choices[i])\n # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res)\n # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -> list[list[int]]:\n \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\n state = [] # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort() # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n start = 0 # \u904d\u5386\u8d77\u59cb\u70b9\n res = [] # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res)\n return res\n
subset_sum_ii.java/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.push_back(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choices.size(); i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push_back(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop_back();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector<vector<int>> subsetSumII(vector<int> &nums, int target) {\n vector<int> state; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n vector<vector<int>> res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.cs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List<Integer> state, int target, int[] choices, int start, List<List<Integer>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.add(new ArrayList<>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.remove(state.size() - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<Integer>> subsetSumII(int[] nums, int target) {\n List<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.go/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid Backtrack(List<int> state, int target, int[] choices, int start, List<List<int>> res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.Add(new List<int>(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choices.Length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.Add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n Backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.RemoveAt(state.Count - 1);\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<int>> SubsetSumII(int[] nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n Backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.swift/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n newState := append([]int{}, *state...)\n *res = append(*res, newState)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for i := start; i < len(*choices); i++ {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target-(*choices)[i] < 0 {\n break\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if i > start && (*choices)[i] == (*choices)[i-1] {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n *state = append(*state, (*choices)[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n *state = (*state)[:len(*state)-1]\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\n state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n sort.Ints(nums) // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n start := 0 // \u904d\u5386\u8d77\u59cb\u70b9\n res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrackSubsetSumII(start, target, &state, &nums, &res)\n return res\n}\n
subset_sum_ii.js/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n res.append(state)\n return\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for i in stride(from: start, to: choices.count, by: 1) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0 {\n break\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if i > start, choices[i] == choices[i - 1] {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.append(choices[i])\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state: &state, target: target - choices[i], choices: choices, start: i + 1, res: &res)\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast()\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums: [Int], target: Int) -> [[Int]] {\n var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state: &state, target: target, choices: nums, start: start, res: &res)\n return res\n}\n
subset_sum_ii.ts/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(state, target, choices, start, res) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target === 0) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (let i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] === choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums, target) {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort((a, b) => a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.dart/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(\n state: number[],\n target: number,\n choices: number[],\n start: number,\n res: number[][]\n): void {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target === 0) {\n res.push([...state]);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (let i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] === choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort((a, b) => a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.rs/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(\n List<int> state,\n int target,\n List<int> choices,\n int start,\n List<List<int>> res,\n) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n res.add(List.from(state));\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choices.length; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if (target - choices[i] < 0) {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.add(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state, target - choices[i], choices, i + 1, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.removeLast();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<int>> subsetSumII(List<int> nums, int target) {\n List<int> state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n List<List<int>> res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, res);\n return res;\n}\n
subset_sum_ii.c/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfn backtrack(mut state: Vec<i32>, target: i32, choices: &[i32], start: usize, res: &mut Vec<Vec<i32>>) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if target == 0 {\n res.push(state);\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for i in start..choices.len() {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n if target - choices[i] < 0 {\n break;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if i > start && choices[i] == choices[i - 1] {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state.push(choices[i]);\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(state.clone(), target - choices[i], choices, i, res);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n state.pop();\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfn subset_sum_ii(nums: &mut [i32], target: i32) -> Vec<Vec<i32>> {\n let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n backtrack(state, target, nums, start, &mut res);\n res\n}\n
subset_sum_ii.zig/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n if (target == 0) {\n for (int i = 0; i < stateSize; i++) {\n res[resSize][i] = state[i];\n }\n resColSizes[resSize++] = stateSize;\n return;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n for (int i = start; i < choicesSize; i++) {\n // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\n if (target - choices[i] < 0) {\n continue;\n }\n // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n if (i > start && choices[i] == choices[i - 1]) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n state[stateSize] = choices[i];\n stateSize++;\n // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n backtrack(target - choices[i], choices, choicesSize, i + 1);\n // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n stateSize--;\n }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n qsort(nums, numsSize, sizeof(int), cmp);\n // \u5f00\u59cb\u56de\u6eaf\n backtrack(target, nums, numsSize, 0);\n}\n[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumII}\n\u56fe 13-14 \u5c55\u793a\u4e86\u6570\u7ec4 \\([4, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u8fc7\u7a0b\uff0c\u5171\u5305\u542b\u56db\u79cd\u526a\u679d\u64cd\u4f5c\u3002\u8bf7\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u6ce8\u91ca\u76f8\u7ed3\u5408\uff0c\u7406\u89e3\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u79cd\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002
\u56fe 13-14 \u00a0 \u5b50\u96c6\u548c II \u56de\u6eaf\u8fc7\u7a0b
"},{"location":"chapter_backtracking/summary/","title":"13.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_backtracking/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"- \u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u662f\u7a77\u4e3e\u6cd5\uff0c\u901a\u8fc7\u5bf9\u89e3\u7a7a\u95f4\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6765\u5bfb\u627e\u7b26\u5408\u6761\u4ef6\u7684\u89e3\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u9047\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u5219\u8bb0\u5f55\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u904d\u5386\u5b8c\u6210\u540e\u7ed3\u675f\u3002
- \u56de\u6eaf\u7b97\u6cd5\u7684\u641c\u7d22\u8fc7\u7a0b\u5305\u62ec\u5c1d\u8bd5\u4e0e\u56de\u9000\u4e24\u4e2a\u90e8\u5206\u3002\u5b83\u901a\u8fc7\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u6765\u5c1d\u8bd5\u5404\u79cd\u9009\u62e9\uff0c\u5f53\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u60c5\u51b5\u65f6\uff0c\u5219\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002\u5c1d\u8bd5\u4e0e\u56de\u9000\u662f\u4e24\u4e2a\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002
- \u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u5b83\u4eec\u53ef\u7528\u4e8e\u5b9e\u73b0\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7ed3\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002
- \u56de\u6eaf\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u4e8e\u89e3\u51b3\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u3002\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u867d\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u7b97\u6cd5\u89e3\u51b3\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u6548\u7387\u66f4\u9ad8\u6216\u6548\u679c\u66f4\u597d\u7684\u89e3\u6cd5\u3002
- \u5168\u6392\u5217\u95ee\u9898\u65e8\u5728\u641c\u7d22\u7ed9\u5b9a\u96c6\u5408\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5143\u7d20\u662f\u5426\u88ab\u9009\u62e9\uff0c\u526a\u6389\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\u7684\u641c\u7d22\u5206\u652f\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002
- \u5728\u5168\u6392\u5217\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff0c\u5219\u6700\u7ec8\u7ed3\u679c\u4f1a\u51fa\u73b0\u91cd\u590d\u6392\u5217\u3002\u6211\u4eec\u9700\u8981\u7ea6\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u8fd9\u901a\u5e38\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002
- \u5b50\u96c6\u548c\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u7ed9\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u4e3a\u76ee\u6807\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u800c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u8f93\u51fa\u6240\u6709\u987a\u5e8f\u7684\u7ed3\u679c\uff0c\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002\u6211\u4eec\u5728\u56de\u6eaf\u524d\u5c06\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8bbe\u7f6e\u4e00\u4e2a\u53d8\u91cf\u6765\u6307\u793a\u6bcf\u4e00\u8f6e\u7684\u904d\u5386\u8d77\u59cb\u70b9\uff0c\u4ece\u800c\u5c06\u751f\u6210\u91cd\u590d\u5b50\u96c6\u7684\u641c\u7d22\u5206\u652f\u8fdb\u884c\u526a\u679d\u3002
- \u5bf9\u4e8e\u5b50\u96c6\u548c\u95ee\u9898\uff0c\u6570\u7ec4\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u4f1a\u4ea7\u751f\u91cd\u590d\u96c6\u5408\u3002\u6211\u4eec\u5229\u7528\u6570\u7ec4\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\uff0c\u901a\u8fc7\u5224\u65ad\u76f8\u90bb\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5b9e\u73b0\u526a\u679d\uff0c\u4ece\u800c\u786e\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u4e2d\u4e00\u6b21\u3002
- \\(n\\) \u7687\u540e\u95ee\u9898\u65e8\u5728\u5bfb\u627e\u5c06 \\(n\\) \u4e2a\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76d8\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u4e24\u4e24\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u5bf9\u65b9\u3002\u8be5\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\u6709\u884c\u7ea6\u675f\u3001\u5217\u7ea6\u675f\u3001\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u4e3a\u6ee1\u8db3\u884c\u7ea6\u675f\uff0c\u6211\u4eec\u91c7\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8bc1\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002
- \u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u5904\u7406\u65b9\u5f0f\u7c7b\u4f3c\u3002\u5bf9\u4e8e\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u4ece\u800c\u6307\u793a\u9009\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5bf9\u4e8e\u5bf9\u89d2\u7ebf\u7ea6\u675f\uff0c\u6211\u4eec\u501f\u52a9\u4e24\u4e2a\u6570\u7ec4\u6765\u5206\u522b\u8bb0\u5f55\u8be5\u4e3b\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96be\u70b9\u5728\u4e8e\u627e\u5904\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5bf9\u89d2\u7ebf\u4e0a\u683c\u5b50\u6ee1\u8db3\u7684\u884c\u5217\u7d22\u5f15\u89c4\u5f8b\u3002
\u600e\u4e48\u7406\u89e3\u56de\u6eaf\u548c\u9012\u5f52\u7684\u5173\u7cfb\uff1f
\u603b\u7684\u6765\u770b\uff0c\u56de\u6eaf\u662f\u4e00\u79cd\u201c\u7b97\u6cd5\u7b56\u7565\u201d\uff0c\u800c\u9012\u5f52\u66f4\u50cf\u662f\u4e00\u4e2a\u201c\u5de5\u5177\u201d\u3002
- \u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002\u7136\u800c\uff0c\u56de\u6eaf\u662f\u9012\u5f52\u7684\u5e94\u7528\u573a\u666f\u4e4b\u4e00\uff0c\u662f\u9012\u5f52\u5728\u641c\u7d22\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u3002
- \u9012\u5f52\u7684\u7ed3\u6784\u4f53\u73b0\u4e86\u201c\u5b50\u95ee\u9898\u5206\u89e3\u201d\u7684\u89e3\u9898\u8303\u5f0f\uff0c\u5e38\u7528\u4e8e\u89e3\u51b3\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\uff08\u8bb0\u5fc6\u5316\u9012\u5f52\uff09\u7b49\u95ee\u9898\u3002
Abstract
\u590d\u6742\u5ea6\u5206\u6790\u72b9\u5982\u6d69\u701a\u7684\u7b97\u6cd5\u5b87\u5b99\u4e2d\u7684\u65f6\u7a7a\u5411\u5bfc\u3002
\u5b83\u5e26\u9886\u6211\u4eec\u5728\u65f6\u95f4\u4e0e\u7a7a\u95f4\u8fd9\u4e24\u4e2a\u7ef4\u5ea6\u4e0a\u6df1\u5165\u63a2\u7d22\uff0c\u5bfb\u627e\u66f4\u4f18\u96c5\u7684\u89e3\u51b3\u65b9\u6848\u3002
"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30
- 2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52
- 2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6
- 2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6
- 2.5 \u00a0 \u5c0f\u7ed3
\u5728\u7b97\u6cd5\u4e2d\uff0c\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u662f\u5f88\u5e38\u89c1\u7684\uff0c\u5b83\u4e0e\u590d\u6742\u5ea6\u5206\u6790\u606f\u606f\u76f8\u5173\u3002\u56e0\u6b64\uff0c\u5728\u4ecb\u7ecd\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e4b\u524d\uff0c\u6211\u4eec\u5148\u6765\u4e86\u89e3\u5982\u4f55\u5728\u7a0b\u5e8f\u4e2d\u5b9e\u73b0\u91cd\u590d\u6267\u884c\u4efb\u52a1\uff0c\u5373\u4e24\u79cd\u57fa\u672c\u7684\u7a0b\u5e8f\u63a7\u5236\u7ed3\u6784\uff1a\u8fed\u4ee3\u3001\u9012\u5f52\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","title":"2.2.1 \u00a0 \u8fed\u4ee3","text":"\u300c\u8fed\u4ee3 iteration\u300d\u662f\u4e00\u79cd\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u7684\u63a7\u5236\u7ed3\u6784\u3002\u5728\u8fed\u4ee3\u4e2d\uff0c\u7a0b\u5e8f\u4f1a\u5728\u6ee1\u8db3\u4e00\u5b9a\u7684\u6761\u4ef6\u4e0b\u91cd\u590d\u6267\u884c\u67d0\u6bb5\u4ee3\u7801\uff0c\u76f4\u5230\u8fd9\u4e2a\u6761\u4ef6\u4e0d\u518d\u6ee1\u8db3\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","title":"1. \u00a0 for \u5faa\u73af","text":"for\u5faa\u73af\u662f\u6700\u5e38\u89c1\u7684\u8fed\u4ee3\u5f62\u5f0f\u4e4b\u4e00\uff0c\u9002\u5408\u5728\u9884\u5148\u77e5\u9053\u8fed\u4ee3\u6b21\u6570\u65f6\u4f7f\u7528\u3002\u4ee5\u4e0b\u51fd\u6570\u57fa\u4e8e
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyfor\u5faa\u73af\u5b9e\u73b0\u4e86\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff0c\u6c42\u548c\u7ed3\u679c\u4f7f\u7528\u53d8\u91cfres\u8bb0\u5f55\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0cPython \u4e2drange(a, b)\u5bf9\u5e94\u7684\u533a\u95f4\u662f\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u7684\uff0c\u5bf9\u5e94\u7684\u904d\u5386\u8303\u56f4\u4e3a \\(a, a + 1, \\dots, b-1\\) \uff1a
iteration.cppdef for_loop(n: int) -> int:\n \"\"\"for \u5faa\u73af\"\"\"\n res = 0\n # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in range(1, n + 1):\n res += i\n return res\n
iteration.java/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; ++i) {\n res += i;\n }\n return res;\n}\n
iteration.cs/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.go/* for \u5faa\u73af */\nint ForLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.swift/* for \u5faa\u73af */\nfunc forLoop(n int) int {\n res := 0\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i := 1; i <= n; i++ {\n res += i\n }\n return res\n}\n
iteration.js/* for \u5faa\u73af */\nfunc forLoop(n: Int) -> Int {\n var res = 0\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in 1 ... n {\n res += i\n }\n return res\n}\n
iteration.ts/* for \u5faa\u73af */\nfunction forLoop(n) {\n let res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.dart/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\n let res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.rs/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n
iteration.c/* for \u5faa\u73af */\nfn for_loop(n: i32) -> i32 {\n let mut res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i in 1..=n {\n res += i;\n }\n res\n} \n
iteration.zig/* for \u5faa\u73af */\nint forLoop(int n) {\n int res = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n res += i;\n }\n return res;\n}\n// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\n var res: i32 = 0;\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for (1..n+1) |i| {\n res = res + @as(i32, @intCast(i));\n }\n return res;\n} \n\u56fe 2-1 \u662f\u8be5\u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe\u3002
\u56fe 2-1 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe
\u6b64\u6c42\u548c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u6210\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u63cf\u8ff0\u7684\u5c31\u662f\u8fd9\u4e2a\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u76f8\u5173\u5185\u5bb9\u5c06\u4f1a\u5728\u4e0b\u4e00\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","title":"2. \u00a0 while \u5faa\u73af","text":"\u4e0e
for\u5faa\u73af\u7c7b\u4f3c\uff0cwhile\u5faa\u73af\u4e5f\u662f\u4e00\u79cd\u5b9e\u73b0\u8fed\u4ee3\u7684\u65b9\u6cd5\u3002\u5728while\u5faa\u73af\u4e2d\uff0c\u7a0b\u5e8f\u6bcf\u8f6e\u90fd\u4f1a\u5148\u68c0\u67e5\u6761\u4ef6\uff0c\u5982\u679c\u6761\u4ef6\u4e3a\u771f\uff0c\u5219\u7ee7\u7eed\u6267\u884c\uff0c\u5426\u5219\u5c31\u7ed3\u675f\u5faa\u73af\u3002\u4e0b\u9762\u6211\u4eec\u7528
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pywhile\u5faa\u73af\u6765\u5b9e\u73b0\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff1a
iteration.cppdef while_loop(n: int) -> int:\n \"\"\"while \u5faa\u73af\"\"\"\n res = 0\n i = 1 # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n:\n res += i\n i += 1 # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n return res\n
iteration.java/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.cs/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.go/* while \u5faa\u73af */\nint WhileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.swift/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\n res := 0\n // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n i := 1\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n for i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++\n }\n return res\n}\n
iteration.js/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -> Int {\n var res = 0\n var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n {\n res += i\n i += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res\n}\n
iteration.ts/* while \u5faa\u73af */\nfunction whileLoop(n) {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.dart/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.rs/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n
iteration.c/* while \u5faa\u73af */\nfn while_loop(n: i32) -> i32 {\n let mut res = 0;\n let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while i <= n {\n res += i;\n i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n res\n}\n
iteration.zig/* while \u5faa\u73af */\nint whileLoop(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += i;\n i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n }\n return res;\n}\n// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\n var res: i32 = 0;\n var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n while (i <= n) {\n res += @intCast(i);\n i += 1;\n }\n return res;\n}\nwhile\u5faa\u73af\u6bd4for\u5faa\u73af\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\u3002\u5728while\u5faa\u73af\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u5730\u8bbe\u8ba1\u6761\u4ef6\u53d8\u91cf\u7684\u521d\u59cb\u5316\u548c\u66f4\u65b0\u6b65\u9aa4\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6761\u4ef6\u53d8\u91cf \\(i\\) \u6bcf\u8f6e\u8fdb\u884c\u4e24\u6b21\u66f4\u65b0\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u4e0d\u592a\u65b9\u4fbf\u7528
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyfor\u5faa\u73af\u5b9e\u73b0\uff1a
iteration.cppdef while_loop_ii(n: int) -> int:\n \"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\n res = 0\n i = 1 # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n # \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n:\n res += i\n # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1\n i *= 2\n return res\n
iteration.java/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.cs/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.go/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 2, 4, 5...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1; \n i *= 2;\n }\n return res;\n}\n
iteration.swift/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n res := 0\n // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n i := 1\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n for i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++\n i *= 2\n }\n return res\n}\n
iteration.js/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -> Int {\n var res = 0\n var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n {\n res += i\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1\n i *= 2\n }\n return res\n}\n
iteration.ts/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.dart/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n let res = 0;\n let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.rs/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n
iteration.c/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -> i32 {\n let mut res = 0;\n let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while i <= n {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1;\n i *= 2;\n }\n res\n}\n
iteration.zig/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n int res = 0;\n int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += i;\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i++;\n i *= 2;\n }\n return res;\n}\n// while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n var res: i32 = 0;\n var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n while (i <= n) {\n res += @intCast(i);\n // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n i += 1;\n i *= 2;\n }\n return res;\n}\n\u603b\u7684\u6765\u8bf4\uff0c
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","title":"3. \u00a0 \u5d4c\u5957\u5faa\u73af","text":"for\u5faa\u73af\u7684\u4ee3\u7801\u66f4\u52a0\u7d27\u51d1\uff0cwhile\u5faa\u73af\u66f4\u52a0\u7075\u6d3b\uff0c\u4e24\u8005\u90fd\u53ef\u4ee5\u5b9e\u73b0\u8fed\u4ee3\u7ed3\u6784\u3002\u9009\u62e9\u4f7f\u7528\u54ea\u4e00\u4e2a\u5e94\u8be5\u6839\u636e\u7279\u5b9a\u95ee\u9898\u7684\u9700\u6c42\u6765\u51b3\u5b9a\u3002\u6211\u4eec\u53ef\u4ee5\u5728\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\u5185\u5d4c\u5957\u53e6\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\uff0c\u4e0b\u9762\u4ee5
PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.pyfor\u5faa\u73af\u4e3a\u4f8b\uff1a
iteration.cppdef nested_for_loop(n: int) -> str:\n \"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\n res = \"\"\n # \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in range(1, n + 1):\n # \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in range(1, n + 1):\n res += f\"({i}, {j}), \"\n return res\n
iteration.java/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\n ostringstream res;\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; ++i) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; ++j) {\n res << \"(\" << i << \", \" << j << \"), \";\n }\n }\n return res.str();\n}\n
iteration.cs/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n StringBuilder res = new StringBuilder();\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res.append(\"(\" + i + \", \" + j + \"), \");\n }\n }\n return res.toString();\n}\n
iteration.go/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\n StringBuilder res = new();\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res.Append($\"({i}, {j}), \");\n }\n }\n return res.ToString();\n}\n
iteration.swift/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\n res := \"\"\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i := 1; i <= n; i++ {\n for j := 1; j <= n; j++ {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n res += fmt.Sprintf(\"(%d, %d), \", i, j)\n }\n }\n return res\n}\n
iteration.js/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -> String {\n var res = \"\"\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in 1 ... n {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in 1 ... n {\n res.append(\"(\\(i), \\(j)), \")\n }\n }\n return res\n}\n
iteration.ts/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\n let res = '';\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (let j = 1; j <= n; j++) {\n res += `(${i}, ${j}), `;\n }\n }\n return res;\n}\n
iteration.dart/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\n let res = '';\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (let i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (let j = 1; j <= n; j++) {\n res += `(${i}, ${j}), `;\n }\n }\n return res;\n}\n
iteration.rs/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n String res = \"\";\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n res += \"($i, $j), \";\n }\n }\n return res;\n}\n
iteration.c/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -> String {\n let mut res = vec![];\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for i in 1..=n {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for j in 1..=n {\n res.push(format!(\"({}, {}), \", i, j));\n }\n }\n res.join(\"\")\n}\n
iteration.zig/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n // n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\n int size = n * n * 26 + 1;\n char *res = malloc(size * sizeof(char));\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (int i = 1; i <= n; i++) {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (int j = 1; j <= n; j++) {\n char tmp[26];\n snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n strncat(res, tmp, size - strlen(res) - 1);\n }\n }\n return res;\n}\n// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n var res = std.ArrayList(u8).init(allocator);\n defer res.deinit();\n var buffer: [20]u8 = undefined;\n // \u5faa\u73af i = 1, 2, ..., n-1, n\n for (1..n+1) |i| {\n // \u5faa\u73af j = 1, 2, ..., n-1, n\n for (1..n+1) |j| {\n var _str = try std.fmt.bufPrint(&buffer, \"({d}, {d}), \", .{i, j});\n try res.appendSlice(_str);\n }\n }\n return res.toOwnedSlice();\n}\n\u56fe 2-2 \u662f\u8be5\u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe\u3002
\u56fe 2-2 \u00a0 \u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe
\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e \\(n^2\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u548c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u201c\u5e73\u65b9\u5173\u7cfb\u201d\u3002
\u6211\u4eec\u53ef\u4ee5\u7ee7\u7eed\u6dfb\u52a0\u5d4c\u5957\u5faa\u73af\uff0c\u6bcf\u4e00\u6b21\u5d4c\u5957\u90fd\u662f\u4e00\u6b21\u201c\u5347\u7ef4\u201d\uff0c\u5c06\u4f1a\u4f7f\u65f6\u95f4\u590d\u6742\u5ea6\u63d0\u9ad8\u81f3\u201c\u7acb\u65b9\u5173\u7cfb\u201d\u201c\u56db\u6b21\u65b9\u5173\u7cfb\u201d\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","title":"2.2.2 \u00a0 \u9012\u5f52","text":"\u300c\u9012\u5f52 recursion\u300d\u662f\u4e00\u79cd\u7b97\u6cd5\u7b56\u7565\uff0c\u901a\u8fc7\u51fd\u6570\u8c03\u7528\u81ea\u8eab\u6765\u89e3\u51b3\u95ee\u9898\u3002\u5b83\u4e3b\u8981\u5305\u542b\u4e24\u4e2a\u9636\u6bb5\u3002
- \u9012\uff1a\u7a0b\u5e8f\u4e0d\u65ad\u6df1\u5165\u5730\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u4f20\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\uff0c\u76f4\u5230\u8fbe\u5230\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u3002
- \u5f52\uff1a\u89e6\u53d1\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u540e\uff0c\u7a0b\u5e8f\u4ece\u6700\u6df1\u5c42\u7684\u9012\u5f52\u51fd\u6570\u5f00\u59cb\u9010\u5c42\u8fd4\u56de\uff0c\u6c47\u805a\u6bcf\u4e00\u5c42\u7684\u7ed3\u679c\u3002
\u800c\u4ece\u5b9e\u73b0\u7684\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u4ee3\u7801\u4e3b\u8981\u5305\u542b\u4e09\u4e2a\u8981\u7d20\u3002
- \u7ec8\u6b62\u6761\u4ef6\uff1a\u7528\u4e8e\u51b3\u5b9a\u4ec0\u4e48\u65f6\u5019\u7531\u201c\u9012\u201d\u8f6c\u201c\u5f52\u201d\u3002
- \u9012\u5f52\u8c03\u7528\uff1a\u5bf9\u5e94\u201c\u9012\u201d\uff0c\u51fd\u6570\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u8f93\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\u3002
- \u8fd4\u56de\u7ed3\u679c\uff1a\u5bf9\u5e94\u201c\u5f52\u201d\uff0c\u5c06\u5f53\u524d\u9012\u5f52\u5c42\u7ea7\u7684\u7ed3\u679c\u8fd4\u56de\u81f3\u4e0a\u4e00\u5c42\u3002
\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u53ea\u9700\u8c03\u7528\u51fd\u6570
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyrecur(n)\uff0c\u5c31\u53ef\u4ee5\u5b8c\u6210 \\(1 + 2 + \\dots + n\\) \u7684\u8ba1\u7b97\uff1a
recursion.cppdef recur(n: int) -> int:\n \"\"\"\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if n == 1:\n return 1\n # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n res = recur(n - 1)\n # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n
recursion.java/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.cs/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.go/* \u9012\u5f52 */\nint Recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = Recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.swift/* \u9012\u5f52 */\nfunc recur(n int) int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n res := recur(n - 1)\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n}\n
recursion.js/* \u9012\u5f52 */\nfunc recur(n: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n let res = recur(n: n - 1)\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res\n}\n
recursion.ts/* \u9012\u5f52 */\nfunction recur(n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n const res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.dart/* \u9012\u5f52 */\nfunction recur(n: number): number {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n const res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.rs/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1) return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n
recursion.c/* \u9012\u5f52 */\nfn recur(n: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 1 {\n return 1;\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n let res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n n + res\n}\n
recursion.zig/* \u9012\u5f52 */\nint recur(int n) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1)\n return 1;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n int res = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 1) {\n return 1;\n }\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n var res: i32 = recur(n - 1);\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n return n + res;\n}\n\u56fe 2-3 \u5c55\u793a\u4e86\u8be5\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b\u3002
\u56fe 2-3 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b
\u867d\u7136\u4ece\u8ba1\u7b97\u89d2\u5ea6\u770b\uff0c\u8fed\u4ee3\u4e0e\u9012\u5f52\u53ef\u4ee5\u5f97\u5230\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u4f46\u5b83\u4eec\u4ee3\u8868\u4e86\u4e24\u79cd\u5b8c\u5168\u4e0d\u540c\u7684\u601d\u8003\u548c\u89e3\u51b3\u95ee\u9898\u7684\u8303\u5f0f\u3002
- \u8fed\u4ee3\uff1a\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u4ece\u6700\u57fa\u7840\u7684\u6b65\u9aa4\u5f00\u59cb\uff0c\u7136\u540e\u4e0d\u65ad\u91cd\u590d\u6216\u7d2f\u52a0\u8fd9\u4e9b\u6b65\u9aa4\uff0c\u76f4\u5230\u4efb\u52a1\u5b8c\u6210\u3002
- \u9012\u5f52\uff1a\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u8fd9\u4e9b\u5b50\u95ee\u9898\u548c\u539f\u95ee\u9898\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f\u3002\u63a5\u4e0b\u6765\u5c06\u5b50\u95ee\u9898\u7ee7\u7eed\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u57fa\u672c\u60c5\u51b5\u65f6\u505c\u6b62\uff08\u57fa\u672c\u60c5\u51b5\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff09\u3002
\u4ee5\u4e0a\u8ff0\u6c42\u548c\u51fd\u6570\u4e3a\u4f8b\uff0c\u8bbe\u95ee\u9898 \\(f(n) = 1 + 2 + \\dots + n\\) \u3002
- \u8fed\u4ee3\uff1a\u5728\u5faa\u73af\u4e2d\u6a21\u62df\u6c42\u548c\u8fc7\u7a0b\uff0c\u4ece \\(1\\) \u904d\u5386\u5230 \\(n\\) \uff0c\u6bcf\u8f6e\u6267\u884c\u6c42\u548c\u64cd\u4f5c\uff0c\u5373\u53ef\u6c42\u5f97 \\(f(n)\\) \u3002
- \u9012\u5f52\uff1a\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(f(n) = n + f(n-1)\\) \uff0c\u4e0d\u65ad\uff08\u9012\u5f52\u5730\uff09\u5206\u89e3\u4e0b\u53bb\uff0c\u76f4\u81f3\u57fa\u672c\u60c5\u51b5 \\(f(1) = 1\\) \u65f6\u7ec8\u6b62\u3002
\u9012\u5f52\u51fd\u6570\u6bcf\u6b21\u8c03\u7528\u81ea\u8eab\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u4e3a\u65b0\u5f00\u542f\u7684\u51fd\u6570\u5206\u914d\u5185\u5b58\uff0c\u4ee5\u5b58\u50a8\u5c40\u90e8\u53d8\u91cf\u3001\u8c03\u7528\u5730\u5740\u548c\u5176\u4ed6\u4fe1\u606f\u7b49\u3002\u8fd9\u5c06\u5bfc\u81f4\u4e24\u65b9\u9762\u7684\u7ed3\u679c\u3002
- \u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u90fd\u5b58\u50a8\u5728\u79f0\u4e3a\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u7684\u5185\u5b58\u533a\u57df\u4e2d\uff0c\u76f4\u81f3\u51fd\u6570\u8fd4\u56de\u540e\u624d\u4f1a\u88ab\u91ca\u653e\u3002\u56e0\u6b64\uff0c\u9012\u5f52\u901a\u5e38\u6bd4\u8fed\u4ee3\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002
- \u9012\u5f52\u8c03\u7528\u51fd\u6570\u4f1a\u4ea7\u751f\u989d\u5916\u7684\u5f00\u9500\u3002\u56e0\u6b64\u9012\u5f52\u901a\u5e38\u6bd4\u5faa\u73af\u7684\u65f6\u95f4\u6548\u7387\u66f4\u4f4e\u3002
\u5982\u56fe 2-4 \u6240\u793a\uff0c\u5728\u89e6\u53d1\u7ec8\u6b62\u6761\u4ef6\u524d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684\u9012\u5f52\u51fd\u6570\uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002
\u56fe 2-4 \u00a0 \u9012\u5f52\u8c03\u7528\u6df1\u5ea6
\u5728\u5b9e\u9645\u4e2d\uff0c\u7f16\u7a0b\u8bed\u8a00\u5141\u8bb8\u7684\u9012\u5f52\u6df1\u5ea6\u901a\u5e38\u662f\u6709\u9650\u7684\uff0c\u8fc7\u6df1\u7684\u9012\u5f52\u53ef\u80fd\u5bfc\u81f4\u6808\u6ea2\u51fa\u9519\u8bef\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","title":"2. \u00a0 \u5c3e\u9012\u5f52","text":"\u6709\u8da3\u7684\u662f\uff0c\u5982\u679c\u51fd\u6570\u5728\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u6b65\u624d\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5219\u8be5\u51fd\u6570\u53ef\u4ee5\u88ab\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u4f18\u5316\uff0c\u4f7f\u5176\u5728\u7a7a\u95f4\u6548\u7387\u4e0a\u4e0e\u8fed\u4ee3\u76f8\u5f53\u3002\u8fd9\u79cd\u60c5\u51b5\u88ab\u79f0\u4e3a\u300c\u5c3e\u9012\u5f52 tail recursion\u300d\u3002
- \u666e\u901a\u9012\u5f52\uff1a\u5f53\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u7684\u51fd\u6570\u540e\uff0c\u9700\u8981\u7ee7\u7eed\u6267\u884c\u4ee3\u7801\uff0c\u56e0\u6b64\u7cfb\u7edf\u9700\u8981\u4fdd\u5b58\u4e0a\u4e00\u5c42\u8c03\u7528\u7684\u4e0a\u4e0b\u6587\u3002
- \u5c3e\u9012\u5f52\uff1a\u9012\u5f52\u8c03\u7528\u662f\u51fd\u6570\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u4e2a\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u540e\uff0c\u65e0\u987b\u7ee7\u7eed\u6267\u884c\u5176\u4ed6\u64cd\u4f5c\uff0c\u56e0\u6b64\u7cfb\u7edf\u65e0\u987b\u4fdd\u5b58\u4e0a\u4e00\u5c42\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u3002
\u4ee5\u8ba1\u7b97 \\(1 + 2 + \\dots + n\\) \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u7ed3\u679c\u53d8\u91cf
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyres\u8bbe\u4e3a\u51fd\u6570\u53c2\u6570\uff0c\u4ece\u800c\u5b9e\u73b0\u5c3e\u9012\u5f52\uff1a
recursion.cppdef tail_recur(n, res):\n \"\"\"\u5c3e\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if n == 0:\n return res\n # \u5c3e\u9012\u5f52\u8c03\u7528\n return tail_recur(n - 1, res + n)\n
recursion.java/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.cs/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.go/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return TailRecur(n - 1, res + n);\n}\n
recursion.swift/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n-1, res+n)\n}\n
recursion.js/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n: n - 1, res: res + n)\n}\n
recursion.ts/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.dart/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n === 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.rs/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0) return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n
recursion.c/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if n == 0 {\n return res;\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n tail_recur(n - 1, res + n)\n}\n
recursion.zig/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0)\n return res;\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6\n if (n == 0) {\n return res;\n }\n // \u5c3e\u9012\u5f52\u8c03\u7528\n return tailRecur(n - 1, res + n);\n}\n\u5c3e\u9012\u5f52\u7684\u6267\u884c\u8fc7\u7a0b\u5982\u56fe 2-5 \u6240\u793a\u3002\u5bf9\u6bd4\u666e\u901a\u9012\u5f52\u548c\u5c3e\u9012\u5f52\uff0c\u4e24\u8005\u7684\u6c42\u548c\u64cd\u4f5c\u7684\u6267\u884c\u70b9\u662f\u4e0d\u540c\u7684\u3002
- \u666e\u901a\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u6bcf\u5c42\u8fd4\u56de\u540e\u90fd\u8981\u518d\u6267\u884c\u4e00\u6b21\u6c42\u548c\u64cd\u4f5c\u3002
- \u5c3e\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u53ea\u9700\u5c42\u5c42\u8fd4\u56de\u3002
\u56fe 2-5 \u00a0 \u5c3e\u9012\u5f52\u8fc7\u7a0b
Tip
\u8bf7\u6ce8\u610f\uff0c\u8bb8\u591a\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u5e76\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\u3002\u4f8b\u5982\uff0cPython \u9ed8\u8ba4\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u56e0\u6b64\u5373\u4f7f\u51fd\u6570\u662f\u5c3e\u9012\u5f52\u5f62\u5f0f\uff0c\u4ecd\u7136\u53ef\u80fd\u4f1a\u9047\u5230\u6808\u6ea2\u51fa\u95ee\u9898\u3002
"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","title":"3. \u00a0 \u9012\u5f52\u6811","text":"\u5f53\u5904\u7406\u4e0e\u201c\u5206\u6cbb\u201d\u76f8\u5173\u7684\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u9012\u5f52\u5f80\u5f80\u6bd4\u8fed\u4ee3\u7684\u601d\u8def\u66f4\u52a0\u76f4\u89c2\u3001\u4ee3\u7801\u66f4\u52a0\u6613\u8bfb\u3002\u4ee5\u201c\u6590\u6ce2\u90a3\u5951\u6570\u5217\u201d\u4e3a\u4f8b\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u5217 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\) \uff0c\u6c42\u8be5\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u3002
\u8bbe\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u4e3a \\(f(n)\\) \uff0c\u6613\u5f97\u4e24\u4e2a\u7ed3\u8bba\u3002
- \u6570\u5217\u7684\u524d\u4e24\u4e2a\u6570\u5b57\u4e3a \\(f(1) = 0\\) \u548c \\(f(2) = 1\\) \u3002
- \u6570\u5217\u4e2d\u7684\u6bcf\u4e2a\u6570\u5b57\u662f\u524d\u4e24\u4e2a\u6570\u5b57\u7684\u548c\uff0c\u5373 \\(f(n) = f(n - 1) + f(n - 2)\\) \u3002
\u6309\u7167\u9012\u63a8\u5173\u7cfb\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5c06\u524d\u4e24\u4e2a\u6570\u5b57\u4f5c\u4e3a\u7ec8\u6b62\u6761\u4ef6\uff0c\u4fbf\u53ef\u5199\u51fa\u9012\u5f52\u4ee3\u7801\u3002\u8c03\u7528
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.pyfib(n)\u5373\u53ef\u5f97\u5230\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\uff1a
recursion.cppdef fib(n: int) -> int:\n \"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 or n == 2:\n return n - 1\n # \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n res = fib(n - 1) + fib(n - 2)\n # \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n
recursion.java/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.cs/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.go/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = Fib(n - 1) + Fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.swift/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n res := fib(n-1) + fib(n-2)\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n}\n
recursion.js/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -> Int {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n let res = fib(n: n - 1) + fib(n: n - 2)\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res\n}\n
recursion.ts/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n === 1 || n === 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n const res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.dart/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n === 1 || n === 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n const res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.rs/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2) return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n
recursion.c/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -> i32 {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if n == 1 || n == 2 {\n return n - 1;\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n let res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c\n res\n}\n
recursion.zig/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 || n == 2)\n return n - 1;\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n int res = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n if (n == 1 or n == 2) {\n return n - 1;\n }\n // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n var res: i32 = fib(n - 1) + fib(n - 2);\n // \u8fd4\u56de\u7ed3\u679c f(n)\n return res;\n}\n\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u6211\u4eec\u5728\u51fd\u6570\u5185\u9012\u5f52\u8c03\u7528\u4e86\u4e24\u4e2a\u51fd\u6570\uff0c\u8fd9\u610f\u5473\u7740\u4ece\u4e00\u4e2a\u8c03\u7528\u4ea7\u751f\u4e86\u4e24\u4e2a\u8c03\u7528\u5206\u652f\u3002\u5982\u56fe 2-6 \u6240\u793a\uff0c\u8fd9\u6837\u4e0d\u65ad\u9012\u5f52\u8c03\u7528\u4e0b\u53bb\uff0c\u6700\u7ec8\u5c06\u4ea7\u751f\u4e00\u68f5\u5c42\u6570\u4e3a \\(n\\) \u7684\u300c\u9012\u5f52\u6811 recursion tree\u300d\u3002
\u56fe 2-6 \u00a0 \u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u9012\u5f52\u6811
\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9012\u5f52\u4f53\u73b0\u4e86\u201c\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u5b50\u95ee\u9898\u201d\u7684\u601d\u7ef4\u8303\u5f0f\uff0c\u8fd9\u79cd\u5206\u6cbb\u7b56\u7565\u81f3\u5173\u91cd\u8981\u3002
- \u4ece\u7b97\u6cd5\u89d2\u5ea6\u770b\uff0c\u641c\u7d22\u3001\u6392\u5e8f\u3001\u56de\u6eaf\u3001\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u7b49\u8bb8\u591a\u91cd\u8981\u7b97\u6cd5\u7b56\u7565\u76f4\u63a5\u6216\u95f4\u63a5\u5730\u5e94\u7528\u4e86\u8fd9\u79cd\u601d\u7ef4\u65b9\u5f0f\u3002
- \u4ece\u6570\u636e\u7ed3\u6784\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u5929\u7136\u9002\u5408\u5904\u7406\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u76f8\u5173\u95ee\u9898\uff0c\u56e0\u4e3a\u5b83\u4eec\u975e\u5e38\u9002\u5408\u7528\u5206\u6cbb\u601d\u60f3\u8fdb\u884c\u5206\u6790\u3002
\u603b\u7ed3\u4ee5\u4e0a\u5185\u5bb9\uff0c\u5982\u8868 2-1 \u6240\u793a\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5b9e\u73b0\u3001\u6027\u80fd\u548c\u9002\u7528\u6027\u4e0a\u6709\u6240\u4e0d\u540c\u3002
\u8868 2-1 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52\u7279\u70b9\u5bf9\u6bd4
\u8fed\u4ee3 \u9012\u5f52 \u5b9e\u73b0\u65b9\u5f0f \u5faa\u73af\u7ed3\u6784 \u51fd\u6570\u8c03\u7528\u81ea\u8eab \u65f6\u95f4\u6548\u7387 \u6548\u7387\u901a\u5e38\u8f83\u9ad8\uff0c\u65e0\u51fd\u6570\u8c03\u7528\u5f00\u9500 \u6bcf\u6b21\u51fd\u6570\u8c03\u7528\u90fd\u4f1a\u4ea7\u751f\u5f00\u9500 \u5185\u5b58\u4f7f\u7528 \u901a\u5e38\u4f7f\u7528\u56fa\u5b9a\u5927\u5c0f\u7684\u5185\u5b58\u7a7a\u95f4 \u7d2f\u79ef\u51fd\u6570\u8c03\u7528\u53ef\u80fd\u4f7f\u7528\u5927\u91cf\u7684\u6808\u5e27\u7a7a\u95f4 \u9002\u7528\u95ee\u9898 \u9002\u7528\u4e8e\u7b80\u5355\u5faa\u73af\u4efb\u52a1\uff0c\u4ee3\u7801\u76f4\u89c2\u3001\u53ef\u8bfb\u6027\u597d \u9002\u7528\u4e8e\u5b50\u95ee\u9898\u5206\u89e3\uff0c\u5982\u6811\u3001\u56fe\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u7b49\uff0c\u4ee3\u7801\u7ed3\u6784\u7b80\u6d01\u3001\u6e05\u6670Tip
\u5982\u679c\u611f\u89c9\u4ee5\u4e0b\u5185\u5bb9\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5728\u8bfb\u5b8c\u201c\u6808\u201d\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\u3002
\u90a3\u4e48\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5177\u6709\u4ec0\u4e48\u5185\u5728\u8054\u7cfb\u5462\uff1f\u4ee5\u4e0a\u8ff0\u9012\u5f52\u51fd\u6570\u4e3a\u4f8b\uff0c\u6c42\u548c\u64cd\u4f5c\u5728\u9012\u5f52\u7684\u201c\u5f52\u201d\u9636\u6bb5\u8fdb\u884c\u3002\u8fd9\u610f\u5473\u7740\u6700\u521d\u88ab\u8c03\u7528\u7684\u51fd\u6570\u5b9e\u9645\u4e0a\u662f\u6700\u540e\u5b8c\u6210\u5176\u6c42\u548c\u64cd\u4f5c\u7684\uff0c\u8fd9\u79cd\u5de5\u4f5c\u673a\u5236\u4e0e\u6808\u7684\u201c\u5148\u5165\u540e\u51fa\u201d\u539f\u5219\u5f02\u66f2\u540c\u5de5\u3002
\u4e8b\u5b9e\u4e0a\uff0c\u201c\u8c03\u7528\u6808\u201d\u548c\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u8fd9\u7c7b\u9012\u5f52\u672f\u8bed\u5df2\u7ecf\u6697\u793a\u4e86\u9012\u5f52\u4e0e\u6808\u4e4b\u95f4\u7684\u5bc6\u5207\u5173\u7cfb\u3002
- \u9012\uff1a\u5f53\u51fd\u6570\u88ab\u8c03\u7528\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5728\u201c\u8c03\u7528\u6808\u201d\u4e0a\u4e3a\u8be5\u51fd\u6570\u5206\u914d\u65b0\u7684\u6808\u5e27\uff0c\u7528\u4e8e\u5b58\u50a8\u51fd\u6570\u7684\u5c40\u90e8\u53d8\u91cf\u3001\u53c2\u6570\u3001\u8fd4\u56de\u5730\u5740\u7b49\u6570\u636e\u3002
- \u5f52\uff1a\u5f53\u51fd\u6570\u5b8c\u6210\u6267\u884c\u5e76\u8fd4\u56de\u65f6\uff0c\u5bf9\u5e94\u7684\u6808\u5e27\u4f1a\u88ab\u4ece\u201c\u8c03\u7528\u6808\u201d\u4e0a\u79fb\u9664\uff0c\u6062\u590d\u4e4b\u524d\u51fd\u6570\u7684\u6267\u884c\u73af\u5883\u3002
\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u8c03\u7528\u6808\u7684\u884c\u4e3a\uff0c\u4ece\u800c\u5c06\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u5f62\u5f0f\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py
recursion.cppdef for_loop_recur(n: int) -> int:\n \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n # \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack = []\n res = 0\n # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in range(n, 0, -1):\n # \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.append(i)\n # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while stack:\n # \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop()\n # res = 1+2+3+...+n\n return res\n
recursion.java/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack<int> stack;\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.empty()) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.top();\n stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.cs/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n Stack<Integer> stack = new Stack<>();\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.isEmpty()) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.go/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n Stack<int> stack = new();\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.Push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.Count > 0) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.Pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.swift/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n stack := list.New()\n res := 0\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i := n; i > 0; i-- {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.PushBack(i)\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n for stack.Len() != 0 {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.Back().Value.(int)\n stack.Remove(stack.Back())\n }\n // res = 1+2+3+...+n\n return res\n}\n
recursion.js/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -> Int {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n var stack: [Int] = []\n var res = 0\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in stride(from: n, to: 0, by: -1) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.append(i)\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while !stack.isEmpty {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.removeLast()\n }\n // res = 1+2+3+...+n\n return res\n}\n
recursion.ts/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n const stack = [];\n let res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (let i = 1; i <= n; i++) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.length) { \n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.dart/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \n const stack: number[] = [];\n let res: number = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (let i = 1; i <= n; i++) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (stack.length) { \n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.rs/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n List<int> stack = [];\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.add(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (!stack.isEmpty) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.removeLast();\n }\n // res = 1+2+3+...+n\n return res;\n}\n
recursion.c/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -> i32 {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n let mut stack = Vec::new();\n let mut res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for i in (1..=n).rev() {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack.push(i);\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while !stack.is_empty() {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack.pop().unwrap();\n }\n // res = 1+2+3+...+n\n res\n}\n
recursion.zig/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n int stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\n int top = -1; // \u6808\u9876\u7d22\u5f15\n int res = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n for (int i = n; i > 0; i--) {\n // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n stack[1 + top++] = i;\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n while (top >= 0) {\n // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n res += stack[top--];\n }\n // res = 1+2+3+...+n\n return res;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n var stack: [n]i32 = undefined;\n var res: i32 = 0;\n // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n var i: usize = n;\n while (i > 0) {\n stack[i - 1] = @intCast(i);\n i -= 1;\n }\n // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n var index: usize = n;\n while (index > 0) {\n index -= 1;\n res += stack[index];\n }\n // res = 1+2+3+...+n\n return res;\n}\n\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u5f53\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u540e\uff0c\u4ee3\u7801\u53d8\u5f97\u66f4\u52a0\u590d\u6742\u4e86\u3002\u5c3d\u7ba1\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4e92\u76f8\u8f6c\u5316\uff0c\u4f46\u4e0d\u4e00\u5b9a\u503c\u5f97\u8fd9\u6837\u505a\uff0c\u6709\u4ee5\u4e0b\u4e24\u70b9\u539f\u56e0\u3002
- \u8f6c\u5316\u540e\u7684\u4ee3\u7801\u53ef\u80fd\u66f4\u52a0\u96be\u4ee5\u7406\u89e3\uff0c\u53ef\u8bfb\u6027\u66f4\u5dee\u3002
- \u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\u7684\u884c\u4e3a\u53ef\u80fd\u975e\u5e38\u56f0\u96be\u3002
\u603b\u4e4b\uff0c\u9009\u62e9\u8fed\u4ee3\u8fd8\u662f\u9012\u5f52\u53d6\u51b3\u4e8e\u7279\u5b9a\u95ee\u9898\u7684\u6027\u8d28\u3002\u5728\u7f16\u7a0b\u5b9e\u8df5\u4e2d\uff0c\u6743\u8861\u4e24\u8005\u7684\u4f18\u52a3\u5e76\u6839\u636e\u60c5\u5883\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u81f3\u5173\u91cd\u8981\u3002
"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":"\u5728\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u8ffd\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\u3002
- \u627e\u5230\u95ee\u9898\u89e3\u6cd5\uff1a\u7b97\u6cd5\u9700\u8981\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u5185\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002
- \u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\uff1a\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u5c3d\u53ef\u80fd\u9ad8\u6548\u7684\u7b97\u6cd5\u3002
\u4e5f\u5c31\u662f\u8bf4\uff0c\u5728\u80fd\u591f\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u5df2\u6210\u4e3a\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\uff0c\u5b83\u5305\u62ec\u4ee5\u4e0b\u4e24\u4e2a\u7ef4\u5ea6\u3002
- \u65f6\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002
- \u7a7a\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u7684\u5927\u5c0f\u3002
\u7b80\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bbe\u8ba1\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u3002\u800c\u6709\u6548\u5730\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u53ea\u6709\u8fd9\u6837\u6211\u4eec\u624d\u80fd\u5c06\u5404\u79cd\u7b97\u6cd5\u8fdb\u884c\u5bf9\u6bd4\uff0c\u8fdb\u800c\u6307\u5bfc\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u4f18\u5316\u8fc7\u7a0b\u3002
\u6548\u7387\u8bc4\u4f30\u65b9\u6cd5\u4e3b\u8981\u5206\u4e3a\u4e24\u79cd\uff1a\u5b9e\u9645\u6d4b\u8bd5\u3001\u7406\u8bba\u4f30\u7b97\u3002
"},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1 \u00a0 \u5b9e\u9645\u6d4b\u8bd5","text":"\u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5
A\u548c\u7b97\u6cd5B\uff0c\u5b83\u4eec\u90fd\u80fd\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u8fd9\u4e24\u4e2a\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u8fd0\u884c\u8fd9\u4e24\u4e2a\u7b97\u6cd5\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u5b83\u4eec\u7684\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u4e5f\u5b58\u5728\u8f83\u5927\u7684\u5c40\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u7b97\u6cd5\u7684\u6027\u80fd\u3002\u6bd4\u5982\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5
A\u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5B\u77ed\uff1b\u4f46\u5728\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u53ef\u80fd\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u8fdb\u884c\u6d4b\u8bd5\uff0c\u7edf\u8ba1\u5e73\u5747\u6548\u7387\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u8868\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5
"},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2 \u00a0 \u7406\u8bba\u4f30\u7b97","text":"A\u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5B\u77ed\uff1b\u800c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u53ef\u80fd\u6070\u6070\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u5f97\u5230\u6709\u8bf4\u670d\u529b\u7684\u7ed3\u8bba\uff0c\u6211\u4eec\u9700\u8981\u6d4b\u8bd5\u5404\u79cd\u89c4\u6a21\u7684\u8f93\u5165\u6570\u636e\uff0c\u800c\u8fd9\u9700\u8981\u8017\u8d39\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u3002\u7531\u4e8e\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u8f83\u5927\u7684\u5c40\u9650\u6027\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\u6765\u8bc4\u4f30\u7b97\u6cd5\u7684\u6548\u7387\u3002\u8fd9\u79cd\u4f30\u7b97\u65b9\u6cd5\u88ab\u79f0\u4e3a\u300c\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 asymptotic complexity analysis\u300d\uff0c\u7b80\u79f0\u300c\u590d\u6742\u5ea6\u5206\u6790\u300d\u3002
\u590d\u6742\u5ea6\u5206\u6790\u80fd\u591f\u4f53\u73b0\u7b97\u6cd5\u8fd0\u884c\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u5b83\u63cf\u8ff0\u4e86\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\uff0c\u7b97\u6cd5\u6267\u884c\u6240\u9700\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u5b9a\u4e49\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\u3002
- \u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u201d\u5206\u522b\u5bf9\u5e94\u300c\u65f6\u95f4\u590d\u6742\u5ea6 time complexity\u300d\u548c\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u3002
- \u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\u201d\u610f\u5473\u7740\u590d\u6742\u5ea6\u53cd\u6620\u4e86\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u4f53\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002
- \u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u5173\u6ce8\u7684\u4e0d\u662f\u8fd0\u884c\u65f6\u95f4\u6216\u5360\u7528\u7a7a\u95f4\u7684\u5177\u4f53\u503c\uff0c\u800c\u662f\u65f6\u95f4\u6216\u7a7a\u95f4\u589e\u957f\u7684\u201c\u5feb\u6162\u201d\u3002
\u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\uff0c\u4f53\u73b0\u5728\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u9762\u3002
- \u5b83\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002
- \u5b83\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002
Tip
\u5982\u679c\u4f60\u4ecd\u5bf9\u590d\u6742\u5ea6\u7684\u6982\u5ff5\u611f\u5230\u56f0\u60d1\uff0c\u65e0\u987b\u62c5\u5fc3\uff0c\u6211\u4eec\u4f1a\u5728\u540e\u7eed\u7ae0\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002
\u590d\u6742\u5ea6\u5206\u6790\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u628a\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u4f7f\u6211\u4eec\u53ef\u4ee5\u8861\u91cf\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u5bf9\u6bd4\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002
\u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u770b\uff0c\u590d\u6742\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9002\u5408\u4f5c\u4e3a\u6700\u5148\u4ecb\u7ecd\u7684\u5185\u5bb9\u3002\u7136\u800c\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u96be\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002
\u7efc\u4e0a\u6240\u8ff0\uff0c\u5efa\u8bae\u4f60\u5728\u6df1\u5165\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5148\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u5efa\u7acb\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u4ee5\u4fbf\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002
"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u5e38\u7c7b\u4f3c\uff0c\u53ea\u9700\u5c06\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u66ff\u6362\u4e3a\u201c\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u201d\u3002
"},{"location":"chapter_computational_complexity/space_complexity/#241","title":"2.4.1 \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002
- \u8f93\u5165\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\u3002
- \u6682\u5b58\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\u3002
- \u8f93\u51fa\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\u3002
\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u7edf\u8ba1\u8303\u56f4\u662f\u201c\u6682\u5b58\u7a7a\u95f4\u201d\u52a0\u4e0a\u201c\u8f93\u51fa\u7a7a\u95f4\u201d\u3002
\u6682\u5b58\u7a7a\u95f4\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002
- \u6682\u5b58\u6570\u636e\uff1a\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5404\u79cd\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u7b49\u3002
- \u6808\u5e27\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u5728\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\u90fd\u4f1a\u5728\u6808\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u540e\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002
- \u6307\u4ee4\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8ba1\u3002
\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5e8f\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u7edf\u8ba1\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u8f93\u51fa\u6570\u636e\u4e09\u90e8\u5206\uff0c\u5982\u56fe 2-15 \u6240\u793a\u3002
\u56fe 2-15 \u00a0 \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4
\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZigclass Node:\n \"\"\"\u7c7b\"\"\"\n def __init__(self, x: int):\n self.val: int = x # \u8282\u70b9\u503c\n self.next: Node | None = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\ndef function() -> int:\n \"\"\"\u51fd\u6570\"\"\"\n # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n\ndef algorithm(n) -> int: # \u8f93\u5165\u6570\u636e\n A = 0 # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\n b = 0 # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n node = Node(0) # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n c = function() # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return A + b + c # \u8f93\u51fa\u6570\u636e\n/* \u7ed3\u6784\u4f53 */\nstruct Node {\n int val;\n Node *next;\n Node(int x) : val(x), next(nullptr) {}\n};\n\n/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n const int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n Node* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n int c = func(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n int val;\n Node next;\n Node(int x) { val = x; }\n}\n\n/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n final int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n Node node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n int c = function(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node(int x) {\n int val = x;\n Node next;\n}\n\n/* \u51fd\u6570 */\nint Function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint Algorithm(int n) { // \u8f93\u5165\u6570\u636e\n const int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n Node node = new(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n int c = Function(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7ed3\u6784\u4f53 */\ntype node struct {\n val int\n next *node\n}\n\n/* \u521b\u5efa node \u7ed3\u6784\u4f53 */\nfunc newNode(val int) *node {\n return &node{val: val}\n}\n\n/* \u51fd\u6570 */\nfunc function() int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n}\n\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\n const a = 0 // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n b := 0 // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n newNode(0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n c := function() // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n var val: Int\n var next: Node?\n\n init(x: Int) {\n val = x\n }\n}\n\n/* \u51fd\u6570 */\nfunc function() -> Int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n}\n\nfunc algorithm(n: Int) -> Int { // \u8f93\u5165\u6570\u636e\n let a = 0 // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n var b = 0 // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n let node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n let c = function() // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n val;\n next;\n constructor(val) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = null; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\nfunction algorithm(n) { // \u8f93\u5165\u6570\u636e\n const a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n let b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n const node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n const c = constFunc(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n val: number;\n next: Node | null;\n constructor(val?: number) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.next = null; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\n const a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n let b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n const node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n const c = constFunc(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u7c7b */\nclass Node {\n int val;\n Node next;\n Node(this.val, [this.next]);\n}\n\n/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n const int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n Node node = Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n int c = function(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u7ed3\u6784\u4f53 */\nstruct Node {\n val: i32,\n next: Option<Rc<RefCell<Node>>>,\n}\n\n/* \u521b\u5efa Node \u7ed3\u6784\u4f53 */\nimpl Node {\n fn new(val: i32) -> Self {\n Self { val: val, next: None }\n }\n}\n\n/* \u51fd\u6570 */\nfn function() -> i32 { \n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nfn algorithm(n: i32) -> i32 { // \u8f93\u5165\u6570\u636e\n const a: i32 = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n let mut b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n let node = Node::new(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n let c = function(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n const int a = 0; // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n int b = 0; // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n int c = func(); // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n return a + b + c; // \u8f93\u51fa\u6570\u636e\n}\n
"},{"location":"chapter_computational_complexity/space_complexity/#242","title":"2.4.2 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"\n\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u9700\u5c06\u7edf\u8ba1\u5bf9\u8c61\u4ece\u201c\u64cd\u4f5c\u6570\u91cf\u201d\u8f6c\u4e3a\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002
\u800c\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u3002\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u9879\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u786e\u4fdd\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002
\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\u3002
- \u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\uff1a\u5f53 \\(n < 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u5f53 \\(n > 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4
nums\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002 - \u4ee5\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\uff1a\u4f8b\u5982\uff0c\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u5360\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4
nums\u65f6\uff0c\u7a0b\u5e8f\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
def algorithm(n: int):\n a = 0 # O(1)\n b = [0] * 10000 # O(1)\n if n > 10:\n nums = [0] * n # O(n)\nvoid algorithm(int n) {\n int a = 0; // O(1)\n vector<int> b(10000); // O(1)\n if (n > 10)\n vector<int> nums(n); // O(n)\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n int[] b = new int[10000]; // O(1)\n if (n > 10)\n int[] nums = new int[n]; // O(n)\n}\nvoid Algorithm(int n) {\n int a = 0; // O(1)\n int[] b = new int[10000]; // O(1)\n if (n > 10) {\n int[] nums = new int[n]; // O(n)\n }\n}\nfunc algorithm(n int) {\n a := 0 // O(1)\n b := make([]int, 10000) // O(1)\n var nums []int\n if n > 10 {\n nums := make([]int, n) // O(n)\n }\n fmt.Println(a, b, nums)\n}\nfunc algorithm(n: Int) {\n let a = 0 // O(1)\n let b = Array(repeating: 0, count: 10000) // O(1)\n if n > 10 {\n let nums = Array(repeating: 0, count: n) // O(n)\n }\n}\nfunction algorithm(n) {\n const a = 0; // O(1)\n const b = new Array(10000); // O(1)\n if (n > 10) {\n const nums = new Array(n); // O(n)\n }\n}\nfunction algorithm(n: number): void {\n const a = 0; // O(1)\n const b = new Array(10000); // O(1)\n if (n > 10) {\n const nums = new Array(n); // O(n)\n }\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n List<int> b = List.filled(10000, 0); // O(1)\n if (n > 10) {\n List<int> nums = List.filled(n, 0); // O(n)\n }\n}\nfn algorithm(n: i32) {\n let a = 0; // O(1)\n let b = [0; 10000]; // O(1)\n if n > 10 {\n let nums = vec![0; n as usize]; // O(n)\n }\n}\nvoid algorithm(int n) {\n int a = 0; // O(1)\n int b[10000]; // O(1)\n if (n > 10)\n int nums[n] = {0}; // O(n)\n}\n\n\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef function() -> int:\n # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n\ndef loop(n: int):\n \"\"\"\u5faa\u73af O(1)\"\"\"\n for _ in range(n):\n function()\n\ndef recur(n: int) -> int:\n \"\"\"\u9012\u5f52 O(n)\"\"\"\n if n == 1: return\n return recur(n - 1)\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nint Function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid Loop(int n) {\n for (int i = 0; i < n; i++) {\n Function();\n }\n}\n/* \u9012\u5f52 O(n) */\nint Recur(int n) {\n if (n == 1) return 1;\n return Recur(n - 1);\n}\nfunc function() int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\n for i := 0; i < n; i++ {\n function()\n }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\n if n == 1 {\n return\n }\n recur(n - 1)\n}\n@discardableResult\nfunc function() -> Int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\n for _ in 0 ..< n {\n function()\n }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\n if n == 1 {\n return\n }\n recur(n: n - 1)\n}\nfunction constFunc() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\n if (n === 1) return;\n return recur(n - 1);\n}\nfunction constFunc(): number {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\n if (n === 1) return;\n return recur(n - 1);\n}\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\nfn function() -> i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nfn loop(n: i32) {\n for i in 0..n {\n function();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(n: i32) {\n if n == 1 {\n return;\n }\n recur(n - 1);\n}\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n if (n == 1) return;\n return recur(n - 1);\n}\n\n\u51fd\u6570
loop()\u548crecur()\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u4f46\u7a7a\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u3002- \u51fd\u6570
loop()\u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21function()\uff0c\u6bcf\u8f6e\u4e2d\u7684function()\u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002 - \u9012\u5f52\u51fd\u6570
recur()\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684recur()\uff0c\u4ece\u800c\u5360\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002
\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u56fe 2-16 \u5c55\u793a\u4e86\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09\u3002
\\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n^2) < O(2^n) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} \\end{aligned} \\]\u56fe 2-16 \u00a0 \u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b
"},{"location":"chapter_computational_complexity/space_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"\u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002
\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u56e0\u6b64\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef function() -> int:\n \"\"\"\u51fd\u6570\"\"\"\n # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n\ndef constant(n: int):\n \"\"\"\u5e38\u6570\u9636\"\"\"\n # \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n a = 0\n nums = [0] * 10000\n node = ListNode(0)\n # \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in range(n):\n c = 0\n # \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in range(n):\n function()\n
space_complexity.java/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const int a = 0;\n int b = 0;\n vector<int> nums(10000);\n ListNode node(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n
space_complexity.cs/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n final int a = 0;\n int b = 0;\n int[] nums = new int[10000];\n ListNode node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n function();\n }\n}\n
space_complexity.go/* \u51fd\u6570 */\nint Function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n int a = 0;\n int b = 0;\n int[] nums = new int[10000];\n ListNode node = new(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n Function();\n }\n}\n
space_complexity.swift/* \u51fd\u6570 */\nfunc function() int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0\n b := 0\n nums := make([]int, 10000)\n ListNode := newNode(0)\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n var c int\n for i := 0; i < n; i++ {\n c = 0\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for i := 0; i < n; i++ {\n function()\n }\n fmt.Println(a, b, nums, c, ListNode)\n}\n
space_complexity.js/* \u51fd\u6570 */\n@discardableResult\nfunc function() -> Int {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n let a = 0\n var b = 0\n let nums = Array(repeating: 0, count: 10000)\n let node = ListNode(x: 0)\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in 0 ..< n {\n let c = 0\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for _ in 0 ..< n {\n function()\n }\n}\n
space_complexity.ts/* \u51fd\u6570 */\nfunction constFunc() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0;\n const b = 0;\n const nums = new Array(10000);\n const node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n const c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n
space_complexity.dart/* \u51fd\u6570 */\nfunction constFunc(): number {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a = 0;\n const b = 0;\n const nums = new Array(10000);\n const node = new ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n const c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (let i = 0; i < n; i++) {\n constFunc();\n }\n}\n
space_complexity.rs/* \u51fd\u6570 */\nint function() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n final int a = 0;\n int b = 0;\n List<int> nums = List.filled(10000, 0);\n ListNode node = ListNode(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (var i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (var i = 0; i < n; i++) {\n function();\n }\n}\n
space_complexity.c/* \u51fd\u6570 */\nfn function() ->i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const A: i32 = 0;\n let b = 0;\n let nums = vec![0; 10000];\n let node = ListNode::new(0);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for i in 0..n {\n let c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for i in 0..n {\n function();\n }\n}\n
space_complexity.zig/* \u51fd\u6570 */\nint func() {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const int a = 0;\n int b = 0;\n int nums[1000];\n ListNode *node = newListNode(0);\n free(node);\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n int c = 0;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n for (int i = 0; i < n; i++) {\n func();\n }\n}\n
"},{"location":"chapter_computational_complexity/space_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"// \u51fd\u6570\nfn function() i32 {\n // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n return 0;\n}\n\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n const a: i32 = 0;\n var b: i32 = 0;\n var nums = [_]i32{0}**10000;\n var node = inc.ListNode(i32){.val = 0};\n var i: i32 = 0;\n // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n while (i < n) : (i += 1) {\n var c: i32 = 0;\n _ = c;\n }\n // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n i = 0;\n while (i < n) : (i += 1) {\n _ = function();\n }\n _ = a;\n _ = b;\n _ = nums;\n _ = node;\n}\n\u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef linear(n: int):\n \"\"\"\u7ebf\u6027\u9636\"\"\"\n # \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n nums = [0] * n\n # \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n hmap = dict[int, str]()\n for i in range(n):\n hmap[i] = str(i)\n
space_complexity.java/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n vector<int> nums(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n vector<ListNode> nodes;\n for (int i = 0; i < n; i++) {\n nodes.push_back(ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n unordered_map<int, string> map;\n for (int i = 0; i < n; i++) {\n map[i] = to_string(i);\n }\n}\n
space_complexity.cs/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int[] nums = new int[n];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n nodes.add(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Map<Integer, String> map = new HashMap<>();\n for (int i = 0; i < n; i++) {\n map.put(i, String.valueOf(i));\n }\n}\n
space_complexity.go/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int[] nums = new int[n];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = [];\n for (int i = 0; i < n; i++) {\n nodes.Add(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Dictionary<int, string> map = [];\n for (int i = 0; i < n; i++) {\n map.Add(i, i.ToString());\n }\n}\n
space_complexity.swift/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n _ = make([]int, n)\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var nodes []*node\n for i := 0; i < n; i++ {\n nodes = append(nodes, newNode(i))\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n m := make(map[int]string, n)\n for i := 0; i < n; i++ {\n m[i] = strconv.Itoa(i)\n }\n}\n
space_complexity.js/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n let nums = Array(repeating: 0, count: n)\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let nodes = (0 ..< n).map { ListNode(x: $0) }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, \"\\($0)\") })\n}\n
space_complexity.ts/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n const nums = new Array(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const nodes = [];\n for (let i = 0; i < n; i++) {\n nodes.push(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const map = new Map();\n for (let i = 0; i < n; i++) {\n map.set(i, i.toString());\n }\n}\n
space_complexity.dart/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n const nums = new Array(n);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const nodes: ListNode[] = [];\n for (let i = 0; i < n; i++) {\n nodes.push(new ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n const map = new Map();\n for (let i = 0; i < n; i++) {\n map.set(i, i.toString());\n }\n}\n
space_complexity.rs/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n List<int> nums = List.filled(n, 0);\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n List<ListNode> nodes = [];\n for (var i = 0; i < n; i++) {\n nodes.add(ListNode(i));\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n Map<int, String> map = HashMap();\n for (var i = 0; i < n; i++) {\n map.putIfAbsent(i, () => i.toString());\n }\n}\n
space_complexity.c/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n let mut nums = vec![0; n as usize];\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let mut nodes = Vec::new();\n for i in 0..n {\n nodes.push(ListNode::new(i))\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n let mut map = HashMap::new();\n for i in 0..n {\n map.insert(i, i.to_string());\n }\n}\n
space_complexity.zig/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n int key;\n int val;\n UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n int *nums = malloc(sizeof(int) * n);\n free(nums);\n\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n ListNode **nodes = malloc(sizeof(ListNode *) * n);\n for (int i = 0; i < n; i++) {\n nodes[i] = newListNode(i);\n }\n // \u5185\u5b58\u91ca\u653e\n for (int i = 0; i < n; i++) {\n free(nodes[i]);\n }\n free(nodes);\n\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n HashTable *h = NULL;\n for (int i = 0; i < n; i++) {\n HashTable *tmp = malloc(sizeof(HashTable));\n tmp->key = i;\n tmp->val = i;\n HASH_ADD_INT(h, key, tmp);\n }\n\n // \u5185\u5b58\u91ca\u653e\n HashTable *curr, *tmp;\n HASH_ITER(hh, h, curr, tmp) {\n HASH_DEL(h, curr);\n free(curr);\n }\n}\n// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n var nums = [_]i32{0}**n;\n // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n defer nodes.deinit();\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n try nodes.append(i);\n }\n // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n defer map.deinit();\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n defer std.heap.page_allocator.free(string);\n try map.put(i, string);\n }\n _ = nums;\n}\n\u5982\u56fe 2-17 \u6240\u793a\uff0c\u6b64\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5373\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.pylinear_recur()\u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\uff1a
space_complexity.cppdef linear_recur(n: int):\n \"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n print(\"\u9012\u5f52 n =\", n)\n if n == 1:\n return\n linear_recur(n - 1)\n
space_complexity.java/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n cout << \"\u9012\u5f52 n = \" << n << endl;\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n
space_complexity.cs/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n System.out.println(\"\u9012\u5f52 n = \" + n);\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n
space_complexity.go/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\n Console.WriteLine(\"\u9012\u5f52 n = \" + n);\n if (n == 1) return;\n LinearRecur(n - 1);\n}\n
space_complexity.swift/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\n fmt.Println(\"\u9012\u5f52 n =\", n)\n if n == 1 {\n return\n }\n spaceLinearRecur(n - 1)\n}\n
space_complexity.js/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\n print(\"\u9012\u5f52 n = \\(n)\")\n if n == 1 {\n return\n }\n linearRecur(n: n - 1)\n}\n
space_complexity.ts/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\n console.log(`\u9012\u5f52 n = ${n}`);\n if (n === 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.dart/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\n console.log(`\u9012\u5f52 n = ${n}`);\n if (n === 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.rs/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n print('\u9012\u5f52 n = $n');\n if (n == 1) return;\n linearRecur(n - 1);\n}\n
space_complexity.c/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\n println!(\"\u9012\u5f52 n = {}\", n);\n if n == 1 {return};\n linear_recur(n - 1);\n}\n
space_complexity.zig/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n printf(\"\u9012\u5f52 n = %d\\r\\n\", n);\n if (n == 1)\n return;\n linearRecur(n - 1);\n}\n// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\n std.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\n if (n == 1) return;\n linearRecur(n - 1);\n}\n\u56fe 2-17 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6
"},{"location":"chapter_computational_complexity/space_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"\u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u77e9\u9635\u548c\u56fe\uff0c\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef quadratic(n: int):\n \"\"\"\u5e73\u65b9\u9636\"\"\"\n # \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n num_matrix = [[0] * n for _ in range(n)]\n
space_complexity.java/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n vector<vector<int>> numMatrix;\n for (int i = 0; i < n; i++) {\n vector<int> tmp;\n for (int j = 0; j < n; j++) {\n tmp.push_back(0);\n }\n numMatrix.push_back(tmp);\n }\n}\n
space_complexity.cs/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n int[][] numMatrix = new int[n][n];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<Integer>> numList = new ArrayList<>();\n for (int i = 0; i < n; i++) {\n List<Integer> tmp = new ArrayList<>();\n for (int j = 0; j < n; j++) {\n tmp.add(0);\n }\n numList.add(tmp);\n }\n}\n
space_complexity.go/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n int[,] numMatrix = new int[n, n];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numList = [];\n for (int i = 0; i < n; i++) {\n List<int> tmp = [];\n for (int j = 0; j < n; j++) {\n tmp.Add(0);\n }\n numList.Add(tmp);\n }\n}\n
space_complexity.swift/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n numMatrix := make([][]int, n)\n for i := 0; i < n; i++ {\n numMatrix[i] = make([]int, n)\n }\n}\n
space_complexity.js/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n
space_complexity.ts/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numMatrix = Array(n)\n .fill(null)\n .map(() => Array(n).fill(null));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numList = [];\n for (let i = 0; i < n; i++) {\n const tmp = [];\n for (let j = 0; j < n; j++) {\n tmp.push(0);\n }\n numList.push(tmp);\n }\n}\n
space_complexity.dart/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numMatrix = Array(n)\n .fill(null)\n .map(() => Array(n).fill(null));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n const numList = [];\n for (let i = 0; i < n; i++) {\n const tmp = [];\n for (let j = 0; j < n; j++) {\n tmp.push(0);\n }\n numList.push(tmp);\n }\n}\n
space_complexity.rs/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numMatrix = List.generate(n, (_) => List.filled(n, 0));\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n List<List<int>> numList = [];\n for (var i = 0; i < n; i++) {\n List<int> tmp = [];\n for (int j = 0; j < n; j++) {\n tmp.add(0);\n }\n numList.add(tmp);\n }\n}\n
space_complexity.c/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n let num_matrix = vec![vec![0; n as usize]; n as usize];\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n let mut num_list = Vec::new();\n for i in 0..n {\n let mut tmp = Vec::new();\n for j in 0..n {\n tmp.push(0);\n }\n num_list.push(tmp);\n }\n}\n
space_complexity.zig/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n int **numMatrix = malloc(sizeof(int *) * n);\n for (int i = 0; i < n; i++) {\n int *tmp = malloc(sizeof(int) * n);\n for (int j = 0; j < n; j++) {\n tmp[j] = 0;\n }\n numMatrix[i] = tmp;\n }\n\n // \u5185\u5b58\u91ca\u653e\n for (int i = 0; i < n; i++) {\n free(numMatrix[i]);\n }\n free(numMatrix);\n}\n// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n defer nodes.deinit();\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n defer tmp.deinit();\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n try tmp.append(0);\n }\n try nodes.append(tmp);\n }\n}\n\u5982\u56fe 2-18 \u6240\u793a\uff0c\u8be5\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5728\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(n / 2\\) \uff0c\u56e0\u6b64\u603b\u4f53\u5360\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef quadratic_recur(n: int) -> int:\n \"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n <= 0:\n return 0\n # \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n nums = [0] * n\n return quadratic_recur(n - 1)\n
space_complexity.java/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n vector<int> nums(n);\n cout << \"\u9012\u5f52 n = \" << n << \" \u4e2d\u7684 nums \u957f\u5ea6 = \" << nums.size() << endl;\n return quadraticRecur(n - 1);\n}\n
space_complexity.cs/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n int[] nums = new int[n];\n System.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\n return quadraticRecur(n - 1);\n}\n
space_complexity.go/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\n if (n <= 0) return 0;\n int[] nums = new int[n];\n Console.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\n return QuadraticRecur(n - 1);\n}\n
space_complexity.swift/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n if n <= 0 {\n return 0\n }\n nums := make([]int, n)\n fmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\n return spaceQuadraticRecur(n - 1)\n}\n
space_complexity.js/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -> Int {\n if n <= 0 {\n return 0\n }\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n let nums = Array(repeating: 0, count: n)\n print(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\n return quadraticRecur(n: n - 1)\n}\n
space_complexity.ts/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\n if (n <= 0) return 0;\n const nums = new Array(n);\n console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n return quadraticRecur(n - 1);\n}\n
space_complexity.dart/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\n if (n <= 0) return 0;\n const nums = new Array(n);\n console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n return quadraticRecur(n - 1);\n}\n
space_complexity.rs/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0) return 0;\n List<int> nums = List.filled(n, 0);\n print('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\n return quadraticRecur(n - 1);\n}\n
space_complexity.c/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -> i32 {\n if n <= 0 {return 0};\n // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n let nums = vec![0; n as usize];\n println!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\n return quadratic_recur(n - 1);\n}\n
space_complexity.zig/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n if (n <= 0)\n return 0;\n int *nums = malloc(sizeof(int) * n);\n printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\n int res = quadraticRecur(n - 1);\n free(nums);\n return res;\n}\n// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n if (n <= 0) return 0;\n var nums = [_]i32{0}**n;\n std.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\n return quadraticRecur(n - 1);\n}\n\u56fe 2-18 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6
"},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"\u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u89c2\u5bdf\u56fe 2-19 \uff0c\u5c42\u6570\u4e3a \\(n\\) \u7684\u201c\u6ee1\u4e8c\u53c9\u6811\u201d\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u5360\u7528 \\(O(2^n)\\) \u7a7a\u95f4\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py
space_complexity.cppdef build_tree(n: int) -> TreeNode | None:\n \"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\n if n == 0:\n return None\n root = TreeNode(0)\n root.left = build_tree(n - 1)\n root.right = build_tree(n - 1)\n return root\n
space_complexity.java/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n if (n == 0)\n return nullptr;\n TreeNode *root = new TreeNode(0);\n root->left = buildTree(n - 1);\n root->right = buildTree(n - 1);\n return root;\n}\n
space_complexity.cs/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\n if (n == 0)\n return null;\n TreeNode root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.go/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\n if (n == 0) return null;\n TreeNode root = new(0) {\n left = BuildTree(n - 1),\n right = BuildTree(n - 1)\n };\n return root;\n}\n
space_complexity.swift/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\n if n == 0 {\n return nil\n }\n root := newTreeNode(0)\n root.left = buildTree(n - 1)\n root.right = buildTree(n - 1)\n return root\n}\n
space_complexity.js/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -> TreeNode? {\n if n == 0 {\n return nil\n }\n let root = TreeNode(x: 0)\n root.left = buildTree(n: n - 1)\n root.right = buildTree(n: n - 1)\n return root\n}\n
space_complexity.ts/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\n if (n === 0) return null;\n const root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.dart/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n if (n === 0) return null;\n const root = new TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.rs/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\n if (n == 0) return null;\n TreeNode root = TreeNode(0);\n root.left = buildTree(n - 1);\n root.right = buildTree(n - 1);\n return root;\n}\n
space_complexity.c/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -> Option<Rc<RefCell<TreeNode>>> {\n if n == 0 {return None};\n let root = TreeNode::new(0);\n root.borrow_mut().left = build_tree(n - 1);\n root.borrow_mut().right = build_tree(n - 1);\n return Some(root);\n}\n
space_complexity.zig/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n if (n == 0)\n return NULL;\n TreeNode *root = newTreeNode(0);\n root->left = buildTree(n - 1);\n root->right = buildTree(n - 1);\n return root;\n}\n// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n if (n == 0) return null;\n const root = try mem_allocator.create(inc.TreeNode(i32));\n root.init(0);\n root.left = try buildTree(mem_allocator, n - 1);\n root.right = try buildTree(mem_allocator, n - 1);\n return root;\n}\n\u56fe 2-19 \u00a0 \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6
"},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"\u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u3002\u4f8b\u5982\u5f52\u5e76\u6392\u5e8f\uff0c\u8f93\u5165\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u6bcf\u8f6e\u9012\u5f52\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6808\u5e27\u7a7a\u95f4\u3002
\u518d\u4f8b\u5982\u5c06\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\uff0c\u8f93\u5165\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\log_{10} n + 1\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\log_{10} n + 1\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n + 1) = O(\\log n)\\) \u3002
"},{"location":"chapter_computational_complexity/space_complexity/#244","title":"2.4.4 \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u8fbe\u5230\u6700\u4f18\u3002\u7136\u800c\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u975e\u5e38\u56f0\u96be\u3002
\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u5c06\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff1b\u53cd\u4e4b\uff0c\u5219\u79f0\u4e3a\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002
\u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\uff0c\u56e0\u6b64\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u5f53\u7136\uff0c\u5728\u6570\u636e\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u63a7\u5236\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u975e\u5e38\u91cd\u8981\u3002
"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_computational_complexity/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"\u7b97\u6cd5\u6548\u7387\u8bc4\u4f30
- \u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e24\u4e2a\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\u3002
- \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6d88\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5f71\u54cd\uff0c\u4e14\u4f1a\u8017\u8d39\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002
- \u590d\u6742\u5ea6\u5206\u6790\u53ef\u4ee5\u6d88\u9664\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u80fd\u591f\u63ed\u793a\u7b97\u6cd5\u5728\u4e0d\u540c\u6570\u636e\u89c4\u6a21\u4e0b\u7684\u6548\u7387\u3002
\u65f6\u95f4\u590d\u6742\u5ea6
- \u65f6\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f93\u5165\u7684\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u3002
- \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u5411\u6b63\u65e0\u7a77\u65f6\uff0c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u7684\u589e\u957f\u7ea7\u522b\u3002
- \u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002
- \u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n \\log n)\\)\u3001\\(O(n^2)\\)\u3001\\(O(2^n)\\) \u548c \\(O(n!)\\) \u7b49\u3002
- \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u4e00\u822c\u9700\u8981\u6ee1\u8db3\u4e25\u683c\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002
- \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53cd\u6620\u7b97\u6cd5\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u5206\u5e03\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002
\u7a7a\u95f4\u590d\u6742\u5ea6
- \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u4f5c\u7528\u7c7b\u4f3c\u4e8e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\u3002
- \u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u7eb3\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u6307\u4ee4\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u901a\u5e38\u4ec5\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u5f71\u54cd\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
- \u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u523b\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
- \u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n^2)\\) \u548c \\(O(2^n)\\) \u7b49\u3002
\u5c3e\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u5417\uff1f
\u7406\u8bba\u4e0a\uff0c\u5c3e\u9012\u5f52\u51fd\u6570\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u8fc7\u7edd\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\u3001Python\u3001C++\u3001Go\u3001C# \u7b49\uff09\u4e0d\u652f\u6301\u81ea\u52a8\u4f18\u5316\u5c3e\u9012\u5f52\uff0c\u56e0\u6b64\u901a\u5e38\u8ba4\u4e3a\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \u3002
\u51fd\u6570\u548c\u65b9\u6cd5\u8fd9\u4e24\u4e2a\u672f\u8bed\u7684\u533a\u522b\u662f\u4ec0\u4e48\uff1f
\u300c\u51fd\u6570 function\u300d\u53ef\u4ee5\u88ab\u72ec\u7acb\u6267\u884c\uff0c\u6240\u6709\u53c2\u6570\u90fd\u4ee5\u663e\u5f0f\u4f20\u9012\u3002\u300c\u65b9\u6cd5 method\u300d\u4e0e\u4e00\u4e2a\u5bf9\u8c61\u5173\u8054\uff0c\u88ab\u9690\u5f0f\u4f20\u9012\u7ed9\u8c03\u7528\u5b83\u7684\u5bf9\u8c61\uff0c\u80fd\u591f\u5bf9\u7c7b\u7684\u5b9e\u4f8b\u4e2d\u5305\u542b\u7684\u6570\u636e\u8fdb\u884c\u64cd\u4f5c\u3002
\u4e0b\u9762\u4ee5\u51e0\u79cd\u5e38\u89c1\u7684\u7f16\u7a0b\u8bed\u8a00\u4e3a\u4f8b\u6765\u8bf4\u660e\u3002
- C \u8bed\u8a00\u662f\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\u8bed\u8a00\uff0c\u6ca1\u6709\u9762\u5411\u5bf9\u8c61\u7684\u6982\u5ff5\uff0c\u6240\u4ee5\u53ea\u6709\u51fd\u6570\u3002\u4f46\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u521b\u5efa\u7ed3\u6784\u4f53\uff08struct\uff09\u6765\u6a21\u62df\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff0c\u4e0e\u7ed3\u6784\u4f53\u76f8\u5173\u8054\u7684\u51fd\u6570\u5c31\u76f8\u5f53\u4e8e\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u65b9\u6cd5\u3002
- Java \u548c C# \u662f\u9762\u5411\u5bf9\u8c61\u7684\u7f16\u7a0b\u8bed\u8a00\uff0c\u4ee3\u7801\u5757\uff08\u65b9\u6cd5\uff09\u901a\u5e38\u4f5c\u4e3a\u67d0\u4e2a\u7c7b\u7684\u4e00\u90e8\u5206\u3002\u9759\u6001\u65b9\u6cd5\u7684\u884c\u4e3a\u7c7b\u4f3c\u4e8e\u51fd\u6570\uff0c\u56e0\u4e3a\u5b83\u88ab\u7ed1\u5b9a\u5728\u7c7b\u4e0a\uff0c\u4e0d\u80fd\u8bbf\u95ee\u7279\u5b9a\u7684\u5b9e\u4f8b\u53d8\u91cf\u3002
- C++ \u548c Python \u65e2\u652f\u6301\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\uff08\u51fd\u6570\uff09\uff0c\u4e5f\u652f\u6301\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff08\u65b9\u6cd5\uff09\u3002
\u56fe\u89e3\u201c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\uff1f
\u4e0d\u662f\uff0c\u8be5\u56fe\u5c55\u793a\u7684\u662f\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5176\u53cd\u6620\u7684\u662f\u589e\u957f\u8d8b\u52bf\uff0c\u800c\u4e0d\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\u3002
\u5047\u8bbe\u53d6 \\(n = 8\\) \uff0c\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u6bcf\u6761\u66f2\u7ebf\u7684\u503c\u4e0e\u51fd\u6570\u5bf9\u5e94\u4e0d\u4e0a\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6761\u66f2\u7ebf\u90fd\u5305\u542b\u4e00\u4e2a\u5e38\u6570\u9879\uff0c\u7528\u4e8e\u5c06\u53d6\u503c\u8303\u56f4\u538b\u7f29\u5230\u4e00\u4e2a\u89c6\u89c9\u8212\u9002\u7684\u8303\u56f4\u5185\u3002
\u5728\u5b9e\u9645\u4e2d\uff0c\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u4e2a\u65b9\u6cd5\u7684\u201c\u5e38\u6570\u9879\u201d\u590d\u6742\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u65e0\u6cd5\u4ec5\u51ed\u590d\u6742\u5ea6\u6765\u9009\u62e9 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u4f18\u89e3\u6cd5\u3002\u4f46\u5bf9\u4e8e \\(n = 8^5\\) \u5c31\u5f88\u597d\u9009\u4e86\uff0c\u8fd9\u65f6\u589e\u957f\u8d8b\u52bf\u5df2\u7ecf\u5360\u4e3b\u5bfc\u4e86\u3002
"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":"\u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u53cd\u6620\u7b97\u6cd5\u7684\u6548\u7387\u3002\u5982\u679c\u6211\u4eec\u60f3\u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u5e94\u8be5\u5982\u4f55\u64cd\u4f5c\u5462\uff1f
- \u786e\u5b9a\u8fd0\u884c\u5e73\u53f0\uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u56e0\u7d20\u90fd\u4f1a\u5f71\u54cd\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002
- \u8bc4\u4f30\u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c
+\u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c*\u9700\u8981 10 ns \uff0c\u6253\u5370\u64cd\u4f5cprint()\u9700\u8981 5 ns \u7b49\u3002 - \u7edf\u8ba1\u4ee3\u7801\u4e2d\u6240\u6709\u7684\u8ba1\u7b97\u64cd\u4f5c\uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002
\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig# \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int):\n a = 2 # 1 ns\n a = a + 1 # 1 ns\n a = a * 2 # 10 ns\n # \u5faa\u73af n \u6b21\n for _ in range(n): # 1 ns\n print(0) # 5 ns\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n cout << 0 << endl; // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n System.out.println(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid Algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n Console.WriteLine(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\n a := 2 // 1 ns\n a = a + 1 // 1 ns\n a = a * 2 // 10 ns\n // \u5faa\u73af n \u6b21\n for i := 0; i < n; i++ { // 1 ns\n fmt.Println(a) // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\n var a = 2 // 1 ns\n a = a + 1 // 1 ns\n a = a * 2 // 10 ns\n // \u5faa\u73af n \u6b21\n for _ in 0 ..< n { // 1 ns\n print(0) // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\n var a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n console.log(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\n var a: number = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n console.log(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n print(0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: i32) {\n let mut a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for _ in 0..n { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n println!(\"{}\", 0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n int a = 2; // 1 ns\n a = a + 1; // 1 ns\n a = a * 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n printf(\"%d\", 0); // 5 ns\n }\n}\n// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: usize) void {\n var a: i32 = 2; // 1 ns\n a += 1; // 1 ns\n a *= 2; // 10 ns\n // \u5faa\u73af n \u6b21\n for (0..n) |_| { // 1 ns\n std.debug.print(\"{}\\n\", .{0}); // 5 ns\n }\n}\n\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u4e3a \\((6n + 12)\\) ns \uff1a
\\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\]\u4f46\u5b9e\u9645\u4e0a\uff0c\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u5c06\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u56e0\u4e3a\u7b97\u6cd5\u9700\u8981\u5728\u5404\u79cd\u4e0d\u540c\u7684\u5e73\u53f0\u4e0a\u8fd0\u884c\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u7ed9\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#231","title":"2.3.1 \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002
\u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u6bd4\u8f83\u62bd\u8c61\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5
PythonC++JavaC#GoSwiftJSTSDartRustCZigA\u3001B\u548cC\uff1a# \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int):\n print(0)\n# \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int):\n for _ in range(n):\n print(0)\n# \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int):\n for _ in range(1000000):\n print(0)\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n cout << 0 << endl;\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n cout << 0 << endl;\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n cout << 0 << endl;\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n System.out.println(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n System.out.println(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n System.out.println(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmA(int n) {\n Console.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid AlgorithmB(int n) {\n for (int i = 0; i < n; i++) {\n Console.WriteLine(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmC(int n) {\n for (int i = 0; i < 1000000; i++) {\n Console.WriteLine(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\n fmt.Println(0)\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\n for i := 0; i < n; i++ {\n fmt.Println(0)\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\n for i := 0; i < 1000000; i++ {\n fmt.Println(0)\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\n print(0)\n}\n\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\n for _ in 0 ..< n {\n print(0)\n }\n}\n\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\n for _ in 0 ..< 1000000 {\n print(0)\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\n console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\n for (let i = 0; i < n; i++) {\n console.log(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\n for (let i = 0; i < 1000000; i++) {\n console.log(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\n console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\n for (let i = 0; i < n; i++) {\n console.log(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\n for (let i = 0; i < 1000000; i++) {\n console.log(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmA(int n) {\n print(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithmB(int n) {\n for (int i = 0; i < n; i++) {\n print(0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmC(int n) {\n for (int i = 0; i < 1000000; i++) {\n print(0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: i32) {\n println!(\"{}\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) {\n for _ in 0..n {\n println!(\"{}\", 0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) {\n for _ in 0..1000000 {\n println!(\"{}\", 0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n printf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n for (int i = 0; i < n; i++) {\n printf(\"%d\", 0);\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n for (int i = 0; i < 1000000; i++) {\n printf(\"%d\", 0);\n }\n}\n// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: usize) void {\n _ = n;\n std.debug.print(\"{}\\n\", .{0});\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) void {\n for (0..n) |_| {\n std.debug.print(\"{}\\n\", .{0});\n }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) void {\n _ = n;\n for (0..1000000) |_| { \n std.debug.print(\"{}\\n\", .{0});\n }\n}\n\u56fe 2-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4e09\u4e2a\u7b97\u6cd5\u51fd\u6570\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002
- \u7b97\u6cd5
A\u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002 - \u7b97\u6cd5
B\u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u5448\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u9636\u201d\u3002 - \u7b97\u6cd5
C\u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u867d\u7136\u8fd0\u884c\u65f6\u95f4\u5f88\u957f\uff0c\u4f46\u5b83\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64C\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548cA\u76f8\u540c\uff0c\u4ecd\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002
\u56fe 2-7 \u00a0 \u7b97\u6cd5 A\u3001B \u548c C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf
\u76f8\u8f83\u4e8e\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u7279\u70b9\u5462\uff1f
- \u65f6\u95f4\u590d\u6742\u5ea6\u80fd\u591f\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5
B\u7684\u8fd0\u884c\u65f6\u95f4\u5448\u7ebf\u6027\u589e\u957f\uff0c\u5728 \\(n > 1\\) \u65f6\u6bd4\u7b97\u6cd5A\u66f4\u6162\uff0c\u5728 \\(n > 1000000\\) \u65f6\u6bd4\u7b97\u6cd5C\u66f4\u6162\u3002\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u201c\u7ebf\u6027\u9636\u201d\u7684\u7b97\u6cd5\uff0c\u8fd9\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u7684\u542b\u4e49\u3002 - \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7b80\u4fbf\u3002\u663e\u7136\uff0c\u8fd0\u884c\u5e73\u53f0\u548c\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u6b64\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u89c6\u4e3a\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u4ece\u800c\u5c06\u201c\u8ba1\u7b97\u64cd\u4f5c\u8fd0\u884c\u65f6\u95f4\u7edf\u8ba1\u201d\u7b80\u5316\u4e3a\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u7edf\u8ba1\u201d\uff0c\u8fd9\u6837\u4e00\u6765\u4f30\u7b97\u96be\u5ea6\u5c31\u5927\u5927\u964d\u4f4e\u4e86\u3002
- \u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5c3d\u7ba1\u7b97\u6cd5
A\u548cC\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\u5dee\u522b\u5f88\u5927\u3002\u540c\u6837\uff0c\u5c3d\u7ba1\u7b97\u6cd5B\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4C\u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5B\u660e\u663e\u4f18\u4e8e\u7b97\u6cd5C\u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u5224\u65ad\u7b97\u6cd5\u6548\u7387\u7684\u9ad8\u4f4e\u3002\u5f53\u7136\uff0c\u5c3d\u7ba1\u5b58\u5728\u4e0a\u8ff0\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002
\u7ed9\u5b9a\u4e00\u4e2a\u8f93\u5165\u5927\u5c0f\u4e3a \\(n\\) \u7684\u51fd\u6570\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef algorithm(n: int):\n a = 1 # +1\n a = a + 1 # +1\n a = a * 2 # +1\n # \u5faa\u73af n \u6b21\n for i in range(n): # +1\n print(0) # +1\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n cout << 0 << endl; // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n System.out.println(0); // +1\n }\n}\nvoid Algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n Console.WriteLine(0); // +1\n }\n}\nfunc algorithm(n int) {\n a := 1 // +1\n a = a + 1 // +1\n a = a * 2 // +1\n // \u5faa\u73af n \u6b21\n for i := 0; i < n; i++ { // +1\n fmt.Println(a) // +1\n }\n}\nfunc algorithm(n: Int) {\n var a = 1 // +1\n a = a + 1 // +1\n a = a * 2 // +1\n // \u5faa\u73af n \u6b21\n for _ in 0 ..< n { // +1\n print(0) // +1\n }\n}\nfunction algorithm(n) {\n var a = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // \u5faa\u73af n \u6b21\n for(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n console.log(0); // +1\n }\n}\nfunction algorithm(n: number): void{\n var a: number = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // \u5faa\u73af n \u6b21\n for(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n console.log(0); // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n print(0); // +1\n }\n}\nfn algorithm(n: i32) {\n let mut a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n\n // \u5faa\u73af n \u6b21\n for _ in 0..n { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n println!(\"{}\", 0); // +1\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +1\n a = a + 1; // +1\n a = a * 2; // +1\n // \u5faa\u73af n \u6b21\n for (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n printf(\"%d\", 0); // +1\n }\n} \nfn algorithm(n: usize) void {\n var a: i32 = 1; // +1\n a += 1; // +1\n a *= 2; // +1\n // \u5faa\u73af n \u6b21\n for (0..n) |_| { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n std.debug.print(\"{}\\n\", .{0}); // +1\n }\n}\n\u8bbe\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\uff0c\u8bb0\u4e3a \\(T(n)\\) \uff0c\u5219\u4ee5\u4e0a\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a\uff1a
\\[ T(n) = 3 + 2n \\]\\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u5176\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u5b83\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002
\u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u79f0\u4e3a\u300c\u5927 \\(O\\) \u8bb0\u53f7 big-\\(O\\) notation\u300d\uff0c\u8868\u793a\u51fd\u6570 \\(T(n)\\) \u7684\u300c\u6e10\u8fd1\u4e0a\u754c asymptotic upper bound\u300d\u3002
\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u672c\u8d28\u4e0a\u662f\u8ba1\u7b97\u201c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\)\u201d\u7684\u6e10\u8fd1\u4e0a\u754c\uff0c\u5b83\u5177\u6709\u660e\u786e\u7684\u6570\u5b66\u5b9a\u4e49\u3002
\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c
\u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n > n_0\\) \uff0c\u5747\u6709 \\(T(n) \\leq c \\cdot f(n)\\) \uff0c\u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a \\(T(n) = O(f(n))\\) \u3002
\u5982\u56fe 2-8 \u6240\u793a\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5bfb\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5f53 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff0c\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\u3002
\u56fe 2-8 \u00a0 \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c
"},{"location":"chapter_computational_complexity/time_complexity/#233","title":"2.3.3 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u65e0\u987b\u62c5\u5fc3\u3002\u6211\u4eec\u53ef\u4ee5\u5148\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u5728\u4e0d\u65ad\u7684\u5b9e\u8df5\u4e2d\uff0c\u5c31\u53ef\u4ee5\u9010\u6e10\u9886\u609f\u5176\u6570\u5b66\u610f\u4e49\u3002
\u6839\u636e\u5b9a\u4e49\uff0c\u786e\u5b9a \\(f(n)\\) \u4e4b\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\u5982\u4f55\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1. \u00a0 \u7b2c\u4e00\u6b65\uff1a\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"\u9488\u5bf9\u4ee3\u7801\uff0c\u9010\u884c\u4ece\u4e0a\u5230\u4e0b\u8ba1\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u7b80\u5316\u6280\u5de7\u3002
- \u5ffd\u7565 \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\u3002\u56e0\u4e3a\u5b83\u4eec\u90fd\u4e0e \\(n\\) \u65e0\u5173\uff0c\u6240\u4ee5\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002
- \u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7b80\u5316\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u6ca1\u6709\u5f71\u54cd\u3002
- \u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u7b2c
1.\u70b9\u548c\u7b2c2.\u70b9\u7684\u6280\u5de7\u3002
\u7ed9\u5b9a\u4e00\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4e0a\u8ff0\u6280\u5de7\u6765\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZigdef algorithm(n: int):\n a = 1 # +0\uff08\u6280\u5de7 1\uff09\n a = a + n # +0\uff08\u6280\u5de7 1\uff09\n # +n\uff08\u6280\u5de7 2\uff09\n for i in range(5 * n + 1):\n print(0)\n # +n*n\uff08\u6280\u5de7 3\uff09\n for i in range(2 * n):\n for j in range(n + 1):\n print(0)\nvoid algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n cout << 0 << endl;\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n cout << 0 << endl;\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n System.out.println(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n System.out.println(0);\n }\n }\n}\nvoid Algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n Console.WriteLine(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n Console.WriteLine(0);\n }\n }\n}\nfunc algorithm(n int) {\n a := 1 // +0\uff08\u6280\u5de7 1\uff09\n a = a + n // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for i := 0; i < 5 * n + 1; i++ {\n fmt.Println(0)\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for i := 0; i < 2 * n; i++ {\n for j := 0; j < n + 1; j++ {\n fmt.Println(0)\n }\n }\n}\nfunc algorithm(n: Int) {\n var a = 1 // +0\uff08\u6280\u5de7 1\uff09\n a = a + n // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for _ in 0 ..< (5 * n + 1) {\n print(0)\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for _ in 0 ..< (2 * n) {\n for _ in 0 ..< (n + 1) {\n print(0)\n }\n }\n}\nfunction algorithm(n) {\n let a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (let i = 0; i < 5 * n + 1; i++) {\n console.log(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (let i = 0; i < 2 * n; i++) {\n for (let j = 0; j < n + 1; j++) {\n console.log(0);\n }\n }\n}\nfunction algorithm(n: number): void {\n let a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (let i = 0; i < 5 * n + 1; i++) {\n console.log(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (let i = 0; i < 2 * n; i++) {\n for (let j = 0; j < n + 1; j++) {\n console.log(0);\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n print(0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n print(0);\n }\n }\n}\nfn algorithm(n: i32) {\n let mut a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n\n // +n\uff08\u6280\u5de7 2\uff09\n for i in 0..(5 * n + 1) {\n println!(\"{}\", 0);\n }\n\n // +n*n\uff08\u6280\u5de7 3\uff09\n for i in 0..(2 * n) {\n for j in 0..(n + 1) {\n println!(\"{}\", 0);\n }\n }\n}\nvoid algorithm(int n) {\n int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n // +n\uff08\u6280\u5de7 2\uff09\n for (int i = 0; i < 5 * n + 1; i++) {\n printf(\"%d\", 0);\n }\n // +n*n\uff08\u6280\u5de7 3\uff09\n for (int i = 0; i < 2 * n; i++) {\n for (int j = 0; j < n + 1; j++) {\n printf(\"%d\", 0);\n }\n }\n}\nfn algorithm(n: usize) void {\n var a: i32 = 1; // +0\uff08\u6280\u5de7 1\uff09\n a = a + @as(i32, @intCast(n)); // +0\uff08\u6280\u5de7 1\uff09\n\n // +n\uff08\u6280\u5de7 2\uff09\n for(0..(5 * n + 1)) |_| {\n std.debug.print(\"{}\\n\", .{0}); \n }\n\n // +n*n\uff08\u6280\u5de7 3\uff09\n for(0..(2 * n)) |_| {\n for(0..(n + 1)) |_| {\n std.debug.print(\"{}\\n\", .{0}); \n }\n }\n}\n\u4ee5\u4e0b\u516c\u5f0f\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\uff0c\u4e24\u8005\u63a8\u7b97\u51fa\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002
\\[ \\begin{aligned} T(n) & = 2n(n + 1) + (5n + 1) + 2 & \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline & = 2n^2 + 7n + 3 \\newline T(n) & = n^2 + n & \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2. \u00a0 \u7b2c\u4e8c\u6b65\uff1a\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"\u65f6\u95f4\u590d\u6742\u5ea6\u7531 \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u53d1\u6325\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u4ed6\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002
\u8868 2-2 \u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u5938\u5f20\u7684\u503c\u662f\u4e3a\u4e86\u5f3a\u8c03\u201c\u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570\u201d\u8fd9\u4e00\u7ed3\u8bba\u3002\u5f53 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u53d8\u5f97\u65e0\u8db3\u8f7b\u91cd\u3002
\u8868 2-2 \u00a0 \u4e0d\u540c\u64cd\u4f5c\u6570\u91cf\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234","title":"2.3.4 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u5982\u56fe 2-9 \u6240\u793a\uff08\u6309\u7167\u4ece\u4f4e\u5230\u9ad8\u7684\u987a\u5e8f\u6392\u5217\uff09\u3002
\\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n \\log n) < O(n^2) < O(2^n) < O(n!) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u7ebf\u6027\u5bf9\u6570\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} < \\text{\u9636\u4e58\u9636} \\end{aligned} \\]\u56fe 2-9 \u00a0 \u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b
"},{"location":"chapter_computational_complexity/time_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"\u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002
\u5728\u4ee5\u4e0b\u51fd\u6570\u4e2d\uff0c\u5c3d\u7ba1\u64cd\u4f5c\u6570\u91cf
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.pysize\u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u4e8e\u5176\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a
time_complexity.cppdef constant(n: int) -> int:\n \"\"\"\u5e38\u6570\u9636\"\"\"\n count = 0\n size = 100000\n for _ in range(size):\n count += 1\n return count\n
time_complexity.java/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.cs/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.go/* \u5e38\u6570\u9636 */\nint Constant(int n) {\n int count = 0;\n int size = 100000;\n for (int i = 0; i < size; i++)\n count++;\n return count;\n}\n
time_complexity.swift/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\n count := 0\n size := 100000\n for i := 0; i < size; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -> Int {\n var count = 0\n let size = 100_000\n for _ in 0 ..< size {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n let count = 0;\n const size = 100000;\n for (let i = 0; i < size; i++) count++;\n return count;\n}\n
time_complexity.dart/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\n let count = 0;\n const size = 100000;\n for (let i = 0; i < size; i++) count++;\n return count;\n}\n
time_complexity.rs/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n for (var i = 0; i < size; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -> i32 {\n _ = n;\n let mut count = 0;\n let size = 100_000;\n for _ in 0..size {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u5e38\u6570\u9636 */\nint constant(int n) {\n int count = 0;\n int size = 100000;\n int i = 0;\n for (int i = 0; i < size; i++) {\n count++;\n }\n return count;\n}\n
"},{"location":"chapter_computational_complexity/time_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n _ = n;\n var count: i32 = 0;\n const size: i32 = 100_000;\n var i: i32 = 0;\n while(i<size) : (i += 1) {\n count += 1;\n }\n return count;\n}\n\u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5355\u5c42\u5faa\u73af\u4e2d\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef linear(n: int) -> int:\n \"\"\"\u7ebf\u6027\u9636\"\"\"\n count = 0\n for _ in range(n):\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++)\n count++;\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\n count := 0\n for i := 0; i < n; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -> Int {\n var count = 0\n for _ in 0 ..< n {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n let count = 0;\n for (let i = 0; i < n; i++) count++;\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\n let count = 0;\n for (let i = 0; i < n; i++) count++;\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (var i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -> i32 {\n let mut count = 0;\n for _ in 0..n {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n int count = 0;\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\n var count: i32 = 0;\n var i: i32 = 0;\n while (i < n) : (i += 1) {\n count += 1;\n }\n return count;\n}\n\u904d\u5386\u6570\u7ec4\u548c\u904d\u5386\u94fe\u8868\u7b49\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef array_traversal(nums: list[int]) -> int:\n \"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\n count = 0\n # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for num in nums:\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector<int> &nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int num : nums) {\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int num : nums) {\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n foreach (int num in nums) {\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\n count := 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for range nums {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -> Int {\n var count = 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for _ in nums {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (let i = 0; i < nums.length; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (let i = 0; i < nums.length; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List<int> nums) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (var _num in nums) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &[i32]) -> i32 {\n let mut count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for _ in nums {\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\n var count: i32 = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n for (nums) |_| {\n count += 1;\n }\n return count;\n}\n\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u9700\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u5177\u4f53\u786e\u5b9a\u3002\u6bd4\u5982\u5728\u7b2c\u4e00\u4e2a\u793a\u4f8b\u4e2d\uff0c\u53d8\u91cf \\(n\\) \u4e3a\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u5728\u7b2c\u4e8c\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u4e3a\u6570\u636e\u5927\u5c0f\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"\u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u56e0\u6b64\u603b\u4f53\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef quadratic(n: int) -> int:\n \"\"\"\u5e73\u65b9\u9636\"\"\"\n count = 0\n # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for i in range(n):\n for j in range(n):\n count += 1\n return count\n
time_complexity.java/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.cs/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.go/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.swift/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\n count := 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for i := 0; i < n; i++ {\n for j := 0; j < n; j++ {\n count++\n }\n }\n return count\n}\n
time_complexity.js/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -> Int {\n var count = 0\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for _ in 0 ..< n {\n for _ in 0 ..< n {\n count += 1\n }\n }\n return count\n}\n
time_complexity.ts/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.dart/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\n let count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.rs/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n
time_complexity.c/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -> i32 {\n let mut count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for _ in 0..n {\n for _ in 0..n {\n count += 1;\n }\n }\n count\n}\n
time_complexity.zig/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n int count = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < n; j++) {\n count++;\n }\n }\n return count;\n}\n// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\n var count: i32 = 0;\n var i: i32 = 0;\n // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n while (i < n) : (i += 1) {\n var j: i32 = 0;\n while (j < n) : (j += 1) {\n count += 1;\n }\n }\n return count;\n}\n\u56fe 2-10 \u5bf9\u6bd4\u4e86\u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u4e09\u79cd\u65f6\u95f4\u590d\u6742\u5ea6\u3002
\u56fe 2-10 \u00a0 \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u4ee5\u5192\u6ce1\u6392\u5e8f\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af\u6267\u884c \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af\u6267\u884c \\(n-1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(n / 2\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O((n - 1) n / 2) = O(n^2)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef bubble_sort(nums: list[int]) -> int:\n \"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\n count = 0 # \u8ba1\u6570\u5668\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in range(len(nums) - 1, 0, -1):\n # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j in range(i):\n if nums[j] > nums[j + 1]:\n # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n tmp: int = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n count += 3 # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n return count\n
time_complexity.java/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector<int> &nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.size() - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.cs/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.go/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.Length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.swift/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n count := 0 // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := len(nums) - 1; i > 0; i-- {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j := 0; j < i; j++ {\n if nums[j] > nums[j+1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n tmp := nums[j]\n nums[j] = nums[j+1]\n nums[j+1] = tmp\n count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count\n}\n
time_complexity.js/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -> Int {\n var count = 0 // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0 ..< i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count\n}\n
time_complexity.ts/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n let count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.dart/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n let count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.rs/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List<int> nums) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (var i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (var j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
time_complexity.c/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &mut [i32]) -> i32 {\n let mut count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in (1..nums.len()).rev() {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0..i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n count\n}\n
time_complexity.zig/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n int count = 0; // \u8ba1\u6570\u5668\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = n - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n
"},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n var count: i32 = 0; // \u8ba1\u6570\u5668 \n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n while (i > 0) : (i -= 1) {\n var j: usize = 0;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n while (j < i) : (j += 1) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n var tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n }\n }\n }\n return count;\n}\n\u751f\u7269\u5b66\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u662f\u6307\u6570\u9636\u589e\u957f\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u53d8\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u53d8\u4e3a \\(4\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002
\u56fe 2-11 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u7ec6\u80de\u5206\u88c2\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef exponential(n: int) -> int:\n \"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n count = 0\n base = 1\n # \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in range(n):\n for _ in range(base):\n count += 1\n base *= 2\n # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n
time_complexity.java/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.cs/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.go/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\n int count = 0, bas = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < bas; j++) {\n count++;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.swift/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\n count, base := 0, 1\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for i := 0; i < n; i++ {\n for j := 0; j < base; j++ {\n count++\n }\n base *= 2\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n}\n
time_complexity.js/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -> Int {\n var count = 0\n var base = 1\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in 0 ..< n {\n for _ in 0 ..< base {\n count += 1\n }\n base *= 2\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count\n}\n
time_complexity.ts/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\n let count = 0,\n base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.dart/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\n let count = 0,\n base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (let i = 0; i < n; i++) {\n for (let j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.rs/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0, base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (var i = 0; i < n; i++) {\n for (var j = 0; j < base; j++) {\n count++;\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n
time_complexity.c/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -> i32 {\n let mut count = 0;\n let mut base = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for _ in 0..n {\n for _ in 0..base {\n count += 1\n }\n base *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n count\n}\n
time_complexity.zig/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n int count = 0;\n int bas = 1;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < bas; j++) {\n count++;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\n var count: i32 = 0;\n var bas: i32 = 1;\n var i: i32 = 0;\n // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n while (i < n) : (i += 1) {\n var j: i32 = 0;\n while (j < bas) : (j += 1) {\n count += 1;\n }\n bas *= 2;\n }\n // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n return count;\n}\n\u56fe 2-11 \u00a0 \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u5176\u9012\u5f52\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u7ecf\u8fc7 \\(n\\) \u6b21\u5206\u88c2\u540e\u505c\u6b62\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef exp_recur(n: int) -> int:\n \"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n == 1:\n return 1\n return exp_recur(n - 1) + exp_recur(n - 1) + 1\n
time_complexity.java/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.cs/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.go/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\n if (n == 1) return 1;\n return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n
time_complexity.swift/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\n if n == 1 {\n return 1\n }\n return expRecur(n-1) + expRecur(n-1) + 1\n}\n
time_complexity.js/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -> Int {\n if n == 1 {\n return 1\n }\n return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n
time_complexity.ts/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\n if (n === 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.dart/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\n if (n === 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.rs/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
time_complexity.c/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -> i32 {\n if n == 1 {\n return 1;\n }\n exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n
time_complexity.zig/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n if (n == 1)\n return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\n if (n == 1) return 1;\n return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n\u6307\u6570\u9636\u589e\u957f\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u7a77\u4e3e\u6cd5\uff08\u66b4\u529b\u641c\u7d22\u3001\u56de\u6eaf\u7b49\uff09\u4e2d\u6bd4\u8f83\u5e38\u89c1\u3002\u5bf9\u4e8e\u6570\u636e\u89c4\u6a21\u8f83\u5927\u7684\u95ee\u9898\uff0c\u6307\u6570\u9636\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff0c\u901a\u5e38\u9700\u8981\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u6216\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u6765\u89e3\u51b3\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"\u4e0e\u6307\u6570\u9636\u76f8\u53cd\uff0c\u5bf9\u6570\u9636\u53cd\u6620\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u60c5\u51b5\u3002\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002
\u56fe 2-12 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_2 n)\\) \uff0c\u7b80\u8bb0\u4e3a \\(O(\\log n)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef logarithmic(n: float) -> int:\n \"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n count = 0\n while n > 1:\n n = n / 2\n count += 1\n return count\n
time_complexity.java/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n /= 2;\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\n count := 0\n for n > 1 {\n n = n / 2\n count++\n }\n return count\n}\n
time_complexity.js/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -> Int {\n var count = 0\n var n = n\n while n > 1 {\n n = n / 2\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\n let count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\n let count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(num n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: f32) -> i32 {\n let mut count = 0;\n while n > 1.0 {\n n = n / 2.0;\n count += 1;\n }\n count\n}\n
time_complexity.zig/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n int count = 0;\n while (n > 1) {\n n = n / 2;\n count++;\n }\n return count;\n}\n// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\n var count: i32 = 0;\n var n_var = n;\n while (n_var > 1)\n {\n n_var = n_var / 2;\n count +=1;\n }\n return count;\n}\n\u56fe 2-12 \u00a0 \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef log_recur(n: float) -> int:\n \"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n <= 1:\n return 0\n return log_recur(n / 2) + 1\n
time_complexity.java/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.cs/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.go/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(float n) {\n if (n <= 1) return 0;\n return LogRecur(n / 2) + 1;\n}\n
time_complexity.swift/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\n if n <= 1 {\n return 0\n }\n return logRecur(n/2) + 1\n}\n
time_complexity.js/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -> Int {\n if n <= 1 {\n return 0\n }\n return logRecur(n: n / 2) + 1\n}\n
time_complexity.ts/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.dart/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.rs/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(num n) {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n
time_complexity.c/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: f32) -> i32 {\n if n <= 1.0 {\n return 0;\n }\n log_recur(n / 2.0) + 1\n}\n
time_complexity.zig/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n if (n <= 1)\n return 0;\n return logRecur(n / 2) + 1;\n}\n// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\n if (n <= 1) return 0;\n return logRecur(n / 2) + 1;\n}\n\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u7b97\u6cd5\u4e2d\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u591a\u201d\u548c\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002\u5b83\u589e\u957f\u7f13\u6162\uff0c\u662f\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\u7684\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002
\\(O(\\log n)\\) \u7684\u5e95\u6570\u662f\u591a\u5c11\uff1f
\u51c6\u786e\u6765\u8bf4\uff0c\u201c\u4e00\u5206\u4e3a \\(m\\)\u201d\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(\\log_m n)\\) \u3002\u800c\u901a\u8fc7\u5bf9\u6570\u6362\u5e95\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5177\u6709\u4e0d\u540c\u5e95\u6570\u3001\u76f8\u7b49\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a
\\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\]\u4e5f\u5c31\u662f\u8bf4\uff0c\u5e95\u6570 \\(m\\) \u53ef\u4ee5\u5728\u4e0d\u5f71\u54cd\u590d\u6742\u5ea6\u7684\u524d\u63d0\u4e0b\u8f6c\u6362\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u7701\u7565\u5e95\u6570 \\(m\\) \uff0c\u5c06\u5bf9\u6570\u9636\u76f4\u63a5\u8bb0\u4e3a \\(O(\\log n)\\) \u3002
"},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","title":"6. \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"\u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef linear_log_recur(n: float) -> int:\n \"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\n if n <= 1:\n return 1\n count: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n for _ in range(n):\n count += 1\n return count\n
time_complexity.java/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.cs/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.go/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(float n) {\n if (n <= 1) return 1;\n int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.swift/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\n if n <= 1 {\n return 1\n }\n count := linearLogRecur(n/2) + linearLogRecur(n/2)\n for i := 0.0; i < n; i++ {\n count++\n }\n return count\n}\n
time_complexity.js/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -> Int {\n if n <= 1 {\n return 1\n }\n var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n for _ in stride(from: 0, to: n, by: 1) {\n count += 1\n }\n return count\n}\n
time_complexity.ts/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\n if (n <= 1) return 1;\n let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (let i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.dart/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\n if (n <= 1) return 1;\n let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (let i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.rs/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(num n) {\n if (n <= 1) return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (var i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n
time_complexity.c/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: f32) -> i32 {\n if n <= 1.0 {\n return 1;\n }\n let mut count = linear_log_recur(n / 2.0) + linear_log_recur(n / 2.0);\n for _ in 0 ..n as i32 {\n count += 1;\n }\n return count\n}\n
time_complexity.zig/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n if (n <= 1)\n return 1;\n int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n for (int i = 0; i < n; i++) {\n count++;\n }\n return count;\n}\n// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\n if (n <= 1) return 1;\n var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n var i: f32 = 0;\n while (i < n) : (i += 1) {\n count += 1;\n }\n return count;\n}\n\u56fe 2-13 \u5c55\u793a\u4e86\u7ebf\u6027\u5bf9\u6570\u9636\u7684\u751f\u6210\u65b9\u5f0f\u3002\u4e8c\u53c9\u6811\u7684\u6bcf\u4e00\u5c42\u7684\u64cd\u4f5c\u603b\u6570\u90fd\u4e3a \\(n\\) \uff0c\u6811\u5171\u6709 \\(\\log_2 n + 1\\) \u5c42\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
\u56fe 2-13 \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#7-on","title":"7. \u00a0 \u9636\u4e58\u9636 \\(O(n!)\\)","text":"\u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u201c\u5168\u6392\u5217\u201d\u95ee\u9898\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6570\u91cf\u4e3a\uff1a
\\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\]\u9636\u4e58\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u5982\u56fe 2-14 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c42\u65f6\u505c\u6b62\u5206\u88c2\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py
time_complexity.cppdef factorial_recur(n: int) -> int:\n \"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n if n == 0:\n return 1\n count = 0\n # \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in range(n):\n count += factorial_recur(n - 1)\n return count\n
time_complexity.java/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.cs/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.go/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\n if (n == 0) return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (int i = 0; i < n; i++) {\n count += FactorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.swift/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\n if n == 0 {\n return 1\n }\n count := 0\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for i := 0; i < n; i++ {\n count += factorialRecur(n - 1)\n }\n return count\n}\n
time_complexity.js/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -> Int {\n if n == 0 {\n return 1\n }\n var count = 0\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in 0 ..< n {\n count += factorialRecur(n: n - 1)\n }\n return count\n}\n
time_complexity.ts/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\n if (n === 0) return 1;\n let count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (let i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.dart/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\n if (n === 0) return 1;\n let count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (let i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.rs/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0) return 1;\n int count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for (var i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n
time_complexity.c/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -> i32 {\n if n == 0 {\n return 1;\n }\n let mut count = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n for _ in 0..n {\n count += factorial_recur(n - 1);\n }\n count\n}\n
time_complexity.zig/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n if (n == 0)\n return 1;\n int count = 0;\n for (int i = 0; i < n; i++) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\n if (n == 0) return 1;\n var count: i32 = 0;\n var i: i32 = 0;\n // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n while (i < n) : (i += 1) {\n count += factorialRecur(n - 1);\n }\n return count;\n}\n\u56fe 2-14 \u00a0 \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6
\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u5f53 \\(n \\geq 4\\) \u65f6\u6052\u6709 \\(n! > 2^n\\) \uff0c\u6240\u4ee5\u9636\u4e58\u9636\u6bd4\u6307\u6570\u9636\u589e\u957f\u5f97\u66f4\u5feb\uff0c\u5728 \\(n\\) \u8f83\u5927\u65f6\u4e5f\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002
"},{"location":"chapter_computational_complexity/time_complexity/#235","title":"2.3.5 \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"\u7b97\u6cd5\u7684\u65f6\u95f4\u6548\u7387\u5f80\u5f80\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u5047\u8bbe\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4
nums\uff0c\u5176\u4e2dnums\u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u6bcf\u4e2a\u6570\u5b57\u53ea\u51fa\u73b0\u4e00\u6b21\uff1b\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff0c\u4efb\u52a1\u76ee\u6807\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002- \u5f53
nums = [?, ?, ..., 1]\uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u9700\u8981\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002 - \u5f53
nums = [1, ?, ?, ...]\uff0c\u5373\u5f53\u9996\u4e2a\u5143\u7d20\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\) \u3002
\u201c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\u3002\u76f8\u5e94\u5730\uff0c\u201c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\uff0c\u7528 \\(\\Omega\\) \u8bb0\u53f7\u8868\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig worst_best_time_complexity.py
worst_best_time_complexity.cppdef random_numbers(n: int) -> list[int]:\n \"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n # \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\n nums = [i for i in range(1, n + 1)]\n # \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n random.shuffle(nums)\n return nums\n\ndef find_one(nums: list[int]) -> int:\n \"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n for i in range(len(nums)):\n # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1:\n return i\n return -1\n
worst_best_time_complexity.java/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector<int> randomNumbers(int n) {\n vector<int> nums(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\n unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector<int> &nums) {\n for (int i = 0; i < nums.size(); i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.cs/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\n Integer[] nums = new Integer[n];\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n Collections.shuffle(Arrays.asList(nums));\n // Integer[] -> int[]\n int[] res = new int[n];\n for (int i = 0; i < n; i++) {\n res[i] = nums[i];\n }\n return res;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n for (int i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.go/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\n int[] nums = new int[n];\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (int i = 0; i < nums.Length; i++) {\n int index = new Random().Next(i, nums.Length);\n (nums[i], nums[index]) = (nums[index], nums[i]);\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n for (int i = 0; i < nums.Length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n
worst_best_time_complexity.swift/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\n nums := make([]int, n)\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for i := 0; i < n; i++ {\n nums[i] = i + 1\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n rand.Shuffle(len(nums), func(i, j int) {\n nums[i], nums[j] = nums[j], nums[i]\n })\n return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n for i := 0; i < len(nums); i++ {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return i\n }\n }\n return -1\n}\n
worst_best_time_complexity.js/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -> [Int] {\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n var nums = Array(1 ... n)\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle()\n return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -> Int {\n for i in nums.indices {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return i\n }\n }\n return -1\n}\n
worst_best_time_complexity.ts/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\n const nums = Array(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (let i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (let i = 0; i < n; i++) {\n const r = Math.floor(Math.random() * (i + 1));\n const temp = nums[i];\n nums[i] = nums[r];\n nums[r] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n for (let i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] === 1) {\n return i;\n }\n }\n return -1;\n}\n
worst_best_time_complexity.dart/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\n const nums = Array(n);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (let i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (let i = 0; i < n; i++) {\n const r = Math.floor(Math.random() * (i + 1));\n const temp = nums[i];\n nums[i] = nums[r];\n nums[r] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n for (let i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] === 1) {\n return i;\n }\n }\n return -1;\n}\n
worst_best_time_complexity.rs/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList<int> randomNumbers(int n) {\n final nums = List.filled(n, 0);\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (var i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle();\n\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List<int> nums) {\n for (var i = 0; i < nums.length; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1) return i;\n }\n\n return -1;\n}\n
worst_best_time_complexity.c/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -> Vec<i32> {\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n let mut nums = (1..=n).collect::<Vec<i32>>();\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n nums.shuffle(&mut thread_rng());\n nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &[i32]) -> Option<usize> {\n for i in 0..nums.len() {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if nums[i] == 1 {\n return Some(i);\n }\n }\n None\n}\n
worst_best_time_complexity.zig/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n // \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\n int *nums = (int *)malloc(n * sizeof(int));\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (int i = 0; i < n; i++) {\n nums[i] = i + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n for (int i = n - 1; i > 0; i--) {\n int j = rand() % (i + 1);\n int temp = nums[i];\n nums[i] = nums[j];\n nums[j] = temp;\n }\n return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n for (int i = 0; i < n; i++) {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (nums[i] == 1)\n return i;\n }\n return -1;\n}\n// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\n var nums: [n]i32 = undefined;\n // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n for (&nums, 0..) |*num, i| {\n num.* = @as(i32, @intCast(i)) + 1;\n }\n // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n const rand = std.crypto.random;\n rand.shuffle(i32, &nums);\n return nums;\n}\n\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n for (nums, 0..) |num, i| {\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n if (num == 1) return @intCast(i);\n }\n return -1;\n}\n\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6211\u4eec\u5728\u5b9e\u9645\u4e2d\u5f88\u5c11\u4f7f\u7528\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u56e0\u4e3a\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u53ef\u80fd\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u800c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u6548\u7387\u5b89\u5168\u503c\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002
\u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u4e8e\u201c\u7279\u6b8a\u7684\u6570\u636e\u5206\u5e03\u201d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u5e76\u4e0d\u80fd\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\u6765\u8868\u793a\u3002
\u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5c31\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(n / 2\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(n / 2) = \\Theta(n)\\) \u3002
\u4f46\u5bf9\u4e8e\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5f80\u5f80\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002
\u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f
\u53ef\u80fd\u7531\u4e8e \\(O\\) \u7b26\u53f7\u8fc7\u4e8e\u6717\u6717\u4e0a\u53e3\uff0c\u56e0\u6b64\u6211\u4eec\u5e38\u5e38\u4f7f\u7528\u5b83\u6765\u8868\u793a\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u4f46\u4ece\u4e25\u683c\u610f\u4e49\u4e0a\u8bb2\uff0c\u8fd9\u79cd\u505a\u6cd5\u5e76\u4e0d\u89c4\u8303\u3002\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u7c7b\u4f3c\u201c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8bf7\u5c06\u5176\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u3002
"},{"location":"chapter_data_structure/","title":"\u7b2c 3 \u7ae0 \u00a0 \u6570\u636e\u7ed3\u6784","text":"Abstract
\u6570\u636e\u7ed3\u6784\u5982\u540c\u4e00\u526f\u7a33\u56fa\u800c\u591a\u6837\u7684\u6846\u67b6\u3002
\u5b83\u4e3a\u6570\u636e\u7684\u6709\u5e8f\u7ec4\u7ec7\u63d0\u4f9b\u4e86\u84dd\u56fe\uff0c\u7b97\u6cd5\u5f97\u4ee5\u5728\u6b64\u57fa\u7840\u4e0a\u751f\u52a8\u8d77\u6765\u3002
"},{"location":"chapter_data_structure/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b
- 3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b
- 3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *
- 3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *
- 3.5 \u00a0 \u5c0f\u7ed3
\u5f53\u8c08\u53ca\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\u65f6\uff0c\u6211\u4eec\u4f1a\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u79cd\u5f62\u5f0f\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u6570\u636e\u7684\u7ec4\u7ec7\u5f62\u5f0f\u5404\u5f02\uff0c\u4f46\u5b83\u4eec\u90fd\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u3002
\u57fa\u672c\u6570\u636e\u7c7b\u578b\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\uff0c\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002
- \u6574\u6570\u7c7b\u578b
byte\u3001short\u3001int\u3001long\u3002 - \u6d6e\u70b9\u6570\u7c7b\u578b
float\u3001double\uff0c\u7528\u4e8e\u8868\u793a\u5c0f\u6570\u3002 - \u5b57\u7b26\u7c7b\u578b
char\uff0c\u7528\u4e8e\u8868\u793a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u6bcd\u3001\u6807\u70b9\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002 - \u5e03\u5c14\u7c7b\u578b
bool\uff0c\u7528\u4e8e\u8868\u793a\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\u5224\u65ad\u3002
\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4ee5\u4e8c\u8fdb\u5236\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u3002\u4e00\u4e2a\u4e8c\u8fdb\u5236\u4f4d\u5373\u4e3a \\(1\\) \u6bd4\u7279\u3002\u5728\u7edd\u5927\u591a\u6570\u73b0\u4ee3\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\\(1\\) \u5b57\u8282\uff08byte\uff09\u7531 \\(8\\) \u6bd4\u7279\uff08bit\uff09\u7ec4\u6210\u3002
\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5176\u5360\u7528\u7684\u7a7a\u95f4\u5927\u5c0f\u3002\u4e0b\u9762\u4ee5 Java \u4e3a\u4f8b\u3002
- \u6574\u6570\u7c7b\u578b
byte\u5360\u7528 \\(1\\) byte = \\(8\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u4e2a\u6570\u5b57\u3002 - \u6574\u6570\u7c7b\u578b
int\u5360\u7528 \\(4\\) bytes = \\(32\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u6570\u5b57\u3002
\u8868 3-1 \u5217\u4e3e\u4e86 Java \u4e2d\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u3002\u6b64\u8868\u683c\u65e0\u987b\u6b7b\u8bb0\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002
\u8868 3-1 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u548c\u53d6\u503c\u8303\u56f4
\u7c7b\u578b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9ed8\u8ba4\u503c \u6574\u6570byte1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\)short2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\)int4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\)long8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570float4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\)double8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26char2 bytes \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14bool1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\)\u8bf7\u6ce8\u610f\uff0c\u8868 3-1 \u9488\u5bf9\u7684\u662f Java \u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u60c5\u51b5\u3002\u6bcf\u79cd\u7f16\u7a0b\u8bed\u8a00\u90fd\u6709\u5404\u81ea\u7684\u6570\u636e\u7c7b\u578b\u5b9a\u4e49\uff0c\u5b83\u4eec\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002
- \u5728 Python \u4e2d\uff0c\u6574\u6570\u7c7b\u578b
int\u53ef\u4ee5\u662f\u4efb\u610f\u5927\u5c0f\uff0c\u53ea\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\uff1b\u6d6e\u70b9\u6570float\u662f\u53cc\u7cbe\u5ea6 64 \u4f4d\uff1b\u6ca1\u6709char\u7c7b\u578b\uff0c\u5355\u4e2a\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32str\u3002 - C \u548c C++ \u672a\u660e\u786e\u89c4\u5b9a\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5927\u5c0f\uff0c\u800c\u56e0\u5b9e\u73b0\u548c\u5e73\u53f0\u5404\u5f02\u3002\u8868 3-1 \u9075\u5faa LP64 \u6570\u636e\u6a21\u578b\uff0c\u5176\u7528\u4e8e\u5305\u62ec Linux \u548c macOS \u5728\u5185\u7684 Unix 64 \u4f4d\u64cd\u4f5c\u7cfb\u7edf\u3002
- \u5b57\u7b26
char\u7684\u5927\u5c0f\u5728 C \u548c C++ \u4e2d\u4e3a 1 \u5b57\u8282\uff0c\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\u53d6\u51b3\u4e8e\u7279\u5b9a\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\uff0c\u8be6\u89c1\u201c\u5b57\u7b26\u7f16\u7801\u201d\u7ae0\u8282\u3002 - \u5373\u4f7f\u8868\u793a\u5e03\u5c14\u91cf\u4ec5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u5185\u5b58\u4e2d\u901a\u5e38\u5b58\u50a8\u4e3a 1 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u673a CPU \u901a\u5e38\u5c06 1 \u5b57\u8282\u4f5c\u4e3a\u6700\u5c0f\u5bfb\u5740\u5185\u5b58\u5355\u5143\u3002
\u90a3\u4e48\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u95f4\u6709\u4ec0\u4e48\u8054\u7cfb\u5462\uff1f\u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u8fd9\u53e5\u8bdd\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\u800c\u975e\u201c\u6570\u636e\u201d\u3002
\u5982\u679c\u60f3\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u6211\u4eec\u81ea\u7136\u4f1a\u60f3\u5230\u4f7f\u7528\u6570\u7ec4\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u7684\u7ebf\u6027\u7ed3\u6784\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u548c\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5b58\u50a8\u7684\u5185\u5bb9\u662f\u6574\u6570
int\u3001\u5c0f\u6570float\u6216\u662f\u5b57\u7b26char\uff0c\u5219\u4e0e\u201c\u6570\u636e\u7ed3\u6784\u201d\u65e0\u5173\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u7528\u76f8\u540c\u7684\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\uff09\u6765\u5b58\u50a8\u4e0e\u8868\u793a\u4e0d\u540c\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u5305\u62ec
PythonC++JavaC#GoSwiftJSTSDartRustCZigint\u3001float\u3001char\u3001bool\u7b49\u3002# \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python \u7684\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python \u7684\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\u5f15\u7528\ndata = [0, 0.0, 'a', False, ListNode(0)]\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers = Array(repeating: Int(), count: 5)\nlet decimals = Array(repeating: Double(), count: 5)\nlet characters = Array(repeating: Character(\"a\"), count: 5)\nlet bools = Array(repeating: Bool(), count: 5)\n// JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\nconst array = [0, 0.0, 'a', false];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nList<int> numbers = List.filled(5, 0);\nList<double> decimals = List.filled(5, 0.0);\nList<String> characters = List.filled(5, 'a');\nList<bool> bools = List.filled(5, false);\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers: Vec<i32> = vec![0; 5];\nlet decimals: Vec<f32> = vec![0.0, 5];\nlet characters: Vec<char> = vec!['0'; 5];\nlet bools: Vec<bool> = vec![false; 5];\n// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n
"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *","text":"\n\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6240\u6709\u6570\u636e\u90fd\u662f\u4ee5\u4e8c\u8fdb\u5236\u6570\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\uff0c\u5b57\u7b26
"},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1 \u00a0 ASCII \u5b57\u7b26\u96c6","text":"char\u4e5f\u4e0d\u4f8b\u5916\u3002\u4e3a\u4e86\u8868\u793a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u5efa\u7acb\u4e00\u5957\u201c\u5b57\u7b26\u96c6\u201d\uff0c\u89c4\u5b9a\u6bcf\u4e2a\u5b57\u7b26\u548c\u4e8c\u8fdb\u5236\u6570\u4e4b\u95f4\u7684\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u540e\uff0c\u8ba1\u7b97\u673a\u5c31\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5b8c\u6210\u4e8c\u8fdb\u5236\u6570\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002\u300cASCII \u7801\u300d\u662f\u6700\u65e9\u51fa\u73b0\u7684\u5b57\u7b26\u96c6\uff0c\u5176\u5168\u79f0\u4e3a American Standard Code for Information Interchange\uff08\u7f8e\u56fd\u6807\u51c6\u4fe1\u606f\u4ea4\u6362\u4ee3\u7801\uff09\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u8fdb\u5236\u6570\uff08\u4e00\u4e2a\u5b57\u8282\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6700\u591a\u80fd\u591f\u8868\u793a 128 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u5982\u56fe 3-6 \u6240\u793a\uff0cASCII \u7801\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5199\u3001\u6570\u5b57 0 ~ 9\u3001\u4e00\u4e9b\u6807\u70b9\u7b26\u53f7\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u7b26\uff08\u5982\u6362\u884c\u7b26\u548c\u5236\u8868\u7b26\uff09\u3002
\u56fe 3-6 \u00a0 ASCII \u7801
\u7136\u800c\uff0cASCII \u7801\u4ec5\u80fd\u591f\u8868\u793a\u82f1\u6587\u3002\u968f\u7740\u8ba1\u7b97\u673a\u7684\u5168\u7403\u5316\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u80fd\u591f\u8868\u793a\u66f4\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u96c6\u300cEASCII\u300d\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u7840\u4e0a\u6269\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u591f\u8868\u793a 256 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002
\u5728\u4e16\u754c\u8303\u56f4\u5185\uff0c\u9646\u7eed\u51fa\u73b0\u4e86\u4e00\u6279\u9002\u7528\u4e8e\u4e0d\u540c\u5730\u533a\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u8fd9\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u4e2a\u5b57\u7b26\u7edf\u4e00\u4e3a ASCII \u7801\uff0c\u540e 128 \u4e2a\u5b57\u7b26\u5b9a\u4e49\u4e0d\u540c\uff0c\u4ee5\u9002\u5e94\u4e0d\u540c\u8bed\u8a00\u7684\u9700\u6c42\u3002
"},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2 \u00a0 GBK \u5b57\u7b26\u96c6","text":"\u540e\u6765\u4eba\u4eec\u53d1\u73b0\uff0cEASCII \u7801\u4ecd\u7136\u65e0\u6cd5\u6ee1\u8db3\u8bb8\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u6570\u91cf\u8981\u6c42\u3002\u6bd4\u5982\u6c49\u5b57\u6709\u8fd1\u5341\u4e07\u4e2a\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u51e0\u5343\u4e2a\u3002\u4e2d\u56fd\u56fd\u5bb6\u6807\u51c6\u603b\u5c40\u4e8e 1980 \u5e74\u53d1\u5e03\u4e86\u300cGB2312\u300d\u5b57\u7b26\u96c6\uff0c\u5176\u6536\u5f55\u4e86 6763 \u4e2a\u6c49\u5b57\uff0c\u57fa\u672c\u6ee1\u8db3\u4e86\u6c49\u5b57\u7684\u8ba1\u7b97\u673a\u5904\u7406\u9700\u8981\u3002
\u7136\u800c\uff0cGB2312 \u65e0\u6cd5\u5904\u7406\u90e8\u5206\u7f55\u89c1\u5b57\u548c\u7e41\u4f53\u5b57\u3002\u300cGBK\u300d\u5b57\u7b26\u96c6\u662f\u5728 GB2312 \u7684\u57fa\u7840\u4e0a\u6269\u5c55\u5f97\u5230\u7684\uff0c\u5b83\u5171\u6536\u5f55\u4e86 21886 \u4e2a\u6c49\u5b57\u3002\u5728 GBK \u7684\u7f16\u7801\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u7b26\u4f7f\u7528\u4e00\u4e2a\u5b57\u8282\u8868\u793a\uff0c\u6c49\u5b57\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u8868\u793a\u3002
"},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3 \u00a0 Unicode \u5b57\u7b26\u96c6","text":"\u968f\u7740\u8ba1\u7b97\u673a\u6280\u672f\u7684\u84ec\u52c3\u53d1\u5c55\uff0c\u5b57\u7b26\u96c6\u4e0e\u7f16\u7801\u6807\u51c6\u767e\u82b1\u9f50\u653e\uff0c\u800c\u8fd9\u5e26\u6765\u4e86\u8bb8\u591a\u95ee\u9898\u3002\u4e00\u65b9\u9762\uff0c\u8fd9\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u4e49\u4e86\u7279\u5b9a\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u65e0\u6cd5\u5728\u591a\u8bed\u8a00\u73af\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u79cd\u8bed\u8a00\u5b58\u5728\u591a\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u5982\u679c\u4e24\u53f0\u8ba1\u7b97\u673a\u4f7f\u7528\u7684\u662f\u4e0d\u540c\u7684\u7f16\u7801\u6807\u51c6\uff0c\u5219\u5728\u4fe1\u606f\u4f20\u9012\u65f6\u5c31\u4f1a\u51fa\u73b0\u4e71\u7801\u3002
\u90a3\u4e2a\u65f6\u4ee3\u7684\u7814\u7a76\u4eba\u5458\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u4e2a\u8db3\u591f\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c06\u4e16\u754c\u8303\u56f4\u5185\u7684\u6240\u6709\u8bed\u8a00\u548c\u7b26\u53f7\u90fd\u6536\u5f55\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u51b3\u8de8\u8bed\u8a00\u73af\u5883\u548c\u4e71\u7801\u95ee\u9898\u4e86\u5417\uff1f\u5728\u8fd9\u79cd\u60f3\u6cd5\u7684\u9a71\u52a8\u4e0b\uff0c\u4e00\u4e2a\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u5e94\u8fd0\u800c\u751f\u3002
\u300cUnicode\u300d\u7684\u4e2d\u6587\u540d\u79f0\u4e3a\u201c\u7edf\u4e00\u7801\u201d\uff0c\u7406\u8bba\u4e0a\u80fd\u5bb9\u7eb3 100 \u591a\u4e07\u4e2a\u5b57\u7b26\u3002\u5b83\u81f4\u529b\u4e8e\u5c06\u5168\u7403\u8303\u56f4\u5185\u7684\u5b57\u7b26\u7eb3\u5165\u7edf\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u79cd\u901a\u7528\u7684\u5b57\u7b26\u96c6\u6765\u5904\u7406\u548c\u663e\u793a\u5404\u79cd\u8bed\u8a00\u6587\u5b57\uff0c\u51cf\u5c11\u56e0\u4e3a\u7f16\u7801\u6807\u51c6\u4e0d\u540c\u800c\u4ea7\u751f\u7684\u4e71\u7801\u95ee\u9898\u3002
\u81ea 1991 \u5e74\u53d1\u5e03\u4ee5\u6765\uff0cUnicode \u4e0d\u65ad\u6269\u5145\u65b0\u7684\u8bed\u8a00\u4e0e\u5b57\u7b26\u3002\u622a\u81f3 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7ecf\u5305\u542b 149186 \u4e2a\u5b57\u7b26\uff0c\u5305\u62ec\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\u3001\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002\u5728\u5e9e\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u7b26\u5360\u7528 3 \u5b57\u8282\u751a\u81f3 4 \u5b57\u8282\u3002
Unicode \u662f\u4e00\u79cd\u901a\u7528\u5b57\u7b26\u96c6\uff0c\u672c\u8d28\u4e0a\u662f\u7ed9\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u4e00\u4e2a\u7f16\u53f7\uff08\u79f0\u4e3a\u201c\u7801\u70b9\u201d\uff09\uff0c\u4f46\u5b83\u5e76\u6ca1\u6709\u89c4\u5b9a\u5728\u8ba1\u7b97\u673a\u4e2d\u5982\u4f55\u5b58\u50a8\u8fd9\u4e9b\u5b57\u7b26\u7801\u70b9\u3002\u6211\u4eec\u4e0d\u7981\u4f1a\u95ee\uff1a\u5f53\u591a\u79cd\u957f\u5ea6\u7684 Unicode \u7801\u70b9\u540c\u65f6\u51fa\u73b0\u5728\u4e00\u4e2a\u6587\u672c\u4e2d\u65f6\uff0c\u7cfb\u7edf\u5982\u4f55\u89e3\u6790\u5b57\u7b26\uff1f\u4f8b\u5982\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u7684\u7f16\u7801\uff0c\u7cfb\u7edf\u5982\u4f55\u786e\u8ba4\u5b83\u662f\u4e00\u4e2a 2 \u5b57\u8282\u7684\u5b57\u7b26\u8fd8\u662f\u4e24\u4e2a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff1f
\u5bf9\u4e8e\u4ee5\u4e0a\u95ee\u9898\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u662f\u5c06\u6240\u6709\u5b57\u7b26\u5b58\u50a8\u4e3a\u7b49\u957f\u7684\u7f16\u7801\u3002\u5982\u56fe 3-7 \u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff0c\u201c\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9ad8\u4f4d\u586b 0 \u5c06\u201cHello \u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u7b26\u90fd\u7f16\u7801\u4e3a 2 \u5b57\u8282\u957f\u5ea6\u3002\u8fd9\u6837\u7cfb\u7edf\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u5b57\u8282\u89e3\u6790\u4e00\u4e2a\u5b57\u7b26\uff0c\u6062\u590d\u8fd9\u4e2a\u77ed\u8bed\u7684\u5185\u5bb9\u4e86\u3002
\u56fe 3-7 \u00a0 Unicode \u7f16\u7801\u793a\u4f8b
\u7136\u800c ASCII \u7801\u5df2\u7ecf\u5411\u6211\u4eec\u8bc1\u660e\uff0c\u7f16\u7801\u82f1\u6587\u53ea\u9700 1 \u5b57\u8282\u3002\u82e5\u91c7\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u672c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u5c06\u4f1a\u662f ASCII \u7f16\u7801\u4e0b\u7684\u4e24\u500d\uff0c\u975e\u5e38\u6d6a\u8d39\u5185\u5b58\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002
"},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4 \u00a0 UTF-8 \u7f16\u7801","text":"\u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u4e3a\u56fd\u9645\u4e0a\u4f7f\u7528\u6700\u5e7f\u6cdb\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u79cd\u53ef\u53d8\u957f\u5ea6\u7684\u7f16\u7801\uff0c\u4f7f\u7528 1 \u5230 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6839\u636e\u5b57\u7b26\u7684\u590d\u6742\u6027\u800c\u53d8\u3002ASCII \u5b57\u7b26\u53ea\u9700 1 \u5b57\u8282\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u814a\u5b57\u6bcd\u9700\u8981 2 \u5b57\u8282\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u9700\u8981 3 \u5b57\u8282\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u7b26\u9700\u8981 4 \u5b57\u8282\u3002
UTF-8 \u7684\u7f16\u7801\u89c4\u5219\u5e76\u4e0d\u590d\u6742\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002
- \u5bf9\u4e8e\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff0c\u5c06\u6700\u9ad8\u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff0c\u5176\u4f59 7 \u4f4d\u8bbe\u7f6e\u4e3a Unicode \u7801\u70b9\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u7b26\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u5360\u636e\u4e86\u524d 128 \u4e2a\u7801\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0cUTF-8 \u7f16\u7801\u53ef\u4ee5\u5411\u4e0b\u517c\u5bb9 ASCII \u7801\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u6765\u89e3\u6790\u5e74\u4ee3\u4e45\u8fdc\u7684 ASCII \u7801\u6587\u672c\u3002
- \u5bf9\u4e8e\u957f\u5ea6\u4e3a \\(n\\) \u5b57\u8282\u7684\u5b57\u7b26\uff08\u5176\u4e2d \\(n > 1\\)\uff09\uff0c\u5c06\u9996\u4e2a\u5b57\u8282\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u7b2c \\(n + 1\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff1b\u4ece\u7b2c\u4e8c\u4e2a\u5b57\u8282\u5f00\u59cb\uff0c\u5c06\u6bcf\u4e2a\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \uff1b\u5176\u4f59\u6240\u6709\u4f4d\u7528\u4e8e\u586b\u5145\u5b57\u7b26\u7684 Unicode \u7801\u70b9\u3002
\u56fe 3-8 \u5c55\u793a\u4e86\u201cHello\u7b97\u6cd5\u201d\u5bf9\u5e94\u7684 UTF-8 \u7f16\u7801\u3002\u89c2\u5bdf\u53d1\u73b0\uff0c\u7531\u4e8e\u6700\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u56e0\u6b64\u7cfb\u7edf\u53ef\u4ee5\u901a\u8fc7\u8bfb\u53d6\u6700\u9ad8\u4f4d \\(1\\) \u7684\u4e2a\u6570\u6765\u89e3\u6790\u51fa\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(n\\) \u3002
\u4f46\u4e3a\u4ec0\u4e48\u8981\u5c06\u5176\u4f59\u6240\u6709\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \u5462\uff1f\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a \\(10\\) \u80fd\u591f\u8d77\u5230\u6821\u9a8c\u7b26\u7684\u4f5c\u7528\u3002\u5047\u8bbe\u7cfb\u7edf\u4ece\u4e00\u4e2a\u9519\u8bef\u7684\u5b57\u8282\u5f00\u59cb\u89e3\u6790\u6587\u672c\uff0c\u5b57\u8282\u5934\u90e8\u7684 \\(10\\) \u80fd\u591f\u5e2e\u52a9\u7cfb\u7edf\u5feb\u901f\u5224\u65ad\u51fa\u5f02\u5e38\u3002
\u4e4b\u6240\u4ee5\u5c06 \\(10\\) \u5f53\u4f5c\u6821\u9a8c\u7b26\uff0c\u662f\u56e0\u4e3a\u5728 UTF-8 \u7f16\u7801\u89c4\u5219\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \u3002\u8fd9\u4e2a\u7ed3\u8bba\u53ef\u4ee5\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\uff1a\u5047\u8bbe\u4e00\u4e2a\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \uff0c\u8bf4\u660e\u8be5\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(1\\) \uff0c\u5bf9\u5e94 ASCII \u7801\u3002\u800c ASCII \u7801\u7684\u6700\u9ad8\u4f4d\u5e94\u8be5\u662f \\(0\\) \uff0c\u4e0e\u5047\u8bbe\u77db\u76fe\u3002
\u56fe 3-8 \u00a0 UTF-8 \u7f16\u7801\u793a\u4f8b
\u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u89c1\u7684\u7f16\u7801\u65b9\u5f0f\u8fd8\u5305\u62ec\u4ee5\u4e0b\u4e24\u79cd\u3002
- UTF-16 \u7f16\u7801\uff1a\u4f7f\u7528 2 \u6216 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u3002\u6240\u6709\u7684 ASCII \u5b57\u7b26\u548c\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u7b26\uff0c\u90fd\u7528 2 \u5b57\u8282\u8868\u793a\uff1b\u5c11\u6570\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u5b57\u8282\u8868\u793a\u3002\u5bf9\u4e8e 2 \u5b57\u8282\u7684\u5b57\u7b26\uff0cUTF-16 \u7f16\u7801\u4e0e Unicode \u7801\u70b9\u76f8\u7b49\u3002
- UTF-32 \u7f16\u7801\uff1a\u6bcf\u4e2a\u5b57\u7b26\u90fd\u4f7f\u7528 4 \u5b57\u8282\u3002\u8fd9\u610f\u5473\u7740 UTF-32 \u6bd4 UTF-8 \u548c UTF-16 \u66f4\u5360\u7528\u7a7a\u95f4\uff0c\u7279\u522b\u662f\u5bf9\u4e8e ASCII \u5b57\u7b26\u5360\u6bd4\u8f83\u9ad8\u7684\u6587\u672c\u3002
\u4ece\u5b58\u50a8\u7a7a\u95f4\u5360\u7528\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u7b26\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 1 \u5b57\u8282\uff1b\u4f7f\u7528 UTF-16 \u7f16\u7801\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u7b26\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u4f1a\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 2 \u5b57\u8282\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u5b57\u8282\u3002
\u4ece\u517c\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8bb8\u591a\u5de5\u5177\u548c\u5e93\u4f18\u5148\u652f\u6301 UTF-8 \u3002
"},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801","text":"\u5bf9\u4e8e\u4ee5\u5f80\u7684\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff0c\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u7684\u5b57\u7b26\u4e32\u90fd\u91c7\u7528 UTF-16 \u6216 UTF-32 \u8fd9\u7c7b\u7b49\u957f\u7684\u7f16\u7801\u3002\u5728\u7b49\u957f\u7f16\u7801\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u7b26\u4e32\u770b\u4f5c\u6570\u7ec4\u6765\u5904\u7406\uff0c\u8fd9\u79cd\u505a\u6cd5\u5177\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002
- \u968f\u673a\u8bbf\u95ee\uff1aUTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\u3002UTF-8 \u662f\u4e00\u79cd\u53d8\u957f\u7f16\u7801\uff0c\u8981\u60f3\u627e\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u4ece\u5b57\u7b26\u4e32\u7684\u5f00\u59cb\u5904\u904d\u5386\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u8fd9\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u3002
- \u5b57\u7b26\u8ba1\u6570\uff1a\u4e0e\u968f\u673a\u8bbf\u95ee\u7c7b\u4f3c\uff0c\u8ba1\u7b97 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8ba1\u7b97 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u9700\u8981\u904d\u5386\u6574\u4e2a\u5b57\u7b26\u4e32\u3002
- \u5b57\u7b26\u4e32\u64cd\u4f5c\uff1a\u5728 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u5f88\u591a\u5b57\u7b26\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u8fde\u63a5\u3001\u63d2\u5165\u3001\u5220\u9664\u7b49\uff09\u66f4\u5bb9\u6613\u8fdb\u884c\u3002\u5728 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u8fdb\u884c\u8fd9\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u989d\u5916\u7684\u8ba1\u7b97\uff0c\u4ee5\u786e\u4fdd\u4e0d\u4f1a\u4ea7\u751f\u65e0\u6548\u7684 UTF-8 \u7f16\u7801\u3002
\u5b9e\u9645\u4e0a\uff0c\u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6848\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5f88\u6709\u8da3\u7684\u8bdd\u9898\uff0c\u6d89\u53ca\u8bb8\u591a\u56e0\u7d20\u3002
- Java \u7684
String\u7c7b\u578b\u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a Java \u8bed\u8a00\u8bbe\u8ba1\u4e4b\u521d\uff0c\u4eba\u4eec\u8ba4\u4e3a 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u7b26\u3002\u7136\u800c\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0d\u6b63\u786e\u7684\u5224\u65ad\u3002\u540e\u6765 Unicode \u89c4\u8303\u6269\u5c55\u5230\u4e86\u8d85\u8fc7 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u7b26\u73b0\u5728\u53ef\u80fd\u7531\u4e00\u5bf9 16 \u4f4d\u7684\u503c\uff08\u79f0\u4e3a\u201c\u4ee3\u7406\u5bf9\u201d\uff09\u8868\u793a\u3002 - JavaScript \u548c TypeScript \u7684\u5b57\u7b26\u4e32\u4f7f\u7528 UTF-16 \u7f16\u7801\u7684\u539f\u56e0\u4e0e Java \u7c7b\u4f3c\u3002\u5f53 1995 \u5e74 Netscape \u516c\u53f8\u9996\u6b21\u63a8\u51fa JavaScript \u8bed\u8a00\u65f6\uff0cUnicode \u8fd8\u5904\u4e8e\u53d1\u5c55\u65e9\u671f\uff0c\u90a3\u65f6\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7f16\u7801\u5c31\u8db3\u4ee5\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u7b26\u4e86\u3002
- C# \u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u4e3b\u8981\u662f\u56e0\u4e3a .NET \u5e73\u53f0\u662f\u7531 Microsoft \u8bbe\u8ba1\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u672f\uff08\u5305\u62ec Windows \u64cd\u4f5c\u7cfb\u7edf\uff09\u90fd\u5e7f\u6cdb\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002
\u7531\u4e8e\u4ee5\u4e0a\u7f16\u7a0b\u8bed\u8a00\u5bf9\u5b57\u7b26\u6570\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u4eec\u4e0d\u5f97\u4e0d\u91c7\u53d6\u201c\u4ee3\u7406\u5bf9\u201d\u7684\u65b9\u5f0f\u6765\u8868\u793a\u8d85\u8fc7 16 \u4f4d\u957f\u5ea6\u7684 Unicode \u5b57\u7b26\u3002\u8fd9\u662f\u4e00\u4e2a\u4e0d\u5f97\u5df2\u4e3a\u4e4b\u7684\u65e0\u5948\u4e4b\u4e3e\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5bf9\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u4e00\u4e2a\u5b57\u7b26\u53ef\u80fd\u5360\u7528 2 \u5b57\u8282\u6216 4 \u5b57\u8282\uff0c\u4ece\u800c\u4e27\u5931\u4e86\u7b49\u957f\u7f16\u7801\u7684\u4f18\u52bf\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5904\u7406\u4ee3\u7406\u5bf9\u9700\u8981\u589e\u52a0\u989d\u5916\u4ee3\u7801\uff0c\u8fd9\u63d0\u9ad8\u4e86\u7f16\u7a0b\u7684\u590d\u6742\u6027\u548c\u8c03\u8bd5\u96be\u5ea6\u3002
\u51fa\u4e8e\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7f16\u7a0b\u8bed\u8a00\u63d0\u51fa\u4e86\u4e00\u4e9b\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6848\u3002
- Python \u4e2d\u7684
str\u4f7f\u7528 Unicode \u7f16\u7801\uff0c\u5e76\u91c7\u7528\u4e00\u79cd\u7075\u6d3b\u7684\u5b57\u7b26\u4e32\u8868\u793a\uff0c\u5b58\u50a8\u7684\u5b57\u7b26\u957f\u5ea6\u53d6\u51b3\u4e8e\u5b57\u7b26\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u7801\u70b9\u3002\u82e5\u5b57\u7b26\u4e32\u4e2d\u5168\u90e8\u662f ASCII \u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff1b\u5982\u679c\u6709\u5b57\u7b26\u8d85\u51fa\u4e86 ASCII \u8303\u56f4\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8bed\u8a00\u5e73\u9762\uff08BMP\uff09\u5185\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff1b\u5982\u679c\u6709\u8d85\u51fa BMP \u7684\u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 4 \u5b57\u8282\u3002 - Go \u8bed\u8a00\u7684
string\u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Go \u8bed\u8a00\u8fd8\u63d0\u4f9b\u4e86rune\u7c7b\u578b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002 - Rust \u8bed\u8a00\u7684 str \u548c String \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Rust \u4e5f\u63d0\u4f9b\u4e86
char\u7c7b\u578b\uff0c\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002
\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8ba8\u8bba\u7684\u90fd\u662f\u5b57\u7b26\u4e32\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u8fd9\u548c\u5b57\u7b26\u4e32\u5982\u4f55\u5728\u6587\u4ef6\u4e2d\u5b58\u50a8\u6216\u5728\u7f51\u7edc\u4e2d\u4f20\u8f93\u662f\u4e0d\u540c\u7684\u95ee\u9898\u3002\u5728\u6587\u4ef6\u5b58\u50a8\u6216\u7f51\u7edc\u4f20\u8f93\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u5b57\u7b26\u4e32\u7f16\u7801\u4e3a UTF-8 \u683c\u5f0f\uff0c\u4ee5\u8fbe\u5230\u6700\u4f18\u7684\u517c\u5bb9\u6027\u548c\u7a7a\u95f4\u6548\u7387\u3002
"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\uff0c\u5b83\u4eec\u53ef\u4ee5\u4ece\u201c\u903b\u8f91\u7ed3\u6784\u201d\u548c\u201c\u7269\u7406\u7ed3\u6784\u201d\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002
"},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1 \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"\u903b\u8f91\u7ed3\u6784\u63ed\u793a\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u5728\u6570\u7ec4\u548c\u94fe\u8868\u4e2d\uff0c\u6570\u636e\u6309\u7167\u4e00\u5b9a\u987a\u5e8f\u6392\u5217\uff0c\u4f53\u73b0\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u800c\u5728\u6811\u4e2d\uff0c\u6570\u636e\u4ece\u9876\u90e8\u5411\u4e0b\u6309\u5c42\u6b21\u6392\u5217\uff0c\u8868\u73b0\u51fa\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u5219\u7531\u8282\u70b9\u548c\u8fb9\u6784\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7684\u7f51\u7edc\u5173\u7cfb\u3002
\u5982\u56fe 3-1 \u6240\u793a\uff0c\u903b\u8f91\u7ed3\u6784\u53ef\u5206\u4e3a\u201c\u7ebf\u6027\u201d\u548c\u201c\u975e\u7ebf\u6027\u201d\u4e24\u5927\u7c7b\u3002\u7ebf\u6027\u7ed3\u6784\u6bd4\u8f83\u76f4\u89c2\uff0c\u6307\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u5448\u7ebf\u6027\u6392\u5217\uff1b\u975e\u7ebf\u6027\u7ed3\u6784\u5219\u76f8\u53cd\uff0c\u5448\u975e\u7ebf\u6027\u6392\u5217\u3002
- \u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3002
- \u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u56fe\u3001\u54c8\u5e0c\u8868\u3002
\u56fe 3-1 \u00a0 \u7ebf\u6027\u6570\u636e\u7ed3\u6784\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784
\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u6811\u5f62\u7ed3\u6784\u548c\u7f51\u72b6\u7ed3\u6784\u3002
- \u7ebf\u6027\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u961f\u5217\u3001\u6808\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u4e00\u7684\u987a\u5e8f\u5173\u7cfb\u3002
- \u6811\u5f62\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u591a\u7684\u5173\u7cfb\u3002
- \u7f51\u72b6\u7ed3\u6784\uff1a\u56fe\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u591a\u5bf9\u591a\u7684\u5173\u7cfb\u3002
\u5f53\u7b97\u6cd5\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u56fe 3-2 \u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8\u4e00\u5b9a\u5927\u5c0f\u7684\u6570\u636e\u3002
\u7cfb\u7edf\u901a\u8fc7\u5185\u5b58\u5730\u5740\u6765\u8bbf\u95ee\u76ee\u6807\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u5982\u56fe 3-2 \u6240\u793a\uff0c\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u4e3a\u8868\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5206\u914d\u7f16\u53f7\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u552f\u4e00\u7684\u5185\u5b58\u5730\u5740\u3002\u6709\u4e86\u8fd9\u4e9b\u5730\u5740\uff0c\u7a0b\u5e8f\u4fbf\u53ef\u4ee5\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002
\u56fe 3-2 \u00a0 \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740
Tip
\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5c06\u5185\u5b58\u6bd4\u4f5c Excel \u8868\u683c\u662f\u4e00\u4e2a\u7b80\u5316\u7684\u7c7b\u6bd4\uff0c\u5b9e\u9645\u5185\u5b58\u7684\u5de5\u4f5c\u673a\u5236\u6bd4\u8f83\u590d\u6742\uff0c\u6d89\u53ca\u5730\u5740\u7a7a\u95f4\u3001\u5185\u5b58\u7ba1\u7406\u3001\u7f13\u5b58\u673a\u5236\u3001\u865a\u62df\u5185\u5b58\u548c\u7269\u7406\u5185\u5b58\u7b49\u6982\u5ff5\u3002
\u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u5171\u4eab\u8d44\u6e90\uff0c\u5f53\u67d0\u5757\u5185\u5b58\u88ab\u67d0\u4e2a\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u5219\u65e0\u6cd5\u88ab\u5176\u4ed6\u7a0b\u5e8f\u540c\u65f6\u4f7f\u7528\u4e86\u3002\u56e0\u6b64\u5728\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bbe\u8ba1\u4e2d\uff0c\u5185\u5b58\u8d44\u6e90\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u8003\u8651\u56e0\u7d20\u3002\u6bd4\u5982\uff0c\u7b97\u6cd5\u6240\u5360\u7528\u7684\u5185\u5b58\u5cf0\u503c\u4e0d\u5e94\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u7a7a\u95f2\u5185\u5b58\uff1b\u5982\u679c\u7f3a\u5c11\u8fde\u7eed\u5927\u5757\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u90a3\u4e48\u6240\u9009\u7528\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u5206\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002
\u5982\u56fe 3-3 \u6240\u793a\uff0c\u7269\u7406\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u53ef\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u70b9\u3002
\u56fe 3-3 \u00a0 \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u5206\u6563\u7a7a\u95f4\u5b58\u50a8
\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e8c\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\uff0c\u6808\u548c\u961f\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff1b\u800c\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u6570\u7ec4\u548c\u94fe\u8868\u3002
- \u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\u3002
- \u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u3002
\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u610f\u5473\u7740\u6b64\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u5bf9\u5e94\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u79f0\u201c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u5bf9\u5176\u957f\u5ea6\u8fdb\u884c\u8c03\u6574\u3002
Tip
\u5982\u679c\u4f60\u611f\u89c9\u7269\u7406\u7ed3\u6784\u7406\u89e3\u8d77\u6765\u6709\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u9605\u8bfb\u4e0b\u4e00\u7ae0\uff0c\u7136\u540e\u518d\u56de\u987e\u672c\u8282\u5185\u5bb9\u3002
"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *","text":"Note
\u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u5e26\u6709 * \u7b26\u53f7\u7684\u662f\u9009\u8bfb\u7ae0\u8282\u3002\u5982\u679c\u4f60\u65f6\u95f4\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u518d\u5355\u72ec\u653b\u514b\u3002
"},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1 \u00a0 \u6574\u6570\u7f16\u7801","text":"\u5728\u4e0a\u4e00\u8282\u7684\u8868\u683c\u4e2d\u6211\u4eec\u53d1\u73b0\uff0c\u6240\u6709\u6574\u6570\u7c7b\u578b\u80fd\u591f\u8868\u793a\u7684\u8d1f\u6570\u90fd\u6bd4\u6b63\u6570\u591a\u4e00\u4e2a\uff0c\u4f8b\u5982
byte\u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \u3002\u8fd9\u4e2a\u73b0\u8c61\u6bd4\u8f83\u53cd\u76f4\u89c9\uff0c\u5b83\u7684\u5185\u5728\u539f\u56e0\u6d89\u53ca\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u76f8\u5173\u77e5\u8bc6\u3002\u9996\u5148\u9700\u8981\u6307\u51fa\uff0c\u6570\u5b57\u662f\u4ee5\u201c\u8865\u7801\u201d\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u7684\u3002\u5728\u5206\u6790\u8fd9\u6837\u505a\u7684\u539f\u56e0\u4e4b\u524d\uff0c\u9996\u5148\u7ed9\u51fa\u4e09\u8005\u7684\u5b9a\u4e49\u3002
- \u539f\u7801\uff1a\u6211\u4eec\u5c06\u6570\u5b57\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u89c6\u4e3a\u7b26\u53f7\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6570\uff0c\\(1\\) \u8868\u793a\u8d1f\u6570\uff0c\u5176\u4f59\u4f4d\u8868\u793a\u6570\u5b57\u7684\u503c\u3002
- \u53cd\u7801\uff1a\u6b63\u6570\u7684\u53cd\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u53cd\u7801\u662f\u5bf9\u5176\u539f\u7801\u9664\u7b26\u53f7\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002
- \u8865\u7801\uff1a\u6b63\u6570\u7684\u8865\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u8865\u7801\u662f\u5728\u5176\u53cd\u7801\u7684\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u3002
\u56fe 3-4 \u5c55\u793a\u4e86\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u4e4b\u95f4\u7684\u8f6c\u6362\u65b9\u6cd5\u3002
\u56fe 3-4 \u00a0 \u539f\u7801\u3001\u53cd\u7801\u4e0e\u8865\u7801\u4e4b\u95f4\u7684\u76f8\u4e92\u8f6c\u6362
\u300c\u539f\u7801 sign-magnitude\u300d\u867d\u7136\u6700\u76f4\u89c2\uff0c\u4f46\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u8d1f\u6570\u7684\u539f\u7801\u4e0d\u80fd\u76f4\u63a5\u7528\u4e8e\u8fd0\u7b97\u3002\u4f8b\u5982\u5728\u539f\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7ed3\u679c\u662f \\(-3\\) \uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002
\\[ \\begin{aligned} & 1 + (-2) \\newline & \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline & = 1000 \\; 0011 \\newline & \\rightarrow -3 \\end{aligned} \\]\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8ba1\u7b97\u673a\u5f15\u5165\u4e86\u300c\u53cd\u7801 1's complement\u300d\u3002\u5982\u679c\u6211\u4eec\u5148\u5c06\u539f\u7801\u8f6c\u6362\u4e3a\u53cd\u7801\uff0c\u5e76\u5728\u53cd\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u6700\u540e\u5c06\u7ed3\u679c\u4ece\u53cd\u7801\u8f6c\u6362\u56de\u539f\u7801\uff0c\u5219\u53ef\u5f97\u5230\u6b63\u786e\u7ed3\u679c \\(-1\\) \u3002
\\[ \\begin{aligned} & 1 + (-2) \\newline & \\rightarrow 0000 \\; 0001 \\; \\text{(\u539f\u7801)} + 1000 \\; 0010 \\; \\text{(\u539f\u7801)} \\newline & = 0000 \\; 0001 \\; \\text{(\u53cd\u7801)} + 1111 \\; 1101 \\; \\text{(\u53cd\u7801)} \\newline & = 1111 \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline & = 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline & \\rightarrow -1 \\end{aligned} \\]\u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b57\u96f6\u7684\u539f\u7801\u6709 \\(+0\\) \u548c \\(-0\\) \u4e24\u79cd\u8868\u793a\u65b9\u5f0f\u3002\u8fd9\u610f\u5473\u7740\u6570\u5b57\u96f6\u5bf9\u5e94\u4e24\u4e2a\u4e0d\u540c\u7684\u4e8c\u8fdb\u5236\u7f16\u7801\uff0c\u8fd9\u53ef\u80fd\u4f1a\u5e26\u6765\u6b67\u4e49\u3002\u6bd4\u5982\u5728\u6761\u4ef6\u5224\u65ad\u4e2d\uff0c\u5982\u679c\u6ca1\u6709\u533a\u5206\u6b63\u96f6\u548c\u8d1f\u96f6\uff0c\u5219\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5224\u65ad\u7ed3\u679c\u51fa\u9519\u3002\u800c\u5982\u679c\u6211\u4eec\u60f3\u5904\u7406\u6b63\u96f6\u548c\u8d1f\u96f6\u6b67\u4e49\uff0c\u5219\u9700\u8981\u5f15\u5165\u989d\u5916\u7684\u5224\u65ad\u64cd\u4f5c\uff0c\u8fd9\u53ef\u80fd\u4f1a\u964d\u4f4e\u8ba1\u7b97\u673a\u7684\u8fd0\u7b97\u6548\u7387\u3002
\\[ \\begin{aligned} +0 & \\rightarrow 0000 \\; 0000 \\newline -0 & \\rightarrow 1000 \\; 0000 \\end{aligned} \\]\u4e0e\u539f\u7801\u4e00\u6837\uff0c\u53cd\u7801\u4e5f\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\uff0c\u56e0\u6b64\u8ba1\u7b97\u673a\u8fdb\u4e00\u6b65\u5f15\u5165\u4e86\u300c\u8865\u7801 2's complement\u300d\u3002\u6211\u4eec\u5148\u6765\u89c2\u5bdf\u4e00\u4e0b\u8d1f\u96f6\u7684\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u8f6c\u6362\u8fc7\u7a0b\uff1a
\\[ \\begin{aligned} -0 \\rightarrow \\; & 1000 \\; 0000 \\; \\text{(\u539f\u7801)} \\newline = \\; & 1111 \\; 1111 \\; \\text{(\u53cd\u7801)} \\newline = 1 \\; & 0000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline \\end{aligned} \\]\u5728\u8d1f\u96f6\u7684\u53cd\u7801\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u4f1a\u4ea7\u751f\u8fdb\u4f4d\uff0c\u4f46
byte\u7c7b\u578b\u7684\u957f\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u51fa\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u4f1a\u88ab\u820d\u5f03\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8d1f\u96f6\u7684\u8865\u7801\u4e3a \\(0000 \\; 0000\\) \uff0c\u4e0e\u6b63\u96f6\u7684\u8865\u7801\u76f8\u540c\u3002\u8fd9\u610f\u5473\u7740\u5728\u8865\u7801\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u4e2a\u96f6\uff0c\u6b63\u8d1f\u96f6\u6b67\u4e49\u4ece\u800c\u5f97\u5230\u89e3\u51b3\u3002\u8fd8\u5269\u6700\u540e\u4e00\u4e2a\u7591\u60d1\uff1a
byte\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u6765\u7684\u4e00\u4e2a\u8d1f\u6570 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u4eec\u6ce8\u610f\u5230\uff0c\u533a\u95f4 \\([-127, +127]\\) \u5185\u7684\u6240\u6709\u6574\u6570\u90fd\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\uff0c\u5e76\u4e14\u539f\u7801\u548c\u8865\u7801\u4e4b\u95f4\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\u3002\u7136\u800c\uff0c\u8865\u7801 \\(1000 \\; 0000\\) \u662f\u4e00\u4e2a\u4f8b\u5916\uff0c\u5b83\u5e76\u6ca1\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3002\u6839\u636e\u8f6c\u6362\u65b9\u6cd5\uff0c\u6211\u4eec\u5f97\u5230\u8be5\u8865\u7801\u7684\u539f\u7801\u4e3a \\(0000 \\; 0000\\) \u3002\u8fd9\u663e\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u4e3a\u8be5\u539f\u7801\u8868\u793a\u6570\u5b57 \\(0\\) \uff0c\u5b83\u7684\u8865\u7801\u5e94\u8be5\u662f\u81ea\u8eab\u3002\u8ba1\u7b97\u673a\u89c4\u5b9a\u8fd9\u4e2a\u7279\u6b8a\u7684\u8865\u7801 \\(1000 \\; 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5b9e\u9645\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u8865\u7801\u4e0b\u7684\u8ba1\u7b97\u7ed3\u679c\u5c31\u662f \\(-128\\) \u3002
\\[ \\begin{aligned} & (-127) + (-1) \\newline & \\rightarrow 1111 \\; 1111 \\; \\text{(\u539f\u7801)} + 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline & = 1000 \\; 0000 \\; \\text{(\u53cd\u7801)} + 1111 \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline & = 1000 \\; 0001 \\; \\text{(\u8865\u7801)} + 1111 \\; 1111 \\; \\text{(\u8865\u7801)} \\newline & = 1000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline & \\rightarrow -128 \\end{aligned} \\]\u4f60\u53ef\u80fd\u5df2\u7ecf\u53d1\u73b0\u4e86\uff0c\u4e0a\u8ff0\u6240\u6709\u8ba1\u7b97\u90fd\u662f\u52a0\u6cd5\u8fd0\u7b97\u3002\u8fd9\u6697\u793a\u7740\u4e00\u4e2a\u91cd\u8981\u4e8b\u5b9e\uff1a\u8ba1\u7b97\u673a\u5185\u90e8\u7684\u786c\u4ef6\u7535\u8def\u4e3b\u8981\u662f\u57fa\u4e8e\u52a0\u6cd5\u8fd0\u7b97\u8bbe\u8ba1\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u52a0\u6cd5\u8fd0\u7b97\u76f8\u5bf9\u4e8e\u5176\u4ed6\u8fd0\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u51cf\u6cd5\uff09\u6765\u8bf4\uff0c\u786c\u4ef6\u5b9e\u73b0\u8d77\u6765\u66f4\u7b80\u5355\uff0c\u66f4\u5bb9\u6613\u8fdb\u884c\u5e76\u884c\u5316\u5904\u7406\uff0c\u8fd0\u7b97\u901f\u5ea6\u66f4\u5feb\u3002
\u8bf7\u6ce8\u610f\uff0c\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u8ba1\u7b97\u673a\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u901a\u8fc7\u5c06\u52a0\u6cd5\u4e0e\u4e00\u4e9b\u57fa\u672c\u903b\u8f91\u8fd0\u7b97\u7ed3\u5408\uff0c\u8ba1\u7b97\u673a\u80fd\u591f\u5b9e\u73b0\u5404\u79cd\u5176\u4ed6\u7684\u6570\u5b66\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u51cf\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8ba1\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u51cf\u6cd5\u3002
\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u8ba1\u7b97\u673a\u4f7f\u7528\u8865\u7801\u7684\u539f\u56e0\uff1a\u57fa\u4e8e\u8865\u7801\u8868\u793a\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u7528\u540c\u6837\u7684\u7535\u8def\u548c\u64cd\u4f5c\u6765\u5904\u7406\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\u6765\u5904\u7406\u51cf\u6cd5\uff0c\u5e76\u4e14\u65e0\u987b\u7279\u522b\u5904\u7406\u6b63\u8d1f\u96f6\u7684\u6b67\u4e49\u95ee\u9898\u3002\u8fd9\u5927\u5927\u7b80\u5316\u4e86\u786c\u4ef6\u8bbe\u8ba1\uff0c\u63d0\u9ad8\u4e86\u8fd0\u7b97\u6548\u7387\u3002
\u8865\u7801\u7684\u8bbe\u8ba1\u975e\u5e38\u7cbe\u5999\uff0c\u56e0\u7bc7\u5e45\u5173\u7cfb\u6211\u4eec\u5c31\u5148\u4ecb\u7ecd\u5230\u8fd9\u91cc\uff0c\u5efa\u8bae\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8fdb\u4e00\u6b65\u6df1\u5165\u4e86\u89e3\u3002
"},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2 \u00a0 \u6d6e\u70b9\u6570\u7f16\u7801","text":"\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\uff1a
int\u548cfloat\u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 bytes \uff0c\u4f46\u4e3a\u4ec0\u4e48float\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8eint\uff1f\u8fd9\u975e\u5e38\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u6309\u7406\u8bf4float\u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570
\\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\]float\u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u8bb0\u4e00\u4e2a 32-bit \u957f\u5ea6\u7684\u4e8c\u8fdb\u5236\u6570\u4e3a\uff1a\u6839\u636e IEEE 754 \u6807\u51c6\uff0c32-bit \u957f\u5ea6\u7684
float\u7531\u4ee5\u4e0b\u4e09\u4e2a\u90e8\u5206\u6784\u6210\u3002- \u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 bit \uff0c\u5bf9\u5e94 \\(b_{31}\\) \u3002
- \u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 bits \uff0c\u5bf9\u5e94 \\(b_{30} b_{29} \\ldots b_{23}\\) \u3002
- \u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 23 bits \uff0c\u5bf9\u5e94 \\(b_{22} b_{21} \\ldots b_0\\) \u3002
\u4e8c\u8fdb\u5236\u6570
\\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\]float\u5bf9\u5e94\u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u4e3a\uff1a\u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a
\\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\]\u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a\uff1a
\\[ \\begin{aligned} \\mathrm{S} \\in & \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = & (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\]\u56fe 3-5 \u00a0 IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u7684\u8ba1\u7b97\u793a\u4f8b
\u89c2\u5bdf\u56fe 3-5 \uff0c\u7ed9\u5b9a\u4e00\u4e2a\u793a\u4f8b\u6570\u636e \\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u5219\u6709\uff1a
\\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\]\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u95ee\u9898\uff1a
float\u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8eint\u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0cfloat\u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002\u5c3d\u7ba1\u6d6e\u70b9\u6570
float\u6269\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578bint\u5c06\u5168\u90e8 32 \u4f4d\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570float\u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002\u5982\u8868 3-2 \u6240\u793a\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002
\u8868 3-2 \u00a0 \u6307\u6570\u4f4d\u542b\u4e49
\u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\)\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u6d6e\u70b9\u6570\u7684\u7cbe\u5ea6\u3002\u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126}\\) \uff0c\u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23}\\) \u3002
\u53cc\u7cbe\u5ea6
"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_data_structure/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"double\u4e5f\u91c7\u7528\u7c7b\u4f3c\u4e8efloat\u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u505a\u8d58\u8ff0\u3002- \u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u800c\u7269\u7406\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002
- \u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u5305\u62ec\u7ebf\u6027\u3001\u6811\u72b6\u548c\u7f51\u72b6\u7b49\u3002\u901a\u5e38\u6211\u4eec\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002
- \u5f53\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u88ab\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u3002\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u62e5\u6709\u5bf9\u5e94\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u901a\u8fc7\u8fd9\u4e9b\u5185\u5b58\u5730\u5740\u8bbf\u95ee\u6570\u636e\u3002
- \u7269\u7406\u7ed3\u6784\u4e3b\u8981\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e24\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002
- \u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5305\u62ec\u6574\u6570
byte\u3001short\u3001int\u3001long\uff0c\u6d6e\u70b9\u6570float\u3001double\uff0c\u5b57\u7b26char\u548c\u5e03\u5c14boolean\u3002\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002 - \u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7f16\u7801\u6570\u5b57\u7684\u4e09\u79cd\u65b9\u6cd5\uff0c\u5b83\u4eec\u4e4b\u95f4\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\u3002\u6574\u6570\u7684\u539f\u7801\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u53f7\u4f4d\uff0c\u5176\u4f59\u4f4d\u662f\u6570\u5b57\u7684\u503c\u3002
- \u6574\u6570\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u8865\u7801\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\u3002\u5728\u8865\u7801\u8868\u793a\u4e0b\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u5bf9\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\u4e00\u89c6\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u4e3a\u51cf\u6cd5\u64cd\u4f5c\u5355\u72ec\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\uff0c\u5e76\u4e14\u4e0d\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u7684\u95ee\u9898\u3002
- \u6d6e\u70b9\u6570\u7684\u7f16\u7801\u7531 1 \u4f4d\u7b26\u53f7\u4f4d\u30018 \u4f4d\u6307\u6570\u4f4d\u548c 23 \u4f4d\u5206\u6570\u4f4d\u6784\u6210\u3002\u7531\u4e8e\u5b58\u5728\u6307\u6570\u4f4d\uff0c\u56e0\u6b64\u6d6e\u70b9\u6570\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e\u6574\u6570\uff0c\u4ee3\u4ef7\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002
- ASCII \u7801\u662f\u6700\u65e9\u51fa\u73b0\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u957f\u5ea6\u4e3a 1 \u5b57\u8282\uff0c\u5171\u6536\u5f55 127 \u4e2a\u5b57\u7b26\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u5f55\u4e24\u4e07\u591a\u4e2a\u6c49\u5b57\u3002Unicode \u81f4\u529b\u4e8e\u63d0\u4f9b\u4e00\u4e2a\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u6536\u5f55\u4e16\u754c\u5185\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u4ece\u800c\u89e3\u51b3\u7531\u4e8e\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5bfc\u81f4\u7684\u4e71\u7801\u95ee\u9898\u3002
- UTF-8 \u662f\u6700\u53d7\u6b22\u8fce\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u79cd\u53d8\u957f\u7684\u7f16\u7801\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u6269\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5b58\u50a8\u7a7a\u95f4\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u957f\u7684\u7f16\u7801\u65b9\u6cd5\u3002\u5728\u7f16\u7801\u4e2d\u6587\u65f6\uff0cUTF-16 \u6bd4 UTF-8 \u7684\u5360\u7528\u7a7a\u95f4\u66f4\u5c0f\u3002Java \u548c C# \u7b49\u7f16\u7a0b\u8bed\u8a00\u9ed8\u8ba4\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002
\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1f
\u54c8\u5e0c\u8868\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u800c\u4e3a\u4e86\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\uff0c\u6211\u4eec\u53ef\u80fd\u4f1a\u4f7f\u7528\u201c\u94fe\u5f0f\u5730\u5740\u201d\uff08\u540e\u7eed\u201c\u54c8\u5e0c\u51b2\u7a81\u201d\u7ae0\u8282\u4f1a\u8bb2\uff09\uff1a\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u6876\u6307\u5411\u4e00\u4e2a\u94fe\u8868\uff0c\u5f53\u94fe\u8868\u957f\u5ea6\u8d85\u8fc7\u4e00\u5b9a\u9608\u503c\u65f6\uff0c\u53c8\u53ef\u80fd\u88ab\u8f6c\u5316\u4e3a\u6811\uff08\u901a\u5e38\u4e3a\u7ea2\u9ed1\u6811\uff09\u3002 \u4ece\u5b58\u50a8\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u54c8\u5e0c\u8868\u7684\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u6876\u69fd\u4f4d\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u503c\uff0c\u4e5f\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u94fe\u8868\u6216\u4e00\u68f5\u6811\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\u3001\u94fe\u8868\uff09\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff08\u6811\uff09\u3002
char\u7c7b\u578b\u7684\u957f\u5ea6\u662f 1 byte \u5417\uff1fchar\u7c7b\u578b\u7684\u957f\u5ea6\u7531\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u7f16\u7801\u65b9\u6cd5\u51b3\u5b9a\u3002\u4f8b\u5982\uff0cJava\u3001JavaScript\u3001TypeScript\u3001C# \u90fd\u91c7\u7528 UTF-16 \u7f16\u7801\uff08\u4fdd\u5b58 Unicode \u7801\u70b9\uff09\uff0c\u56e0\u6b64 char \u7c7b\u578b\u7684\u957f\u5ea6\u4e3a 2 bytes\u3002\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d \u662f\u5426\u6709\u6b67\u4e49\uff1f\u56e0\u4e3a\u6808\u4e5f\u53ef\u4ee5\u8fdb\u884c\u51fa\u6808\u548c\u5165\u6808\u7b49\u64cd\u4f5c\uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u662f\u201c\u52a8\u6001\u201d\u7684\u3002
\u6808\u786e\u5b9e\u53ef\u4ee5\u5b9e\u73b0\u52a8\u6001\u7684\u6570\u636e\u64cd\u4f5c\uff0c\u4f46\u6570\u636e\u7ed3\u6784\u4ecd\u7136\u662f\u201c\u9759\u6001\u201d\uff08\u957f\u5ea6\u4e0d\u53ef\u53d8\uff09\u7684\u3002\u5c3d\u7ba1\u57fa\u4e8e\u6570\u7ec4\u7684\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u52a8\u6001\u5730\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\uff0c\u4f46\u5b83\u4eec\u7684\u5bb9\u91cf\u662f\u56fa\u5b9a\u7684\u3002\u5982\u679c\u6570\u636e\u91cf\u8d85\u51fa\u4e86\u9884\u5206\u914d\u7684\u5927\u5c0f\uff0c\u5c31\u9700\u8981\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u65e7\u6570\u7ec4\u7684\u5185\u5bb9\u590d\u5236\u5230\u65b0\u6570\u7ec4\u4e2d\u3002
\u5728\u6784\u5efa\u6808\uff08\u961f\u5217\uff09\u7684\u65f6\u5019\uff0c\u672a\u6307\u5b9a\u5b83\u7684\u5927\u5c0f\uff0c\u4e3a\u4ec0\u4e48\u5b83\u4eec\u662f\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\u5462\uff1f
\u5728\u9ad8\u7ea7\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e0\u987b\u4eba\u5de5\u6307\u5b9a\u6808\uff08\u961f\u5217\uff09\u7684\u521d\u59cb\u5bb9\u91cf\uff0c\u8fd9\u4e2a\u5de5\u4f5c\u7531\u7c7b\u5185\u90e8\u81ea\u52a8\u5b8c\u6210\u3002\u4f8b\u5982\uff0cJava \u7684 ArrayList \u7684\u521d\u59cb\u5bb9\u91cf\u901a\u5e38\u4e3a 10\u3002\u53e6\u5916\uff0c\u6269\u5bb9\u64cd\u4f5c\u4e5f\u662f\u81ea\u52a8\u5b9e\u73b0\u7684\u3002\u8be6\u89c1\u540e\u7eed\u7684\u201c\u5217\u8868\u201d\u7ae0\u8282\u3002
"},{"location":"chapter_divide_and_conquer/","title":"\u7b2c 12 \u7ae0 \u00a0 \u5206\u6cbb","text":"Abstract
\u96be\u9898\u88ab\u9010\u5c42\u62c6\u89e3\uff0c\u6bcf\u4e00\u6b21\u7684\u62c6\u89e3\u90fd\u4f7f\u5b83\u53d8\u5f97\u66f4\u4e3a\u7b80\u5355\u3002
\u5206\u800c\u6cbb\u4e4b\u63ed\u793a\u4e86\u4e00\u4e2a\u91cd\u8981\u7684\u4e8b\u5b9e\uff1a\u4ece\u7b80\u5355\u505a\u8d77\uff0c\u4e00\u5207\u90fd\u4e0d\u518d\u590d\u6742\u3002
"},{"location":"chapter_divide_and_conquer/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5
- 12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565
- 12.3 \u00a0 \u6784\u5efa\u6811\u95ee\u9898
- 12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898
- 12.5 \u00a0 \u5c0f\u7ed3
\u6211\u4eec\u5df2\u7ecf\u5b66\u8fc7\uff0c\u641c\u7d22\u7b97\u6cd5\u5206\u4e3a\u4e24\u5927\u7c7b\u3002
- \u66b4\u529b\u641c\u7d22\uff1a\u5b83\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
- \u81ea\u9002\u5e94\u641c\u7d22\uff1a\u5b83\u5229\u7528\u7279\u6709\u7684\u6570\u636e\u7ec4\u7ec7\u5f62\u5f0f\u6216\u5148\u9a8c\u4fe1\u606f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002
\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u548c\u6811\u3002
- \u4e8c\u5206\u67e5\u627e\u7684\u6bcf\u4e00\u6b65\u90fd\u5c06\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\u5206\u89e3\u4e3a\u4e00\u4e2a\u5c0f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u4e00\u76f4\u6301\u7eed\u5230\u6570\u7ec4\u4e3a\u7a7a\u6216\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002
- \u6811\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4ee3\u8868\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u5806\u7b49\u6570\u636e\u7ed3\u6784\u4e2d\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(\\log n)\\) \u3002
\u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u7b56\u7565\u5982\u4e0b\u6240\u793a\u3002
- \u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4e8c\u5206\u67e5\u627e\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\u5206\u89e3\u4e3a\u5b50\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\uff0c\u8fd9\u662f\u901a\u8fc7\u6bd4\u8f83\u4e2d\u95f4\u5143\u7d20\u548c\u76ee\u6807\u5143\u7d20\u6765\u5b9e\u73b0\u7684\u3002
- \u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u5904\u7406\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5b83\u4e0d\u53d7\u5176\u4ed6\u5b50\u95ee\u9898\u7684\u5f71\u54cd\u3002
- \u5b50\u95ee\u9898\u7684\u89e3\u65e0\u987b\u5408\u5e76\uff1a\u4e8c\u5206\u67e5\u627e\u65e8\u5728\u67e5\u627e\u4e00\u4e2a\u7279\u5b9a\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u3002\u5f53\u5b50\u95ee\u9898\u5f97\u5230\u89e3\u51b3\u65f6\uff0c\u539f\u95ee\u9898\u4e5f\u4f1a\u540c\u65f6\u5f97\u5230\u89e3\u51b3\u3002
\u5206\u6cbb\u80fd\u591f\u63d0\u5347\u641c\u7d22\u6548\u7387\uff0c\u672c\u8d28\u4e0a\u662f\u56e0\u4e3a\u66b4\u529b\u641c\u7d22\u6bcf\u8f6e\u53ea\u80fd\u6392\u9664\u4e00\u4e2a\u9009\u9879\uff0c\u800c\u5206\u6cbb\u641c\u7d22\u6bcf\u8f6e\u53ef\u4ee5\u6392\u9664\u4e00\u534a\u9009\u9879\u3002
"},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","title":"1. \u00a0 \u57fa\u4e8e\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e","text":"\u5728\u4e4b\u524d\u7684\u7ae0\u8282\u4e2d\uff0c\u4e8c\u5206\u67e5\u627e\u662f\u57fa\u4e8e\u9012\u63a8\uff08\u8fed\u4ee3\uff09\u5b9e\u73b0\u7684\u3002\u73b0\u5728\u6211\u4eec\u57fa\u4e8e\u5206\u6cbb\uff08\u9012\u5f52\uff09\u6765\u5b9e\u73b0\u5b83\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4
nums\uff0c\u5176\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\uff0c\u8bf7\u67e5\u627e\u5143\u7d20target\u3002\u4ece\u5206\u6cbb\u89d2\u5ea6\uff0c\u6211\u4eec\u5c06\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u8bb0\u4e3a \\(f(i, j)\\) \u3002
\u4ee5\u539f\u95ee\u9898 \\(f(0, n-1)\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u8fdb\u884c\u4e8c\u5206\u67e5\u627e\u3002
- \u8ba1\u7b97\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u7684\u4e2d\u70b9 \\(m\\) \uff0c\u6839\u636e\u5b83\u6392\u9664\u4e00\u534a\u641c\u7d22\u533a\u95f4\u3002
- \u9012\u5f52\u6c42\u89e3\u89c4\u6a21\u51cf\u5c0f\u4e00\u534a\u7684\u5b50\u95ee\u9898\uff0c\u53ef\u80fd\u4e3a \\(f(i, m-1)\\) \u6216 \\(f(m+1, j)\\) \u3002
- \u5faa\u73af\u7b2c
1.\u6b65\u548c\u7b2c2.\u6b65\uff0c\u76f4\u81f3\u627e\u5230target\u6216\u533a\u95f4\u4e3a\u7a7a\u65f6\u8fd4\u56de\u3002
\u56fe 12-4 \u5c55\u793a\u4e86\u5728\u6570\u7ec4\u4e2d\u4e8c\u5206\u67e5\u627e\u5143\u7d20 \\(6\\) \u7684\u5206\u6cbb\u8fc7\u7a0b\u3002
\u56fe 12-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u8fc7\u7a0b
\u5728\u5b9e\u73b0\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_recur.pydfs()\u6765\u6c42\u89e3\u95ee\u9898 \\(f(i, j)\\) \uff1a
binary_search_recur.cppdef dfs(nums: list[int], target: int, i: int, j: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j)\"\"\"\n # \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if i > j:\n return -1\n # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n m = (i + j) // 2\n if nums[m] < target:\n # \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j)\n elif nums[m] > target:\n # \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1)\n else:\n # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n\ndef binary_search(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\"\"\"\n n = len(nums)\n # \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1)\n
binary_search_recur.java/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(vector<int> &nums, int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) / 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(vector<int> &nums, int target) {\n int n = nums.size();\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.cs/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) / 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int[] nums, int target) {\n int n = nums.length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.go/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) / 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return DFS(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return DFS(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint BinarySearch(int[] nums, int target) {\n int n = nums.Length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return DFS(nums, target, 0, n - 1);\n}\n
binary_search_recur.swift/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n // \u5982\u679c\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u6ca1\u6709\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if i > j {\n return -1\n }\n // \u8ba1\u7b97\u7d22\u5f15\u4e2d\u70b9\n m := i + ((j - i) >> 1)\n //\u5224\u65ad\u4e2d\u70b9\u4e0e\u76ee\u6807\u5143\u7d20\u5927\u5c0f\n if nums[m] < target {\n // \u5c0f\u4e8e\u5219\u9012\u5f52\u53f3\u534a\u6570\u7ec4\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m+1, j)\n } else if nums[m] > target {\n // \u5c0f\u4e8e\u5219\u9012\u5f52\u5de6\u534a\u6570\u7ec4\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m-1)\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums []int, target int) int {\n n := len(nums)\n return dfs(nums, target, 0, n-1)\n}\n
binary_search_recur.js/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -> Int {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if i > j {\n return -1\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n let m = (i + j) / 2\n if nums[m] < target {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums: nums, target: target, i: m + 1, j: j)\n } else if nums[m] > target {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums: nums, target: target, i: i, j: m - 1)\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums: [Int], target: Int) -> Int {\n let n = nums.count\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums: nums, target: target, i: 0, j: n - 1)\n}\n
binary_search_recur.ts/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums, target, i, j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n const m = i + ((j - i) >> 1);\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums, target) {\n const n = nums.length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.dart/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n const m = i + ((j - i) >> 1);\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums: number[], target: number): number {\n const n = nums.length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.rs/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(List<int> nums, int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) ~/ 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(List<int> nums, int target) {\n int n = nums.length;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
binary_search_recur.c/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfn dfs(nums: &[i32], target: i32, i: i32, j: i32) -> i32 {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if i > j { return -1; }\n let m: i32 = (i + j) / 2;\n if nums[m as usize] < target {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if nums[m as usize] > target {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfn binary_search(nums: &[i32], target: i32) -> i32 {\n let n = nums.len() as i32;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n dfs(nums, target, 0, n - 1)\n}\n
binary_search_recur.zig/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n if (i > j) {\n return -1;\n }\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n int m = (i + j) / 2;\n if (nums[m] < target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n return dfs(nums, target, m + 1, j);\n } else if (nums[m] > target) {\n // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n return dfs(nums, target, i, m - 1);\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int nums[], int target, int numsSize) {\n int n = numsSize;\n // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n return dfs(nums, target, 0, n - 1);\n}\n
"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","title":"12.3 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u95ee\u9898","text":"[class]{}-[func]{dfs}\n\n[class]{}-[func]{binarySearch}\nQuestion
\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\u7684\u524d\u5e8f\u904d\u5386
preorder\u548c\u4e2d\u5e8f\u904d\u5386inorder\uff0c\u8bf7\u4ece\u4e2d\u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u8fd4\u56de\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u3002\u5047\u8bbe\u4e8c\u53c9\u6811\u4e2d\u6ca1\u6709\u503c\u91cd\u590d\u7684\u8282\u70b9\u3002\u56fe 12-5 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u793a\u4f8b\u6570\u636e
"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","title":"1. \u00a0 \u5224\u65ad\u662f\u5426\u4e3a\u5206\u6cbb\u95ee\u9898","text":"\u539f\u95ee\u9898\u5b9a\u4e49\u4e3a\u4ece
preorder\u548cinorder\u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u662f\u4e00\u4e2a\u5178\u578b\u7684\u5206\u6cbb\u95ee\u9898\u3002- \u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4ece\u5206\u6cbb\u7684\u89d2\u5ea6\u5207\u5165\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\u3001\u6784\u5efa\u53f3\u5b50\u6811\uff0c\u52a0\u4e0a\u4e00\u6b65\u64cd\u4f5c\uff1a\u521d\u59cb\u5316\u6839\u8282\u70b9\u3002\u800c\u5bf9\u4e8e\u6bcf\u68f5\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u590d\u7528\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u5c06\u5176\u5212\u5206\u4e3a\u66f4\u5c0f\u7684\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u7a7a\u5b50\u6811\uff09\u65f6\u7ec8\u6b62\u3002
- \u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u4ea4\u96c6\u3002\u5728\u6784\u5efa\u5de6\u5b50\u6811\u65f6\uff0c\u6211\u4eec\u53ea\u9700\u5173\u6ce8\u4e2d\u5e8f\u904d\u5386\u548c\u524d\u5e8f\u904d\u5386\u4e2d\u4e0e\u5de6\u5b50\u6811\u5bf9\u5e94\u7684\u90e8\u5206\u3002\u53f3\u5b50\u6811\u540c\u7406\u3002
- \u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e00\u65e6\u5f97\u5230\u4e86\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u5b83\u4eec\u94fe\u63a5\u5230\u6839\u8282\u70b9\u4e0a\uff0c\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u8fd9\u9053\u9898\u53ef\u4ee5\u4f7f\u7528\u5206\u6cbb\u6765\u6c42\u89e3\uff0c\u4f46\u5982\u4f55\u901a\u8fc7\u524d\u5e8f\u904d\u5386
preorder\u548c\u4e2d\u5e8f\u904d\u5386inorder\u6765\u5212\u5206\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u5462\uff1f\u6839\u636e\u5b9a\u4e49\uff0c
preorder\u548cinorder\u90fd\u53ef\u4ee5\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002- \u524d\u5e8f\u904d\u5386\uff1a
[ \u6839\u8282\u70b9 | \u5de6\u5b50\u6811 | \u53f3\u5b50\u6811 ]\uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94[ 3 | 9 | 2 1 7 ]\u3002 - \u4e2d\u5e8f\u904d\u5386\uff1a
[ \u5de6\u5b50\u6811 | \u6839\u8282\u70b9 \uff5c \u53f3\u5b50\u6811 ]\uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94[ 9 | 3 | 1 2 7 ]\u3002
\u4ee5\u4e0a\u56fe\u6570\u636e\u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u56fe 12-6 \u6240\u793a\u7684\u6b65\u9aa4\u5f97\u5230\u5212\u5206\u7ed3\u679c\u3002
- \u524d\u5e8f\u904d\u5386\u7684\u9996\u5143\u7d20 3 \u662f\u6839\u8282\u70b9\u7684\u503c\u3002
- \u67e5\u627e\u6839\u8282\u70b9 3 \u5728
inorder\u4e2d\u7684\u7d22\u5f15\uff0c\u5229\u7528\u8be5\u7d22\u5f15\u53ef\u5c06inorder\u5212\u5206\u4e3a[ 9 | 3 \uff5c 1 2 7 ]\u3002 - \u6839\u636e
inorder\u5212\u5206\u7ed3\u679c\uff0c\u6613\u5f97\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u5206\u522b\u4e3a 1 \u548c 3 \uff0c\u4ece\u800c\u53ef\u5c06preorder\u5212\u5206\u4e3a[ 3 | 9 | 2 1 7 ]\u3002
\u56fe 12-6 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u5212\u5206\u5b50\u6811
"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","title":"3. \u00a0 \u57fa\u4e8e\u53d8\u91cf\u63cf\u8ff0\u5b50\u6811\u533a\u95f4","text":"\u6839\u636e\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u6211\u4eec\u5df2\u7ecf\u5f97\u5230\u6839\u8282\u70b9\u3001\u5de6\u5b50\u6811\u3001\u53f3\u5b50\u6811\u5728
preorder\u548cinorder\u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002\u800c\u4e3a\u4e86\u63cf\u8ff0\u8fd9\u4e9b\u7d22\u5f15\u533a\u95f4\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u3002- \u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728
preorder\u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(i\\) \u3002 - \u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728
inorder\u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(m\\) \u3002 - \u5c06\u5f53\u524d\u6811\u5728
inorder\u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u8bb0\u4e3a \\([l, r]\\) \u3002
\u5982\u8868 12-1 \u6240\u793a\uff0c\u901a\u8fc7\u4ee5\u4e0a\u53d8\u91cf\u5373\u53ef\u8868\u793a\u6839\u8282\u70b9\u5728
preorder\u4e2d\u7684\u7d22\u5f15\uff0c\u4ee5\u53ca\u5b50\u6811\u5728inorder\u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002\u8868 12-1 \u00a0 \u6839\u8282\u70b9\u548c\u5b50\u6811\u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u7d22\u5f15
\u6839\u8282\u70b9\u5728preorder\u4e2d\u7684\u7d22\u5f15 \u5b50\u6811\u5728inorder\u4e2d\u7684\u7d22\u5f15\u533a\u95f4 \u5f53\u524d\u6811 \\(i\\) \\([l, r]\\) \u5de6\u5b50\u6811 \\(i + 1\\) \\([l, m-1]\\) \u53f3\u5b50\u6811 \\(i + 1 + (m - l)\\) \\([m+1, r]\\)\u8bf7\u6ce8\u610f\uff0c\u53f3\u5b50\u6811\u6839\u8282\u70b9\u7d22\u5f15\u4e2d\u7684 \\((m-l)\\) \u7684\u542b\u4e49\u662f\u201c\u5de6\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u201d\uff0c\u5efa\u8bae\u914d\u5408\u56fe 12-7 \u7406\u89e3\u3002
\u56fe 12-7 \u00a0 \u6839\u8282\u70b9\u548c\u5de6\u53f3\u5b50\u6811\u7684\u7d22\u5f15\u533a\u95f4\u8868\u793a
"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","title":"4. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u4e3a\u4e86\u63d0\u5347\u67e5\u8be2 \\(m\\) \u7684\u6548\u7387\uff0c\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig build_tree.pyhmap\u6765\u5b58\u50a8\u6570\u7ec4inorder\u4e2d\u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff1a
build_tree.cppdef dfs(\n preorder: list[int],\n inorder_map: dict[int, int],\n i: int,\n l: int,\n r: int,\n) -> TreeNode | None:\n \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb\"\"\"\n # \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if r - l < 0:\n return None\n # \u521d\u59cb\u5316\u6839\u8282\u70b9\n root = TreeNode(preorder[i])\n # \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n m = inorder_map[preorder[i]]\n # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n # \u8fd4\u56de\u6839\u8282\u70b9\n return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -> TreeNode | None:\n \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\"\"\"\n # \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n inorder_map = {val: i for i, val in enumerate(inorder)}\n root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n return root\n
build_tree.java/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(vector<int> &preorder, unordered_map<int, int> &inorderMap, int i, int l, int r) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0)\n return NULL;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode *root = new TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap[preorder[i]];\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root->left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root->right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(vector<int> &preorder, vector<int> &inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n unordered_map<int, int> inorderMap;\n for (int i = 0; i < inorder.size(); i++) {\n inorderMap[inorder[i]] = i;\n }\n TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n return root;\n}\n
build_tree.cs/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode dfs(int[] preorder, Map<Integer, Integer> inorderMap, int i, int l, int r) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0)\n return null;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode root = new TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap.get(preorder[i]);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n Map<Integer, Integer> inorderMap = new HashMap<>();\n for (int i = 0; i < inorder.length; i++) {\n inorderMap.put(inorder[i], i);\n }\n TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n return root;\n}\n
build_tree.go/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? DFS(int[] preorder, Dictionary<int, int> inorderMap, int i, int l, int r) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0)\n return null;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode root = new(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap[preorder[i]];\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n Dictionary<int, int> inorderMap = [];\n for (int i = 0; i < inorder.Length; i++) {\n inorderMap.TryAdd(inorder[i], i);\n }\n TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n return root;\n}\n
build_tree.swift/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if r-l < 0 {\n return nil\n }\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n root := NewTreeNode(preorder[i])\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n m := inorderMap[preorder[i]]\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n inorderMap := make(map[int]int, len(inorder))\n for i := 0; i < len(inorder); i++ {\n inorderMap[inorder[i]] = i\n }\n\n root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n return root\n}\n
build_tree.js/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -> TreeNode? {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if r - l < 0 {\n return nil\n }\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n let root = TreeNode(x: preorder[i])\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n let m = inorderMap[preorder[i]]!\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -> TreeNode? {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n return dfs(preorder: preorder, inorderMap: inorderMap, i: 0, l: 0, r: inorder.count - 1)\n}\n
build_tree.ts/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(preorder, inorderMap, i, l, r) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0) return null;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n const root = new TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n const m = inorderMap.get(preorder[i]);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder, inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n let inorderMap = new Map();\n for (let i = 0; i < inorder.length; i++) {\n inorderMap.set(inorder[i], i);\n }\n const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n return root;\n}\n
build_tree.dart/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(\n preorder: number[],\n inorderMap: Map<number, number>,\n i: number,\n l: number,\n r: number\n): TreeNode | null {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0) return null;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n const root: TreeNode = new TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n const m = inorderMap.get(preorder[i]);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n let inorderMap = new Map<number, number>();\n for (let i = 0; i < inorder.length; i++) {\n inorderMap.set(inorder[i], i);\n }\n const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n return root;\n}\n
build_tree.rs/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? dfs(\n List<int> preorder,\n Map<int, int> inorderMap,\n int i,\n int l,\n int r,\n) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0) {\n return null;\n }\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode? root = TreeNode(preorder[i]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap[preorder[i]]!;\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? buildTree(List<int> preorder, List<int> inorder) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n Map<int, int> inorderMap = {};\n for (int i = 0; i < inorder.length; i++) {\n inorderMap[inorder[i]] = i;\n }\n TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n return root;\n}\n
build_tree.c/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfn dfs(preorder: &[i32], inorder_map: &HashMap<i32, i32>, i: i32, l: i32, r: i32) -> Option<Rc<RefCell<TreeNode>>> {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if r - l < 0 { return None; }\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n let root = TreeNode::new(preorder[i as usize]);\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n let m = inorder_map.get(&preorder[i as usize]).unwrap();\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n // \u8fd4\u56de\u6839\u8282\u70b9\n Some(root)\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfn build_tree(preorder: &[i32], inorder: &[i32]) -> Option<Rc<RefCell<TreeNode>>> {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n let mut inorder_map: HashMap<i32, i32> = HashMap::new();\n for i in 0..inorder.len() {\n inorder_map.insert(inorder[i], i as i32);\n }\n let root = dfs(preorder, &inorder_map, 0, 0, inorder.len() as i32 - 1);\n root\n}\n
build_tree.zig/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n if (r - l < 0)\n return NULL;\n // \u521d\u59cb\u5316\u6839\u8282\u70b9\n TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n root->val = preorder[i];\n root->left = NULL;\n root->right = NULL;\n // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n int m = inorderMap[preorder[i]];\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n root->left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n root->right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n // \u8fd4\u56de\u6839\u8282\u70b9\n return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n for (int i = 0; i < inorderSize; i++) {\n inorderMap[inorder[i]] = i;\n }\n TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n free(inorderMap);\n return root;\n}\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{buildTree}\n\u56fe 12-8 \u5c55\u793a\u4e86\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5404\u4e2a\u8282\u70b9\u662f\u5728\u5411\u4e0b\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\uff0c\u800c\u5404\u6761\u8fb9\uff08\u5f15\u7528\uff09\u662f\u5728\u5411\u4e0a\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\u3002
<1><2><3><4><5><6><7><8><9>\u56fe 12-8 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b
\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u5185\u7684\u524d\u5e8f\u904d\u5386
preorder\u548c\u4e2d\u5e8f\u904d\u5386inorder\u7684\u5212\u5206\u7ed3\u679c\u5982\u56fe 12-9 \u6240\u793a\u3002\u56fe 12-9 \u00a0 \u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u7684\u5212\u5206\u7ed3\u679c
\u8bbe\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u521d\u59cb\u5316\u6bcf\u4e00\u4e2a\u8282\u70b9\uff08\u6267\u884c\u4e00\u4e2a\u9012\u5f52\u51fd\u6570
dfs()\uff09\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u54c8\u5e0c\u8868\u5b58\u50a8
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/","title":"12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5","text":"inorder\u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u300c\u5206\u6cbb divide and conquer\u300d\uff0c\u5168\u79f0\u5206\u800c\u6cbb\u4e4b\uff0c\u662f\u4e00\u79cd\u975e\u5e38\u91cd\u8981\u4e14\u5e38\u89c1\u7684\u7b97\u6cd5\u7b56\u7565\u3002\u5206\u6cbb\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff0c\u5305\u62ec\u201c\u5206\u201d\u548c\u201c\u6cbb\u201d\u4e24\u4e2a\u6b65\u9aa4\u3002
- \u5206\uff08\u5212\u5206\u9636\u6bb5\uff09\uff1a\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u6216\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898\u65f6\u7ec8\u6b62\u3002
- \u6cbb\uff08\u5408\u5e76\u9636\u6bb5\uff09\uff1a\u4ece\u5df2\u77e5\u89e3\u7684\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002
\u5982\u56fe 12-1 \u6240\u793a\uff0c\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\u4e4b\u4e00\u3002
- \u5206\uff1a\u9012\u5f52\u5730\u5c06\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u5230\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\uff08\u6700\u5c0f\u5b50\u95ee\u9898\uff09\u3002
- \u6cbb\uff1a\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u6709\u5e8f\u7684\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u7684\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002
\u56fe 12-1 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","title":"12.1.1 \u00a0 \u5982\u4f55\u5224\u65ad\u5206\u6cbb\u95ee\u9898","text":"\u4e00\u4e2a\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u5206\u6cbb\u89e3\u51b3\uff0c\u901a\u5e38\u53ef\u4ee5\u53c2\u8003\u4ee5\u4e0b\u51e0\u4e2a\u5224\u65ad\u4f9d\u636e\u3002
- \u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u539f\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\u6210\u89c4\u6a21\u66f4\u5c0f\u3001\u7c7b\u4f3c\u7684\u5b50\u95ee\u9898\uff0c\u4ee5\u53ca\u80fd\u591f\u4ee5\u76f8\u540c\u65b9\u5f0f\u9012\u5f52\u5730\u8fdb\u884c\u5212\u5206\u3002
- \u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5b50\u95ee\u9898\u4e4b\u95f4\u6ca1\u6709\u91cd\u53e0\uff0c\u4e92\u4e0d\u4f9d\u8d56\uff0c\u53ef\u4ee5\u72ec\u7acb\u89e3\u51b3\u3002
- \u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u539f\u95ee\u9898\u7684\u89e3\u901a\u8fc7\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\u5f97\u6765\u3002
\u663e\u7136\uff0c\u5f52\u5e76\u6392\u5e8f\u6ee1\u8db3\u4ee5\u4e0a\u4e09\u6761\u5224\u65ad\u4f9d\u636e\u3002
- \u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u9012\u5f52\u5730\u5c06\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\u3002
- \u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u53ef\u4ee5\u72ec\u7acb\u5730\u8fdb\u884c\u6392\u5e8f\uff08\u5b50\u95ee\u9898\u53ef\u4ee5\u72ec\u7acb\u8fdb\u884c\u6c42\u89e3\uff09\u3002
- \u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u53ef\u4ee5\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002
\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\uff0c\u5f80\u5f80\u8fd8\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u5728\u6392\u5e8f\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u76f8\u8f83\u4e8e\u9009\u62e9\u3001\u5192\u6ce1\u3001\u63d2\u5165\u6392\u5e8f\u66f4\u5feb\uff0c\u5c31\u662f\u56e0\u4e3a\u5b83\u4eec\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\u3002
\u90a3\u4e48\uff0c\u6211\u4eec\u4e0d\u7981\u53d1\u95ee\uff1a\u4e3a\u4ec0\u4e48\u5206\u6cbb\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u5176\u5e95\u5c42\u903b\u8f91\u662f\u4ec0\u4e48\uff1f\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c06\u5927\u95ee\u9898\u5206\u89e3\u4e3a\u591a\u4e2a\u5b50\u95ee\u9898\u3001\u89e3\u51b3\u5b50\u95ee\u9898\u3001\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u5408\u5e76\u4e3a\u539f\u95ee\u9898\u7684\u89e3\uff0c\u8fd9\u51e0\u6b65\u7684\u6548\u7387\u4e3a\u4ec0\u4e48\u6bd4\u76f4\u63a5\u89e3\u51b3\u539f\u95ee\u9898\u7684\u6548\u7387\u66f4\u9ad8\uff1f\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u4ece\u64cd\u4f5c\u6570\u91cf\u548c\u5e76\u884c\u8ba1\u7b97\u4e24\u65b9\u9762\u6765\u8ba8\u8bba\u3002
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","title":"1. \u00a0 \u64cd\u4f5c\u6570\u91cf\u4f18\u5316","text":"\u4ee5\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u4e3a\u4f8b\uff0c\u5176\u5904\u7406\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u9700\u8981 \\(O(n^2)\\) \u65f6\u95f4\u3002\u5047\u8bbe\u6211\u4eec\u6309\u7167\u56fe 12-2 \u6240\u793a\u7684\u65b9\u5f0f\uff0c\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u5219\u5212\u5206\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6bcf\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O((n / 2)^2)\\) \u65f6\u95f4\uff0c\u5408\u5e76\u4e24\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a
\\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\]\u56fe 12-2 \u00a0 \u5212\u5206\u6570\u7ec4\u524d\u540e\u7684\u5192\u6ce1\u6392\u5e8f
\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u8ba1\u7b97\u4ee5\u4e0b\u4e0d\u7b49\u5f0f\uff0c\u5176\u5de6\u8fb9\u548c\u53f3\u8fb9\u5206\u522b\u4e3a\u5212\u5206\u524d\u548c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u603b\u6570\uff1a
\\[ \\begin{aligned} n^2 & > \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n & > 0 \\newline n(n - 4) & > 0 \\end{aligned} \\]\u8fd9\u610f\u5473\u7740\u5f53 \\(n > 4\\) \u65f6\uff0c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u6570\u91cf\u66f4\u5c11\uff0c\u6392\u5e8f\u6548\u7387\u5e94\u8be5\u66f4\u9ad8\u3002\u8bf7\u6ce8\u610f\uff0c\u5212\u5206\u540e\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u7136\u662f\u5e73\u65b9\u9636 \\(O(n^2)\\) \uff0c\u53ea\u662f\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\u4e86\u3002
\u8fdb\u4e00\u6b65\u60f3\uff0c\u5982\u679c\u6211\u4eec\u628a\u5b50\u6570\u7ec4\u4e0d\u65ad\u5730\u518d\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u505c\u6b62\u5212\u5206\u5462\uff1f\u8fd9\u79cd\u601d\u8def\u5b9e\u9645\u4e0a\u5c31\u662f\u201c\u5f52\u5e76\u6392\u5e8f\u201d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
\u518d\u601d\u8003\uff0c\u5982\u679c\u6211\u4eec\u591a\u8bbe\u7f6e\u51e0\u4e2a\u5212\u5206\u70b9\uff0c\u5c06\u539f\u6570\u7ec4\u5e73\u5747\u5212\u5206\u4e3a \\(k\\) \u4e2a\u5b50\u6570\u7ec4\u5462\uff1f\u8fd9\u79cd\u60c5\u51b5\u4e0e\u201c\u6876\u6392\u5e8f\u201d\u975e\u5e38\u7c7b\u4f3c\uff0c\u5b83\u975e\u5e38\u9002\u5408\u6392\u5e8f\u6d77\u91cf\u6570\u636e\uff0c\u7406\u8bba\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n + k)\\) \u3002
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","title":"2. \u00a0 \u5e76\u884c\u8ba1\u7b97\u4f18\u5316","text":"\u6211\u4eec\u77e5\u9053\uff0c\u5206\u6cbb\u751f\u6210\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u56e0\u6b64\u901a\u5e38\u53ef\u4ee5\u5e76\u884c\u89e3\u51b3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8fd8\u6709\u5229\u4e8e\u64cd\u4f5c\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002
\u5e76\u884c\u4f18\u5316\u5728\u591a\u6838\u6216\u591a\u5904\u7406\u5668\u7684\u73af\u5883\u4e2d\u5c24\u5176\u6709\u6548\uff0c\u56e0\u4e3a\u7cfb\u7edf\u53ef\u4ee5\u540c\u65f6\u5904\u7406\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u66f4\u52a0\u5145\u5206\u5730\u5229\u7528\u8ba1\u7b97\u8d44\u6e90\uff0c\u4ece\u800c\u663e\u8457\u51cf\u5c11\u603b\u4f53\u7684\u8fd0\u884c\u65f6\u95f4\u3002
\u6bd4\u5982\u5728\u56fe 12-3 \u6240\u793a\u7684\u201c\u6876\u6392\u5e8f\u201d\u4e2d\uff0c\u6211\u4eec\u5c06\u6d77\u91cf\u7684\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u5219\u53ef\u6240\u6709\u6876\u7684\u6392\u5e8f\u4efb\u52a1\u5206\u6563\u5230\u5404\u4e2a\u8ba1\u7b97\u5355\u5143\uff0c\u5b8c\u6210\u540e\u518d\u5408\u5e76\u7ed3\u679c\u3002
\u56fe 12-3 \u00a0 \u6876\u6392\u5e8f\u7684\u5e76\u884c\u8ba1\u7b97
"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","title":"12.1.3 \u00a0 \u5206\u6cbb\u5e38\u89c1\u5e94\u7528","text":"\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u8bb8\u591a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\u3002
- \u5bfb\u627e\u6700\u8fd1\u70b9\u5bf9\uff1a\u8be5\u7b97\u6cd5\u9996\u5148\u5c06\u70b9\u96c6\u5206\u6210\u4e24\u90e8\u5206\uff0c\u7136\u540e\u5206\u522b\u627e\u51fa\u4e24\u90e8\u5206\u4e2d\u7684\u6700\u8fd1\u70b9\u5bf9\uff0c\u6700\u540e\u627e\u51fa\u8de8\u8d8a\u4e24\u90e8\u5206\u7684\u6700\u8fd1\u70b9\u5bf9\u3002
- \u5927\u6574\u6570\u4e58\u6cd5\uff1a\u4f8b\u5982 Karatsuba \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u6574\u6570\u4e58\u6cd5\u5206\u89e3\u4e3a\u51e0\u4e2a\u8f83\u5c0f\u7684\u6574\u6570\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002
- \u77e9\u9635\u4e58\u6cd5\uff1a\u4f8b\u5982 Strassen \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u77e9\u9635\u4e58\u6cd5\u5206\u89e3\u4e3a\u591a\u4e2a\u5c0f\u77e9\u9635\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002
- \u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u6c49\u8bfa\u5854\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u89e3\u51b3\uff0c\u8fd9\u662f\u5178\u578b\u7684\u5206\u6cbb\u7b56\u7565\u5e94\u7528\u3002
- \u6c42\u89e3\u9006\u5e8f\u5bf9\uff1a\u5728\u4e00\u4e2a\u5e8f\u5217\u4e2d\uff0c\u5982\u679c\u524d\u9762\u7684\u6570\u5b57\u5927\u4e8e\u540e\u9762\u7684\u6570\u5b57\uff0c\u90a3\u4e48\u8fd9\u4e24\u4e2a\u6570\u5b57\u6784\u6210\u4e00\u4e2a\u9006\u5e8f\u5bf9\u3002\u6c42\u89e3\u9006\u5e8f\u5bf9\u95ee\u9898\u53ef\u4ee5\u5229\u7528\u5206\u6cbb\u7684\u601d\u60f3\uff0c\u501f\u52a9\u5f52\u5e76\u6392\u5e8f\u8fdb\u884c\u6c42\u89e3\u3002
\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5728\u7b97\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u4e2d\u5e94\u7528\u975e\u5e38\u5e7f\u6cdb\u3002
- \u4e8c\u5206\u67e5\u627e\uff1a\u4e8c\u5206\u67e5\u627e\u662f\u5c06\u6709\u5e8f\u6570\u7ec4\u4ece\u4e2d\u70b9\u7d22\u5f15\u5904\u5206\u4e3a\u4e24\u90e8\u5206\uff0c\u7136\u540e\u6839\u636e\u76ee\u6807\u503c\u4e0e\u4e2d\u95f4\u5143\u7d20\u503c\u6bd4\u8f83\u7ed3\u679c\uff0c\u51b3\u5b9a\u6392\u9664\u54ea\u4e00\u534a\u533a\u95f4\uff0c\u5e76\u5728\u5269\u4f59\u533a\u95f4\u6267\u884c\u76f8\u540c\u7684\u4e8c\u5206\u64cd\u4f5c\u3002
- \u5f52\u5e76\u6392\u5e8f\uff1a\u672c\u8282\u5f00\u5934\u5df2\u4ecb\u7ecd\uff0c\u4e0d\u518d\u8d58\u8ff0\u3002
- \u5feb\u901f\u6392\u5e8f\uff1a\u5feb\u901f\u6392\u5e8f\u662f\u9009\u53d6\u4e00\u4e2a\u57fa\u51c6\u503c\uff0c\u7136\u540e\u628a\u6570\u7ec4\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u4e00\u4e2a\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5c0f\uff0c\u53e6\u4e00\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5927\uff0c\u518d\u5bf9\u8fd9\u4e24\u90e8\u5206\u8fdb\u884c\u76f8\u540c\u7684\u5212\u5206\u64cd\u4f5c\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e0b\u4e00\u4e2a\u5143\u7d20\u3002
- \u6876\u6392\u5e8f\uff1a\u6876\u6392\u5e8f\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c06\u6570\u636e\u5206\u6563\u5230\u591a\u4e2a\u6876\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u5404\u4e2a\u6876\u7684\u5143\u7d20\u4f9d\u6b21\u53d6\u51fa\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002
- \u6811\uff1a\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u7ea2\u9ed1\u6811\u3001B \u6811\u3001B+ \u6811\u7b49\uff0c\u5b83\u4eec\u7684\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u7b49\u64cd\u4f5c\u90fd\u53ef\u4ee5\u89c6\u4e3a\u5206\u6cbb\u7b56\u7565\u7684\u5e94\u7528\u3002
- \u5806\uff1a\u5806\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u5176\u5404\u79cd\u64cd\u4f5c\uff0c\u5982\u63d2\u5165\u3001\u5220\u9664\u548c\u5806\u5316\uff0c\u5b9e\u9645\u4e0a\u90fd\u9690\u542b\u4e86\u5206\u6cbb\u7684\u601d\u60f3\u3002
- \u54c8\u5e0c\u8868\uff1a\u867d\u7136\u54c8\u5e0c\u8868\u6765\u5e76\u4e0d\u76f4\u63a5\u5e94\u7528\u5206\u6cbb\uff0c\u4f46\u67d0\u4e9b\u54c8\u5e0c\u51b2\u7a81\u89e3\u51b3\u65b9\u6848\u95f4\u63a5\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u4f8b\u5982\uff0c\u94fe\u5f0f\u5730\u5740\u4e2d\u7684\u957f\u94fe\u8868\u4f1a\u88ab\u8f6c\u5316\u4e3a\u7ea2\u9ed1\u6811\uff0c\u4ee5\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002
\u53ef\u4ee5\u770b\u51fa\uff0c\u5206\u6cbb\u662f\u4e00\u79cd\u201c\u6da6\u7269\u7ec6\u65e0\u58f0\u201d\u7684\u7b97\u6cd5\u601d\u60f3\uff0c\u9690\u542b\u5728\u5404\u79cd\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u4e2d\u3002
"},{"location":"chapter_divide_and_conquer/hanota_problem/","title":"12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898","text":"\u5728\u5f52\u5e76\u6392\u5e8f\u548c\u6784\u5efa\u4e8c\u53c9\u6811\u4e2d\uff0c\u6211\u4eec\u90fd\u662f\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a\u539f\u95ee\u9898\u4e00\u534a\u7684\u5b50\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u6c49\u8bfa\u5854\u95ee\u9898\uff0c\u6211\u4eec\u91c7\u7528\u4e0d\u540c\u7684\u5206\u89e3\u7b56\u7565\u3002
Question
\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\uff0c\u8bb0\u4e3a
A\u3001B\u548cC\u3002\u8d77\u59cb\u72b6\u6001\u4e0b\uff0c\u67f1\u5b50A\u4e0a\u5957\u7740 \\(n\\) \u4e2a\u5706\u76d8\uff0c\u5b83\u4eec\u4ece\u4e0a\u5230\u4e0b\u6309\u7167\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u3002\u6211\u4eec\u7684\u4efb\u52a1\u662f\u8981\u628a\u8fd9 \\(n\\) \u4e2a\u5706\u76d8\u79fb\u5230\u67f1\u5b50C\u4e0a\uff0c\u5e76\u4fdd\u6301\u5b83\u4eec\u7684\u539f\u6709\u987a\u5e8f\u4e0d\u53d8\uff08\u5982\u56fe 12-10 \u6240\u793a\uff09\u3002\u5728\u79fb\u52a8\u5706\u76d8\u7684\u8fc7\u7a0b\u4e2d\uff0c\u9700\u8981\u9075\u5b88\u4ee5\u4e0b\u89c4\u5219\u3002- \u5706\u76d8\u53ea\u80fd\u4ece\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u62ff\u51fa\uff0c\u4ece\u53e6\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u653e\u5165\u3002
- \u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\u3002
- \u5c0f\u5706\u76d8\u5fc5\u987b\u65f6\u523b\u4f4d\u4e8e\u5927\u5706\u76d8\u4e4b\u4e0a\u3002
\u56fe 12-10 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u793a\u4f8b
\u6211\u4eec\u5c06\u89c4\u6a21\u4e3a \\(i\\) \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u8bb0\u4f5c \\(f(i)\\) \u3002\u4f8b\u5982 \\(f(3)\\) \u4ee3\u8868\u5c06 \\(3\\) \u4e2a\u5706\u76d8\u4ece
"},{"location":"chapter_divide_and_conquer/hanota_problem/#1","title":"1. \u00a0 \u8003\u8651\u57fa\u672c\u60c5\u51b5","text":"A\u79fb\u52a8\u81f3C\u7684\u6c49\u8bfa\u5854\u95ee\u9898\u3002\u5982\u56fe 12-11 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(1)\\) \uff0c\u5373\u5f53\u53ea\u6709\u4e00\u4e2a\u5706\u76d8\u65f6\uff0c\u6211\u4eec\u5c06\u5b83\u76f4\u63a5\u4ece
<1><2>A\u79fb\u52a8\u81f3C\u5373\u53ef\u3002\u56fe 12-11 \u00a0 \u89c4\u6a21\u4e3a 1 \u95ee\u9898\u7684\u89e3
\u5982\u56fe 12-12 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(2)\\) \uff0c\u5373\u5f53\u6709\u4e24\u4e2a\u5706\u76d8\u65f6\uff0c\u7531\u4e8e\u8981\u65f6\u523b\u6ee1\u8db3\u5c0f\u5706\u76d8\u5728\u5927\u5706\u76d8\u4e4b\u4e0a\uff0c\u56e0\u6b64\u9700\u8981\u501f\u52a9
B\u6765\u5b8c\u6210\u79fb\u52a8\u3002- \u5148\u5c06\u4e0a\u9762\u7684\u5c0f\u5706\u76d8\u4ece
A\u79fb\u81f3B\u3002 - \u518d\u5c06\u5927\u5706\u76d8\u4ece
A\u79fb\u81f3C\u3002 - \u6700\u540e\u5c06\u5c0f\u5706\u76d8\u4ece
B\u79fb\u81f3C\u3002
\u56fe 12-12 \u00a0 \u89c4\u6a21\u4e3a 2 \u95ee\u9898\u7684\u89e3
\u89e3\u51b3\u95ee\u9898 \\(f(2)\\) \u7684\u8fc7\u7a0b\u53ef\u603b\u7ed3\u4e3a\uff1a\u5c06\u4e24\u4e2a\u5706\u76d8\u501f\u52a9
"},{"location":"chapter_divide_and_conquer/hanota_problem/#2","title":"2. \u00a0 \u5b50\u95ee\u9898\u5206\u89e3","text":"B\u4eceA\u79fb\u81f3C\u3002\u5176\u4e2d\uff0cC\u79f0\u4e3a\u76ee\u6807\u67f1\u3001B\u79f0\u4e3a\u7f13\u51b2\u67f1\u3002\u5bf9\u4e8e\u95ee\u9898 \\(f(3)\\) \uff0c\u5373\u5f53\u6709\u4e09\u4e2a\u5706\u76d8\u65f6\uff0c\u60c5\u51b5\u53d8\u5f97\u7a0d\u5fae\u590d\u6742\u4e86\u4e00\u4e9b\u3002
\u56e0\u4e3a\u5df2\u77e5 \\(f(1)\\) \u548c \\(f(2)\\) \u7684\u89e3\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ece\u5206\u6cbb\u89d2\u5ea6\u601d\u8003\uff0c\u5c06
A\u9876\u90e8\u7684\u4e24\u4e2a\u5706\u76d8\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u6267\u884c\u56fe 12-13 \u6240\u793a\u7684\u6b65\u9aa4\u3002\u8fd9\u6837\u4e09\u4e2a\u5706\u76d8\u5c31\u88ab\u987a\u5229\u5730\u4eceA\u79fb\u81f3C\u4e86\u3002- \u4ee4
B\u4e3a\u76ee\u6807\u67f1\u3001C\u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4eceA\u79fb\u81f3B\u3002 - \u5c06
A\u4e2d\u5269\u4f59\u7684\u4e00\u4e2a\u5706\u76d8\u4eceA\u76f4\u63a5\u79fb\u52a8\u81f3C\u3002 - \u4ee4
C\u4e3a\u76ee\u6807\u67f1\u3001A\u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4eceB\u79fb\u81f3C\u3002
\u56fe 12-13 \u00a0 \u89c4\u6a21\u4e3a 3 \u95ee\u9898\u7684\u89e3
\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u5c06\u95ee\u9898 \\(f(3)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(2)\\) \u548c\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u4e4b\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002\u8fd9\u8bf4\u660e\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff0c\u800c\u4e14\u89e3\u53ef\u4ee5\u5408\u5e76\u3002
\u81f3\u6b64\uff0c\u6211\u4eec\u53ef\u603b\u7ed3\u51fa\u56fe 12-14 \u6240\u793a\u7684\u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565\uff1a\u5c06\u539f\u95ee\u9898 \\(f(n)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \u548c\u4e00\u4e2a\u5b50\u95ee\u9898 \\(f(1)\\) \uff0c\u5e76\u6309\u7167\u4ee5\u4e0b\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u3002
- \u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9
C\u4eceA\u79fb\u81f3B\u3002 - \u5c06\u5269\u4f59 \\(1\\) \u4e2a\u5706\u76d8\u4ece
A\u76f4\u63a5\u79fb\u81f3C\u3002 - \u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9
A\u4eceB\u79fb\u81f3C\u3002
\u5bf9\u4e8e\u8fd9\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \uff0c\u53ef\u4ee5\u901a\u8fc7\u76f8\u540c\u7684\u65b9\u5f0f\u8fdb\u884c\u9012\u5f52\u5212\u5206\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u800c \\(f(1)\\) \u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u53ea\u9700\u4e00\u6b21\u79fb\u52a8\u64cd\u4f5c\u5373\u53ef\u3002
\u56fe 12-14 \u00a0 \u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565
"},{"location":"chapter_divide_and_conquer/hanota_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570
PythonC++JavaC#GoSwiftJSTSDartRustCZig hanota.pydfs(i, src, buf, tar)\uff0c\u5b83\u7684\u4f5c\u7528\u662f\u5c06\u67f1src\u9876\u90e8\u7684 \\(i\\) \u4e2a\u5706\u76d8\u501f\u52a9\u7f13\u51b2\u67f1buf\u79fb\u52a8\u81f3\u76ee\u6807\u67f1tar\uff1a
hanota.cppdef move(src: list[int], tar: list[int]):\n \"\"\"\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\"\"\"\n # \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n pan = src.pop()\n # \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i)\"\"\"\n # \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if i == 1:\n move(src, tar)\n return\n # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf)\n # \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar)\n # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898\"\"\"\n n = len(A)\n # \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C)\n
hanota.java/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(vector<int> &src, vector<int> &tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n int pan = src.back();\n src.pop_back();\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.push_back(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, vector<int> &src, vector<int> &buf, vector<int> &tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(vector<int> &A, vector<int> &B, vector<int> &C) {\n int n = A.size();\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.cs/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List<Integer> src, List<Integer> tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n Integer pan = src.remove(src.size() - 1);\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List<Integer> src, List<Integer> buf, List<Integer> tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List<Integer> A, List<Integer> B, List<Integer> C) {\n int n = A.size();\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.go/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid Move(List<int> src, List<int> tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n int pan = src[^1];\n src.RemoveAt(src.Count - 1);\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.Add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid DFS(int i, List<int> src, List<int> buf, List<int> tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n Move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n DFS(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n Move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n DFS(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid SolveHanota(List<int> A, List<int> B, List<int> C) {\n int n = A.Count;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n DFS(n, A, B, C);\n}\n
hanota.swift/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src, tar *list.List) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n pan := src.Back()\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.PushBack(pan.Value)\n // \u79fb\u9664 src \u9876\u90e8\u5706\u76d8\n src.Remove(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if i == 1 {\n move(src, tar)\n return\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfsHanota(i-1, src, tar, buf)\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar)\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfsHanota(i-1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A, B, C *list.List) {\n n := A.Len()\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfsHanota(n, A, B, C)\n}\n
hanota.js/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n let pan = src.popLast()!\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.append(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if i == 1 {\n move(src: &src, tar: &tar)\n return\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i: i - 1, src: &src, buf: &tar, tar: &buf)\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src: &src, tar: &tar)\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i: i - 1, src: &buf, buf: &src, tar: &tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n let n = A.count\n // \u5217\u8868\u5c3e\u90e8\u662f\u67f1\u5b50\u9876\u90e8\n // \u5c06 src \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(i: n, src: &A, buf: &B, tar: &C)\n}\n
hanota.ts/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src, tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n const pan = src.pop();\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i, src, buf, tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i === 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A, B, C) {\n const n = A.length;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.dart/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src: number[], tar: number[]): void {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n const pan = src.pop();\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i === 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n const n = A.length;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.rs/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List<int> src, List<int> tar) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n int pan = src.removeLast();\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List<int> src, List<int> buf, List<int> tar) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n move(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List<int> A, List<int> B, List<int> C) {\n int n = A.length;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.c/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfn move_pan(src: &mut Vec<i32>, tar: &mut Vec<i32>) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n let pan = src.remove(src.len() - 1);\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfn dfs(i: i32, src: &mut Vec<i32>, buf: &mut Vec<i32>, tar: &mut Vec<i32>) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if i == 1 {\n move_pan(src, tar);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, tar, buf);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move_pan(src, tar);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfn solve_hanota(A: &mut Vec<i32>, B: &mut Vec<i32>, C: &mut Vec<i32>) {\n let n = A.len() as i32;\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(n, A, B, C);\n}\n
hanota.zig/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n int pan = src[*srcSize - 1];\n src[*srcSize - 1] = 0;\n (*srcSize)--;\n // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n tar[*tarSize] = pan;\n (*tarSize)++;\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n if (i == 1) {\n move(src, srcSize, tar, tarSize);\n return;\n }\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n move(src, srcSize, tar, tarSize);\n // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n[class]{}-[func]{move}\n\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{solveHanota}\n\u5982\u56fe 12-15 \u6240\u793a\uff0c\u6c49\u8bfa\u5854\u95ee\u9898\u5f62\u6210\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5bf9\u5e94\u4e00\u4e2a\u5f00\u542f\u7684
dfs()\u51fd\u6570\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u56fe 12-15 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u7684\u9012\u5f52\u6811
Quote
\u6c49\u8bfa\u5854\u95ee\u9898\u6e90\u81ea\u4e00\u4e2a\u53e4\u8001\u7684\u4f20\u8bf4\u3002\u5728\u53e4\u5370\u5ea6\u7684\u4e00\u4e2a\u5bfa\u5e99\u91cc\uff0c\u50e7\u4fa3\u4eec\u6709\u4e09\u6839\u9ad8\u5927\u7684\u94bb\u77f3\u67f1\u5b50\uff0c\u4ee5\u53ca \\(64\\) \u4e2a\u5927\u5c0f\u4e0d\u4e00\u7684\u91d1\u5706\u76d8\u3002\u50e7\u4fa3\u4eec\u4e0d\u65ad\u5730\u79fb\u52a8\u5706\u76d8\uff0c\u4ed6\u4eec\u76f8\u4fe1\u5728\u6700\u540e\u4e00\u4e2a\u5706\u76d8\u88ab\u6b63\u786e\u653e\u7f6e\u7684\u90a3\u4e00\u523b\uff0c\u8fd9\u4e2a\u4e16\u754c\u5c31\u4f1a\u7ed3\u675f\u3002
\u7136\u800c\uff0c\u5373\u4f7f\u50e7\u4fa3\u4eec\u6bcf\u79d2\u949f\u79fb\u52a8\u4e00\u6b21\uff0c\u603b\u5171\u9700\u8981\u5927\u7ea6 \\(2^{64} \\approx 1.84\u00d710^{19}\\) \u79d2\uff0c\u5408\u7ea6 \\(5850\\) \u4ebf\u5e74\uff0c\u8fdc\u8fdc\u8d85\u8fc7\u4e86\u73b0\u5728\u5bf9\u5b87\u5b99\u5e74\u9f84\u7684\u4f30\u8ba1\u3002\u6240\u4ee5\uff0c\u5018\u82e5\u8fd9\u4e2a\u4f20\u8bf4\u662f\u771f\u7684\uff0c\u6211\u4eec\u5e94\u8be5\u4e0d\u9700\u8981\u62c5\u5fc3\u4e16\u754c\u672b\u65e5\u7684\u5230\u6765\u3002
"},{"location":"chapter_divide_and_conquer/summary/","title":"12.5 \u00a0 \u5c0f\u7ed3","text":"- \u5206\u6cbb\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u7b97\u6cd5\u8bbe\u8ba1\u7b56\u7565\uff0c\u5305\u62ec\u5206\uff08\u5212\u5206\uff09\u548c\u6cbb\uff08\u5408\u5e76\uff09\u4e24\u4e2a\u9636\u6bb5\uff0c\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002
- \u5224\u65ad\u662f\u5426\u662f\u5206\u6cbb\u7b97\u6cd5\u95ee\u9898\u7684\u4f9d\u636e\u5305\u62ec\uff1a\u95ee\u9898\u80fd\u5426\u5206\u89e3\u3001\u5b50\u95ee\u9898\u662f\u5426\u72ec\u7acb\u3001\u5b50\u95ee\u9898\u80fd\u5426\u5408\u5e76\u3002
- \u5f52\u5e76\u6392\u5e8f\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\uff0c\u5176\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u7b49\u957f\u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u5230\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u5f00\u59cb\u9010\u5c42\u5408\u5e76\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002
- \u5f15\u5165\u5206\u6cbb\u7b56\u7565\u5f80\u5f80\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u7b56\u7565\u51cf\u5c11\u4e86\u64cd\u4f5c\u6570\u91cf\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u540e\u6709\u5229\u4e8e\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002
- \u5206\u6cbb\u65e2\u53ef\u4ee5\u89e3\u51b3\u8bb8\u591a\u7b97\u6cd5\u95ee\u9898\uff0c\u4e5f\u5e7f\u6cdb\u5e94\u7528\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u5904\u5904\u53ef\u89c1\u5176\u8eab\u5f71\u3002
- \u76f8\u8f83\u4e8e\u66b4\u529b\u641c\u7d22\uff0c\u81ea\u9002\u5e94\u641c\u7d22\u6548\u7387\u66f4\u9ad8\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\u3002
- \u4e8c\u5206\u67e5\u627e\u662f\u5206\u6cbb\u7b56\u7565\u7684\u53e6\u4e00\u4e2a\u5178\u578b\u5e94\u7528\uff0c\u5b83\u4e0d\u5305\u542b\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u7684\u6b65\u9aa4\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e\u3002
- \u5728\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u95ee\u9898\u4e2d\uff0c\u6784\u5efa\u6811\uff08\u539f\u95ee\u9898\uff09\u53ef\u4ee5\u5212\u5206\u4e3a\u6784\u5efa\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5212\u5206\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u7684\u7d22\u5f15\u533a\u95f4\u6765\u5b9e\u73b0\u3002
- \u5728\u6c49\u8bfa\u5854\u95ee\u9898\u4e2d\uff0c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(n\\) \u7684\u95ee\u9898\u53ef\u4ee5\u5212\u5206\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a \\(n-1\\) \u7684\u5b50\u95ee\u9898\u548c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(1\\) \u7684\u5b50\u95ee\u9898\u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002
Abstract
\u5c0f\u6eaa\u6c47\u5165\u6cb3\u6d41\uff0c\u6c5f\u6cb3\u6c47\u5165\u5927\u6d77\u3002
\u52a8\u6001\u89c4\u5212\u5c06\u5c0f\u95ee\u9898\u7684\u89e3\u6c47\u96c6\u6210\u5927\u95ee\u9898\u7684\u7b54\u6848\uff0c\u4e00\u6b65\u6b65\u5f15\u9886\u6211\u4eec\u8d70\u5411\u89e3\u51b3\u95ee\u9898\u7684\u5f7c\u5cb8\u3002
"},{"location":"chapter_dynamic_programming/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212
- 14.2 \u00a0 DP \u95ee\u9898\u7279\u6027
- 14.3 \u00a0 DP \u89e3\u9898\u601d\u8def
- 14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898
- 14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898
- 14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898
- 14.7 \u00a0 \u5c0f\u7ed3
\u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u52a8\u6001\u89c4\u5212\u662f\u5982\u4f55\u901a\u8fc7\u5b50\u95ee\u9898\u5206\u89e3\u6765\u6c42\u89e3\u539f\u95ee\u9898\u7684\u3002\u5b9e\u9645\u4e0a\uff0c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u7684\u4fa7\u91cd\u70b9\u4e0d\u540c\u3002
- \u5206\u6cbb\u7b97\u6cd5\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u591a\u4e2a\u76f8\u4e92\u72ec\u7acb\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u6700\u5c0f\u5b50\u95ee\u9898\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6700\u7ec8\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002
- \u52a8\u6001\u89c4\u5212\u4e5f\u5bf9\u95ee\u9898\u8fdb\u884c\u9012\u5f52\u5206\u89e3\uff0c\u4f46\u4e0e\u5206\u6cbb\u7b97\u6cd5\u7684\u4e3b\u8981\u533a\u522b\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u4f9d\u8d56\u7684\uff0c\u5728\u5206\u89e3\u8fc7\u7a0b\u4e2d\u4f1a\u51fa\u73b0\u8bb8\u591a\u91cd\u53e0\u5b50\u95ee\u9898\u3002
- \u56de\u6eaf\u7b97\u6cd5\u5728\u5c1d\u8bd5\u548c\u56de\u9000\u4e2d\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u5e76\u901a\u8fc7\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\u3002\u539f\u95ee\u9898\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\u6784\u6210\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u51b3\u7b56\u6b65\u9aa4\u4e4b\u524d\u7684\u5b50\u5e8f\u5217\u770b\u4f5c\u4e00\u4e2a\u5b50\u95ee\u9898\u3002
\u5b9e\u9645\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5b83\u4eec\u4e0d\u4ec5\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u8fd8\u5177\u6709\u53e6\u5916\u4e24\u5927\u7279\u6027\uff1a\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002
"},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","title":"14.2.1 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784","text":"\u6211\u4eec\u5bf9\u722c\u697c\u68af\u95ee\u9898\u7a0d\u4f5c\u6539\u52a8\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9002\u5408\u5c55\u793a\u6700\u4f18\u5b50\u7ed3\u6784\u6982\u5ff5\u3002
\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7
\u7ed9\u5b9a\u4e00\u4e2a\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u6bcf\u4e00\u9636\u697c\u68af\u4e0a\u90fd\u8d34\u6709\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u4f60\u5728\u8be5\u53f0\u9636\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\u3002\u7ed9\u5b9a\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\u6570\u7ec4 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u4e2a\u53f0\u9636\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\\(cost[0]\\) \u4e3a\u5730\u9762\uff08\u8d77\u59cb\u70b9\uff09\u3002\u8bf7\u8ba1\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u4ef7\u624d\u80fd\u5230\u8fbe\u9876\u90e8\uff1f
\u5982\u56fe 14-6 \u6240\u793a\uff0c\u82e5\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9636\u7684\u4ee3\u4ef7\u5206\u522b\u4e3a \\(1\\)\u3001\\(10\\)\u3001\\(1\\) \uff0c\u5219\u4ece\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7\u4e3a \\(2\\) \u3002
\u56fe 14-6 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7
\u8bbe \\(dp[i]\\) \u4e3a\u722c\u5230\u7b2c \\(i\\) \u9636\u7d2f\u8ba1\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\u7531\u4e8e\u7b2c \\(i\\) \u9636\u53ea\u53ef\u80fd\u4ece \\(i - 1\\) \u9636\u6216 \\(i - 2\\) \u9636\u8d70\u6765\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u4e8e \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u4e3a\u4e86\u5c3d\u53ef\u80fd\u51cf\u5c11\u4ee3\u4ef7\uff0c\u6211\u4eec\u5e94\u8be5\u9009\u62e9\u4e24\u8005\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\uff1a
\\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\]\u8fd9\u4fbf\u53ef\u4ee5\u5f15\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u542b\u4e49\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u662f\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\u7684\u3002
\u672c\u9898\u663e\u7136\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6211\u4eec\u4ece\u4e24\u4e2a\u5b50\u95ee\u9898\u6700\u4f18\u89e3 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u4e2d\u6311\u9009\u51fa\u8f83\u4f18\u7684\u90a3\u4e00\u4e2a\uff0c\u5e76\u7528\u5b83\u6784\u5efa\u51fa\u539f\u95ee\u9898 \\(dp[i]\\) \u7684\u6700\u4f18\u89e3\u3002
\u90a3\u4e48\uff0c\u4e0a\u4e00\u8282\u7684\u722c\u697c\u68af\u9898\u76ee\u6709\u6ca1\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u5462\uff1f\u5b83\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u4e2a\u8ba1\u6570\u95ee\u9898\uff0c\u4f46\u5982\u679c\u6362\u4e00\u79cd\u95ee\u6cd5\uff1a\u201c\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6570\u91cf\u201d\u3002\u6211\u4eec\u610f\u5916\u5730\u53d1\u73b0\uff0c\u867d\u7136\u9898\u76ee\u4fee\u6539\u524d\u540e\u662f\u7b49\u4ef7\u7684\uff0c\u4f46\u6700\u4f18\u5b50\u7ed3\u6784\u6d6e\u73b0\u51fa\u6765\u4e86\uff1a\u7b2c \\(n\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u7b49\u4e8e\u7b2c \\(n-1\\) \u9636\u548c\u7b2c \\(n-2\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8bf4\uff0c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u89e3\u91ca\u65b9\u5f0f\u6bd4\u8f83\u7075\u6d3b\uff0c\u5728\u4e0d\u540c\u95ee\u9898\u4e2d\u4f1a\u6709\u4e0d\u540c\u7684\u542b\u4e49\u3002
\u6839\u636e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72b6\u6001 \\(dp[1] = cost[1]\\) \u548c \\(dp[2] = cost[2]\\) \uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u52a8\u6001\u89c4\u5212\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_cost_climbing_stairs_dp.py
min_cost_climbing_stairs_dp.cppdef min_cost_climbing_stairs_dp(cost: list[int]) -> int:\n \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(cost) - 1\n if n == 1 or n == 2:\n return cost[n]\n # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp = [0] * (n + 1)\n # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1], dp[2] = cost[1], cost[2]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in range(3, n + 1):\n dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n return dp[n]\n
min_cost_climbing_stairs_dp.java/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(vector<int> &cost) {\n int n = cost.size() - 1;\n if (n == 1 || n == 2)\n return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n vector<int> dp(n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.cs/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\n int n = cost.length - 1;\n if (n == 1 || n == 2)\n return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[] dp = new int[n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.go/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDP(int[] cost) {\n int n = cost.Length - 1;\n if (n == 1 || n == 2)\n return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[] dp = new int[n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.swift/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost []int) int {\n n := len(cost) - 1\n if n == 1 || n == 2 {\n return cost[n]\n }\n min := func(a, b int) int {\n if a < b {\n return a\n }\n return b\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp := make([]int, n+1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1]\n dp[2] = cost[2]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n }\n return dp[n]\n}\n
min_cost_climbing_stairs_dp.js/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost: [Int]) -> Int {\n let n = cost.count - 1\n if n == 1 || n == 2 {\n return cost[n]\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = Array(repeating: 0, count: n + 1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1]\n dp[2] = cost[2]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in stride(from: 3, through: n, by: 1) {\n dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n }\n return dp[n]\n}\n
min_cost_climbing_stairs_dp.ts/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost) {\n const n = cost.length - 1;\n if (n === 1 || n === 2) {\n return cost[n];\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = new Array(n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.dart/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost: Array<number>): number {\n const n = cost.length - 1;\n if (n === 1 || n === 2) {\n return cost[n];\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = new Array(n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.rs/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(List<int> cost) {\n int n = cost.length - 1;\n if (n == 1 || n == 2) return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n List<int> dp = List.filled(n + 1, 0);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n
min_cost_climbing_stairs_dp.c/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp(cost: &[i32]) -> i32 {\n let n = cost.len() - 1;\n if n == 1 || n == 2 { return cost[n]; }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n let mut dp = vec![-1; n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in 3..=n {\n dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n dp[n]\n}\n
min_cost_climbing_stairs_dp.zig/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n int n = costSize - 1;\n if (n == 1 || n == 2)\n return cost[n];\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int *dp = calloc(n + 1, sizeof(int));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n }\n int res = dp[n];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDP(comptime cost: []i32) i32 {\n comptime var n = cost.len - 1;\n if (n == 1 or n == 2) {\n return cost[n];\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = [_]i32{-1} ** (n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = cost[1];\n dp[2] = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (3..n + 1) |i| {\n dp[i] = @min(dp[i - 1], dp[i - 2]) + cost[i];\n }\n return dp[n];\n}\n\u56fe 14-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\u3002
\u56fe 14-7 \u00a0 \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
\u672c\u9898\u4e5f\u53ef\u4ee5\u8fdb\u884c\u7a7a\u95f4\u4f18\u5316\uff0c\u5c06\u4e00\u7ef4\u538b\u7f29\u81f3\u96f6\u7ef4\uff0c\u4f7f\u5f97\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_cost_climbing_stairs_dp.py
min_cost_climbing_stairs_dp.cppdef min_cost_climbing_stairs_dp_comp(cost: list[int]) -> int:\n \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(cost) - 1\n if n == 1 or n == 2:\n return cost[n]\n a, b = cost[1], cost[2]\n for i in range(3, n + 1):\n a, b = b, min(a, b) + cost[i]\n return b\n
min_cost_climbing_stairs_dp.java/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(vector<int> &cost) {\n int n = cost.size() - 1;\n if (n == 1 || n == 2)\n return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.cs/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\n int n = cost.length - 1;\n if (n == 1 || n == 2)\n return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = Math.min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.go/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDPComp(int[] cost) {\n int n = cost.Length - 1;\n if (n == 1 || n == 2)\n return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = Math.Min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.swift/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n n := len(cost) - 1\n if n == 1 || n == 2 {\n return cost[n]\n }\n min := func(a, b int) int {\n if a < b {\n return a\n }\n return b\n }\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n a, b := cost[1], cost[2]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n tmp := b\n b = min(a, tmp) + cost[i]\n a = tmp\n }\n return b\n}\n
min_cost_climbing_stairs_dp.js/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -> Int {\n let n = cost.count - 1\n if n == 1 || n == 2 {\n return cost[n]\n }\n var (a, b) = (cost[1], cost[2])\n for i in stride(from: 3, through: n, by: 1) {\n (a, b) = (b, min(a, b) + cost[i])\n }\n return b\n}\n
min_cost_climbing_stairs_dp.ts/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost) {\n const n = cost.length - 1;\n if (n === 1 || n === 2) {\n return cost[n];\n }\n let a = cost[1],\n b = cost[2];\n for (let i = 3; i <= n; i++) {\n const tmp = b;\n b = Math.min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.dart/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost: Array<number>): number {\n const n = cost.length - 1;\n if (n === 1 || n === 2) {\n return cost[n];\n }\n let a = cost[1],\n b = cost[2];\n for (let i = 3; i <= n; i++) {\n const tmp = b;\n b = Math.min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.rs/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(List<int> cost) {\n int n = cost.length - 1;\n if (n == 1 || n == 2) return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
min_cost_climbing_stairs_dp.c/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp_comp(cost: &[i32]) -> i32 {\n let n = cost.len() - 1;\n if n == 1 || n == 2 { return cost[n] };\n let (mut a, mut b) = (cost[1], cost[2]);\n for i in 3..=n {\n let tmp = b;\n b = cmp::min(a, tmp) + cost[i];\n a = tmp;\n }\n b\n}\n
min_cost_climbing_stairs_dp.zig/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n int n = costSize - 1;\n if (n == 1 || n == 2)\n return cost[n];\n int a = cost[1], b = cost[2];\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = myMin(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n
"},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","title":"14.2.2 \u00a0 \u65e0\u540e\u6548\u6027","text":"// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDPComp(cost: []i32) i32 {\n var n = cost.len - 1;\n if (n == 1 or n == 2) {\n return cost[n];\n }\n var a = cost[1];\n var b = cost[2];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (3..n + 1) |i| {\n var tmp = b;\n b = @min(a, tmp) + cost[i];\n a = tmp;\n }\n return b;\n}\n\u65e0\u540e\u6548\u6027\u662f\u52a8\u6001\u89c4\u5212\u80fd\u591f\u6709\u6548\u89e3\u51b3\u95ee\u9898\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5176\u5b9a\u4e49\u4e3a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u786e\u5b9a\u7684\u72b6\u6001\uff0c\u5b83\u7684\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u5f53\u524d\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002
\u4ee5\u722c\u697c\u68af\u95ee\u9898\u4e3a\u4f8b\uff0c\u7ed9\u5b9a\u72b6\u6001 \\(i\\) \uff0c\u5b83\u4f1a\u53d1\u5c55\u51fa\u72b6\u6001 \\(i+1\\) \u548c\u72b6\u6001 \\(i+2\\) \uff0c\u5206\u522b\u5bf9\u5e94\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u8fd9\u4e24\u79cd\u9009\u62e9\u65f6\uff0c\u6211\u4eec\u65e0\u987b\u8003\u8651\u72b6\u6001 \\(i\\) \u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5b83\u4eec\u5bf9\u72b6\u6001 \\(i\\) \u7684\u672a\u6765\u6ca1\u6709\u5f71\u54cd\u3002
\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u7ed9\u722c\u697c\u68af\u95ee\u9898\u6dfb\u52a0\u4e00\u4e2a\u7ea6\u675f\uff0c\u60c5\u51b5\u5c31\u4e0d\u4e00\u6837\u4e86\u3002
\u5e26\u7ea6\u675f\u722c\u697c\u68af
\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u4f46\u4e0d\u80fd\u8fde\u7eed\u4e24\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f
\u5982\u56fe 14-8 \u6240\u793a\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u9636\u4ec5\u5269 \\(2\\) \u79cd\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u8fde\u7eed\u4e09\u6b21\u8df3 \\(1\\) \u9636\u7684\u65b9\u6848\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\uff0c\u56e0\u6b64\u88ab\u820d\u5f03\u3002
\u56fe 14-8 \u00a0 \u5e26\u7ea6\u675f\u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf
\u5728\u8be5\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\uff0c\u90a3\u4e48\u4e0b\u4e00\u8f6e\u5c31\u5fc5\u987b\u8df3 \\(2\\) \u9636\u3002\u8fd9\u610f\u5473\u7740\uff0c\u4e0b\u4e00\u6b65\u9009\u62e9\u4e0d\u80fd\u7531\u5f53\u524d\u72b6\u6001\uff08\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u72ec\u7acb\u51b3\u5b9a\uff0c\u8fd8\u548c\u524d\u4e00\u4e2a\u72b6\u6001\uff08\u4e0a\u8f6e\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u6709\u5173\u3002
\u4e0d\u96be\u53d1\u73b0\uff0c\u6b64\u95ee\u9898\u5df2\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u4e3a \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8bb8\u591a\u201c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\u201d\u65b9\u6848\uff0c\u800c\u4e3a\u4e86\u6ee1\u8db3\u7ea6\u675f\uff0c\u6211\u4eec\u5c31\u4e0d\u80fd\u5c06 \\(dp[i-1]\\) \u76f4\u63a5\u8ba1\u5165 \\(dp[i]\\) \u4e2d\u3002
\u4e3a\u6b64\uff0c\u6211\u4eec\u9700\u8981\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff1a\u72b6\u6001 \\([i, j]\\) \u8868\u793a\u5904\u5728\u7b2c \\(i\\) \u9636\u5e76\u4e14\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(j\\) \u9636\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72b6\u6001\u5b9a\u4e49\u6709\u6548\u5730\u533a\u5206\u4e86\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u8fd8\u662f \\(2\\) \u9636\uff0c\u6211\u4eec\u53ef\u4ee5\u636e\u6b64\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u4ece\u4f55\u800c\u6765\u7684\u3002
- \u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ea\u80fd\u9009\u62e9\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 1]\\) \u53ea\u80fd\u4ece \\(dp[i-1, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002
- \u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(2\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ef\u9009\u62e9\u8df3 \\(1\\) \u9636\u6216\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 2]\\) \u53ef\u4ee5\u4ece \\(dp[i-2, 1]\\) \u6216 \\(dp[i-2, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002
\u5982\u56fe 14-9 \u6240\u793a\uff0c\u5728\u8be5\u5b9a\u4e49\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u65b9\u6848\u6570\u3002\u6b64\u65f6\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a
\\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\]\u56fe 14-9 \u00a0 \u8003\u8651\u7ea6\u675f\u4e0b\u7684\u9012\u63a8\u5173\u7cfb
\u6700\u7ec8\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u4e24\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u9636\u7684\u65b9\u6848\u603b\u6570\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_constraint_dp.py
climbing_stairs_constraint_dp.cppdef climbing_stairs_constraint_dp(n: int) -> int:\n \"\"\"\u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n if n == 1 or n == 2:\n return 1\n # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp = [[0] * 3 for _ in range(n + 1)]\n # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1], dp[1][2] = 1, 0\n dp[2][1], dp[2][2] = 0, 1\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in range(3, n + 1):\n dp[i][1] = dp[i - 1][2]\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n return dp[n][1] + dp[n][2]\n
climbing_stairs_constraint_dp.java/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n vector<vector<int>> dp(n + 1, vector<int>(3, 0));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.cs/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[][] dp = new int[n + 1][3];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.go/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[,] dp = new int[n + 1, 3];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1, 1] = 1;\n dp[1, 2] = 0;\n dp[2, 1] = 0;\n dp[2, 2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i, 1] = dp[i - 1, 2];\n dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n }\n return dp[n, 1] + dp[n, 2];\n}\n
climbing_stairs_constraint_dp.swift/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n int) int {\n if n == 1 || n == 2 {\n return 1\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp := make([][3]int, n+1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1\n dp[1][2] = 0\n dp[2][1] = 0\n dp[2][2] = 1\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n dp[i][1] = dp[i-1][2]\n dp[i][2] = dp[i-2][1] + dp[i-2][2]\n }\n return dp[n][1] + dp[n][2]\n}\n
climbing_stairs_constraint_dp.js/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n: Int) -> Int {\n if n == 1 || n == 2 {\n return 1\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1\n dp[1][2] = 0\n dp[2][1] = 0\n dp[2][2] = 1\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in stride(from: 3, through: n, by: 1) {\n dp[i][1] = dp[i - 1][2]\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n }\n return dp[n][1] + dp[n][2]\n}\n
climbing_stairs_constraint_dp.ts/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n) {\n if (n === 1 || n === 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = Array.from(new Array(n + 1), () => new Array(3));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.dart/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n: number): number {\n if (n === 1 || n === 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = Array.from({ length: n + 1 }, () => new Array(3));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.rs/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(3, 0));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n
climbing_stairs_constraint_dp.c/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_constraint_dp(n: usize) -> i32 {\n if n == 1 || n == 2 { return 1 };\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n let mut dp = vec![vec![-1; 3]; n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in 3..=n {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n dp[n][1] + dp[n][2]\n}\n
climbing_stairs_constraint_dp.zig/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n if (n == 1 || n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(3, sizeof(int));\n }\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n int res = dp[n][1] + dp[n][2];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n free(dp);\n return res;\n}\n// \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsConstraintDP(comptime n: usize) i32 {\n if (n == 1 or n == 2) {\n return 1;\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = [_][3]i32{ [_]i32{ -1, -1, -1 } } ** (n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1][1] = 1;\n dp[1][2] = 0;\n dp[2][1] = 0;\n dp[2][2] = 1;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (3..n + 1) |i| {\n dp[i][1] = dp[i - 1][2];\n dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n }\n return dp[n][1] + dp[n][2];\n}\n\u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u4e8e\u4ec5\u9700\u591a\u8003\u8651\u524d\u9762\u4e00\u4e2a\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u901a\u8fc7\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff0c\u4f7f\u5f97\u95ee\u9898\u91cd\u65b0\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u95ee\u9898\u5177\u6709\u975e\u5e38\u4e25\u91cd\u7684\u201c\u6709\u540e\u6548\u6027\u201d\u3002
\u722c\u697c\u68af\u4e0e\u969c\u788d\u751f\u6210
\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\u3002\u89c4\u5b9a\u5f53\u722c\u5230\u7b2c \\(i\\) \u9636\u65f6\uff0c\u7cfb\u7edf\u81ea\u52a8\u4f1a\u5728\u7b2c \\(2i\\) \u9636\u4e0a\u653e\u4e0a\u969c\u788d\u7269\uff0c\u4e4b\u540e\u6240\u6709\u8f6e\u90fd\u4e0d\u5141\u8bb8\u8df3\u5230\u7b2c \\(2i\\) \u9636\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u4e24\u8f6e\u5206\u522b\u8df3\u5230\u4e86\u7b2c \\(2\\)\u3001\\(3\\) \u9636\u4e0a\uff0c\u5219\u4e4b\u540e\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4\\)\u3001\\(6\\) \u9636\u4e0a\u3002\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f
\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8dc3\u4f9d\u8d56\u8fc7\u53bb\u6240\u6709\u7684\u72b6\u6001\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b21\u8df3\u8dc3\u90fd\u4f1a\u5728\u66f4\u9ad8\u7684\u9636\u68af\u4e0a\u8bbe\u7f6e\u969c\u788d\uff0c\u5e76\u5f71\u54cd\u672a\u6765\u7684\u8df3\u8dc3\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u5f80\u5f80\u96be\u4ee5\u89e3\u51b3\u3002
\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u590d\u6742\u7684\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff08\u4f8b\u5982\u65c5\u884c\u5546\u95ee\u9898\uff09\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u542f\u53d1\u5f0f\u641c\u7d22\u3001\u9057\u4f20\u7b97\u6cd5\u3001\u5f3a\u5316\u5b66\u4e60\u7b49\uff0c\u4ece\u800c\u5728\u6709\u9650\u65f6\u95f4\u5185\u5f97\u5230\u53ef\u7528\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","title":"14.3 \u00a0 \u52a8\u6001\u89c4\u5212\u89e3\u9898\u601d\u8def","text":"\u4e0a\u4e24\u8282\u4ecb\u7ecd\u4e86\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4e3b\u8981\u7279\u5f81\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u4e00\u8d77\u63a2\u7a76\u4e24\u4e2a\u66f4\u52a0\u5b9e\u7528\u7684\u95ee\u9898\u3002
- \u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u95ee\u9898\u662f\u4e0d\u662f\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff1f
- \u6c42\u89e3\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8be5\u4ece\u4f55\u5904\u5165\u624b\uff0c\u5b8c\u6574\u6b65\u9aa4\u662f\u4ec0\u4e48\uff1f
\u603b\u7684\u6765\u8bf4\uff0c\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u90a3\u4e48\u5b83\u901a\u5e38\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u6c42\u89e3\u3002\u7136\u800c\uff0c\u6211\u4eec\u5f88\u96be\u4ece\u95ee\u9898\u63cf\u8ff0\u4e2d\u76f4\u63a5\u63d0\u53d6\u51fa\u8fd9\u4e9b\u7279\u6027\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u653e\u5bbd\u6761\u4ef6\uff0c\u5148\u89c2\u5bdf\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u56de\u6eaf\uff08\u7a77\u4e3e\uff09\u89e3\u51b3\u3002
\u9002\u5408\u7528\u56de\u6eaf\u89e3\u51b3\u7684\u95ee\u9898\u901a\u5e38\u6ee1\u8db3\u201c\u51b3\u7b56\u6811\u6a21\u578b\u201d\uff0c\u8fd9\u79cd\u95ee\u9898\u53ef\u4ee5\u4f7f\u7528\u6811\u5f62\u7ed3\u6784\u6765\u63cf\u8ff0\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\uff0c\u6bcf\u4e00\u6761\u8def\u5f84\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\u3002
\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u95ee\u9898\u5305\u542b\u660e\u786e\u7684\u51b3\u7b56\u6982\u5ff5\uff0c\u5e76\u4e14\u89e3\u662f\u901a\u8fc7\u4e00\u7cfb\u5217\u51b3\u7b56\u4ea7\u751f\u7684\uff0c\u90a3\u4e48\u5b83\u5c31\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u901a\u5e38\u53ef\u4ee5\u4f7f\u7528\u56de\u6eaf\u6765\u89e3\u51b3\u3002
\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8fd8\u6709\u4e00\u4e9b\u5224\u65ad\u7684\u201c\u52a0\u5206\u9879\u201d\u3002
- \u95ee\u9898\u5305\u542b\u6700\u5927\uff08\u5c0f\uff09\u6216\u6700\u591a\uff08\u5c11\uff09\u7b49\u6700\u4f18\u5316\u63cf\u8ff0\u3002
- \u95ee\u9898\u7684\u72b6\u6001\u80fd\u591f\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u3001\u591a\u7ef4\u77e9\u9635\u6216\u6811\u6765\u8868\u793a\uff0c\u5e76\u4e14\u4e00\u4e2a\u72b6\u6001\u4e0e\u5176\u5468\u56f4\u7684\u72b6\u6001\u5b58\u5728\u9012\u63a8\u5173\u7cfb\u3002
\u76f8\u5e94\u5730\uff0c\u4e5f\u5b58\u5728\u4e00\u4e9b\u201c\u51cf\u5206\u9879\u201d\u3002
- \u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e0d\u662f\u627e\u51fa\u6700\u4f18\u89e3\u3002
- \u95ee\u9898\u63cf\u8ff0\u4e2d\u6709\u660e\u663e\u7684\u6392\u5217\u7ec4\u5408\u7684\u7279\u5f81\uff0c\u9700\u8981\u8fd4\u56de\u5177\u4f53\u7684\u591a\u4e2a\u65b9\u6848\u3002
\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u5e76\u5177\u6709\u8f83\u4e3a\u660e\u663e\u7684\u201c\u52a0\u5206\u9879\u201c\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5047\u8bbe\u5b83\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u5e76\u5728\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u9a8c\u8bc1\u5b83\u3002
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","title":"14.3.2 \u00a0 \u95ee\u9898\u6c42\u89e3\u6b65\u9aa4","text":"\u52a8\u6001\u89c4\u5212\u7684\u89e3\u9898\u6d41\u7a0b\u4f1a\u56e0\u95ee\u9898\u7684\u6027\u8d28\u548c\u96be\u5ea6\u800c\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u901a\u5e38\u9075\u5faa\u4ee5\u4e0b\u6b65\u9aa4\uff1a\u63cf\u8ff0\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u5efa\u7acb \\(dp\\) \u8868\uff0c\u63a8\u5bfc\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u7b49\u3002
\u4e3a\u4e86\u66f4\u5f62\u8c61\u5730\u5c55\u793a\u89e3\u9898\u6b65\u9aa4\uff0c\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\u201c\u6700\u5c0f\u8def\u5f84\u548c\u201d\u6765\u4e3e\u4f8b\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a \\(n \\times m\\) \u7684\u4e8c\u7ef4\u7f51\u683c
grid\uff0c\u7f51\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5305\u542b\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u8be5\u5355\u5143\u683c\u7684\u4ee3\u4ef7\u3002\u673a\u5668\u4eba\u4ee5\u5de6\u4e0a\u89d2\u5355\u5143\u683c\u4e3a\u8d77\u59cb\u70b9\uff0c\u6bcf\u6b21\u53ea\u80fd\u5411\u4e0b\u6216\u8005\u5411\u53f3\u79fb\u52a8\u4e00\u6b65\uff0c\u76f4\u81f3\u5230\u8fbe\u53f3\u4e0b\u89d2\u5355\u5143\u683c\u3002\u8bf7\u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230\u53f3\u4e0b\u89d2\u7684\u6700\u5c0f\u8def\u5f84\u548c\u3002\u56fe 14-10 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u7f51\u683c\u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e3a \\(13\\) \u3002
\u56fe 14-10 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u793a\u4f8b\u6570\u636e
\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868
\u672c\u9898\u7684\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u5c31\u662f\u4ece\u5f53\u524d\u683c\u5b50\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u3002\u8bbe\u5f53\u524d\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\([i, j]\\) \uff0c\u5219\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u540e\uff0c\u7d22\u5f15\u53d8\u4e3a \\([i+1, j]\\) \u6216 \\([i, j+1]\\) \u3002\u56e0\u6b64\uff0c\u72b6\u6001\u5e94\u5305\u542b\u884c\u7d22\u5f15\u548c\u5217\u7d22\u5f15\u4e24\u4e2a\u53d8\u91cf\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002
\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u4ece\u8d77\u59cb\u70b9 \\([0, 0]\\) \u8d70\u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\uff0c\u89e3\u8bb0\u4e3a \\(dp[i, j]\\) \u3002
\u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u56fe 14-11 \u6240\u793a\u7684\u4e8c\u7ef4 \\(dp\\) \u77e9\u9635\uff0c\u5176\u5c3a\u5bf8\u4e0e\u8f93\u5165\u7f51\u683c \\(grid\\) \u76f8\u540c\u3002
\u56fe 14-11 \u00a0 \u72b6\u6001\u5b9a\u4e49\u4e0e dp \u8868
Note
\u52a8\u6001\u89c4\u5212\u548c\u56de\u6eaf\u8fc7\u7a0b\u53ef\u4ee5\u63cf\u8ff0\u4e3a\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\uff0c\u800c\u72b6\u6001\u7531\u6240\u6709\u51b3\u7b56\u53d8\u91cf\u6784\u6210\u3002\u5b83\u5e94\u5f53\u5305\u542b\u63cf\u8ff0\u89e3\u9898\u8fdb\u5ea6\u7684\u6240\u6709\u53d8\u91cf\uff0c\u5176\u5305\u542b\u4e86\u8db3\u591f\u7684\u4fe1\u606f\uff0c\u80fd\u591f\u7528\u6765\u63a8\u5bfc\u51fa\u4e0b\u4e00\u4e2a\u72b6\u6001\u3002
\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5b9a\u4e49\u4e00\u4e2a \\(dp\\) \u8868\u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u72b6\u6001\u7684\u6bcf\u4e2a\u72ec\u7acb\u53d8\u91cf\u90fd\u662f \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\\(dp\\) \u8868\u662f\u72b6\u6001\u548c\u5b50\u95ee\u9898\u7684\u89e3\u4e4b\u95f4\u7684\u6620\u5c04\u3002
\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
\u5bf9\u4e8e\u72b6\u6001 \\([i, j]\\) \uff0c\u5b83\u53ea\u80fd\u4ece\u4e0a\u8fb9\u683c\u5b50 \\([i-1, j]\\) \u548c\u5de6\u8fb9\u683c\u5b50 \\([i, j-1]\\) \u8f6c\u79fb\u800c\u6765\u3002\u56e0\u6b64\u6700\u4f18\u5b50\u7ed3\u6784\u4e3a\uff1a\u5230\u8fbe \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u7531 \\([i, j-1]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e0e \\([i-1, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\u51b3\u5b9a\u3002
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u63a8\u51fa\u56fe 14-12 \u6240\u793a\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a
\\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\]\u56fe 14-12 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
Note
\u6839\u636e\u5b9a\u4e49\u597d\u7684 \\(dp\\) \u8868\uff0c\u601d\u8003\u539f\u95ee\u9898\u548c\u5b50\u95ee\u9898\u7684\u5173\u7cfb\uff0c\u627e\u51fa\u901a\u8fc7\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6765\u6784\u9020\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u7684\u65b9\u6cd5\uff0c\u5373\u6700\u4f18\u5b50\u7ed3\u6784\u3002
\u4e00\u65e6\u6211\u4eec\u627e\u5230\u4e86\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002
\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
\u5728\u672c\u9898\u4e2d\uff0c\u5904\u5728\u9996\u884c\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u5de6\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u5904\u5728\u9996\u5217\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u4e0a\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u56e0\u6b64\u9996\u884c \\(i = 0\\) \u548c\u9996\u5217 \\(j = 0\\) \u662f\u8fb9\u754c\u6761\u4ef6\u3002
\u5982\u56fe 14-13 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u662f\u7531\u5176\u5de6\u65b9\u683c\u5b50\u548c\u4e0a\u65b9\u683c\u5b50\u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u5faa\u73af\u6765\u904d\u5386\u77e9\u9635\uff0c\u5916\u5faa\u73af\u904d\u5386\u5404\u884c\uff0c\u5185\u5faa\u73af\u904d\u5386\u5404\u5217\u3002
\u56fe 14-13 \u00a0 \u8fb9\u754c\u6761\u4ef6\u4e0e\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
Note
\u8fb9\u754c\u6761\u4ef6\u5728\u52a8\u6001\u89c4\u5212\u4e2d\u7528\u4e8e\u521d\u59cb\u5316 \\(dp\\) \u8868\uff0c\u5728\u641c\u7d22\u4e2d\u7528\u4e8e\u526a\u679d\u3002
\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f\u7684\u6838\u5fc3\u662f\u8981\u4fdd\u8bc1\u5728\u8ba1\u7b97\u5f53\u524d\u95ee\u9898\u7684\u89e3\u65f6\uff0c\u6240\u6709\u5b83\u4f9d\u8d56\u7684\u66f4\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u90fd\u5df2\u7ecf\u88ab\u6b63\u786e\u5730\u8ba1\u7b97\u51fa\u6765\u3002
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u5df2\u7ecf\u53ef\u4ee5\u76f4\u63a5\u5199\u51fa\u52a8\u6001\u89c4\u5212\u4ee3\u7801\u3002\u7136\u800c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u601d\u60f3\uff0c\u56e0\u6b64\u6309\u7167\u201c\u66b4\u529b\u641c\u7d22 \\(\\rightarrow\\) \u8bb0\u5fc6\u5316\u641c\u7d22 \\(\\rightarrow\\) \u52a8\u6001\u89c4\u5212\u201d\u7684\u987a\u5e8f\u5b9e\u73b0\u66f4\u52a0\u7b26\u5408\u601d\u7ef4\u4e60\u60ef\u3002
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"\u4ece\u72b6\u6001 \\([i, j]\\) \u5f00\u59cb\u641c\u7d22\uff0c\u4e0d\u65ad\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u72b6\u6001 \\([i-1, j]\\) \u548c \\([i, j-1]\\) \uff0c\u9012\u5f52\u51fd\u6570\u5305\u62ec\u4ee5\u4e0b\u8981\u7d20\u3002
- \u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, j]\\) \u3002
- \u8fd4\u56de\u503c\uff1a\u4ece \\([0, 0]\\) \u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c \\(dp[i, j]\\) \u3002
- \u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53 \\(i = 0\\) \u4e14 \\(j = 0\\) \u65f6\uff0c\u8fd4\u56de\u4ee3\u4ef7 \\(grid[0, 0]\\) \u3002
- \u526a\u679d\uff1a\u5f53 \\(i < 0\\) \u65f6\u6216 \\(j < 0\\) \u65f6\u7d22\u5f15\u8d8a\u754c\uff0c\u6b64\u65f6\u8fd4\u56de\u4ee3\u4ef7 \\(+\\infty\\) \uff0c\u4ee3\u8868\u4e0d\u53ef\u884c\u3002
\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py
min_path_sum.cppdef min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -> int:\n \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 and j == 0:\n return grid[0][0]\n # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 or j < 0:\n return inf\n # \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n up = min_path_sum_dfs(grid, i - 1, j)\n left = min_path_sum_dfs(grid, i, j - 1)\n # \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return min(left, up) + grid[i][j]\n
min_path_sum.java/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(vector<vector<int>> &grid, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return INT_MAX;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFS(grid, i - 1, j);\n int left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n
min_path_sum.cs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int[][] grid, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Integer.MAX_VALUE;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFS(grid, i - 1, j);\n int left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return Math.min(left, up) + grid[i][j];\n}\n
min_path_sum.go/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return int.MaxValue;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = MinPathSumDFS(grid, i - 1, j);\n int left = MinPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return Math.Min(left, up) + grid[i][j];\n}\n
min_path_sum.swift/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 && j == 0 {\n return grid[0][0]\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return math.MaxInt\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n up := minPathSumDFS(grid, i-1, j)\n left := minPathSumDFS(grid, i, j-1)\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n
min_path_sum.js/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -> Int {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0, j == 0 {\n return grid[0][0]\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return .max\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return min(left, up) + grid[i][j]\n}\n
min_path_sum.ts/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(grid, i, j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i === 0 && j === 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Infinity;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n const up = minPathSumDFS(grid, i - 1, j);\n const left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return Math.min(left, up) + grid[i][j];\n}\n
min_path_sum.dart/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(\n grid: Array<Array<number>>,\n i: number,\n j: number\n): number {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i === 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Infinity;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n const up = minPathSumDFS(grid, i - 1, j);\n const left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return Math.min(left, up) + grid[i][j];\n}\n
min_path_sum.rs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(List<List<int>> grid, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n return BigInt.from(2).pow(31).toInt();\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFS(grid, i - 1, j);\n int left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return min(left, up) + grid[i][j];\n}\n
min_path_sum.c/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfn min_path_sum_dfs(grid: &Vec<Vec<i32>>, i: i32, j: i32) -> i32 {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 && j == 0 {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return i32::MAX;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n let up = min_path_sum_dfs(grid, i - 1, j);\n let left = min_path_sum_dfs(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n
min_path_sum.zig/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return INT_MAX;\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFS(grid, i - 1, j);\n int left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\nfn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 and j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 or j < 0) {\n return std.math.maxInt(i32);\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n var up = minPathSumDFS(grid, i - 1, j);\n var left = minPathSumDFS(grid, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n return @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n\u56fe 14-14 \u7ed9\u51fa\u4e86\u4ee5 \\(dp[2, 1]\\) \u4e3a\u6839\u8282\u70b9\u7684\u9012\u5f52\u6811\uff0c\u5176\u4e2d\u5305\u542b\u4e00\u4e9b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5176\u6570\u91cf\u4f1a\u968f\u7740\u7f51\u683c
grid\u7684\u5c3a\u5bf8\u53d8\u5927\u800c\u6025\u5267\u589e\u591a\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9020\u6210\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u539f\u56e0\u4e3a\uff1a\u5b58\u5728\u591a\u6761\u8def\u5f84\u53ef\u4ee5\u4ece\u5de6\u4e0a\u89d2\u5230\u8fbe\u67d0\u4e00\u5355\u5143\u683c\u3002
\u56fe 14-14 \u00a0 \u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811
\u6bcf\u4e2a\u72b6\u6001\u90fd\u6709\u5411\u4e0b\u548c\u5411\u53f3\u4e24\u79cd\u9009\u62e9\uff0c\u4ece\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\u603b\u5171\u9700\u8981 \\(m + n - 2\\) \u6b65\uff0c\u6240\u4ee5\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^{m + n})\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u8fd9\u79cd\u8ba1\u7b97\u65b9\u5f0f\u672a\u8003\u8651\u4e34\u8fd1\u7f51\u683c\u8fb9\u754c\u7684\u60c5\u51b5\uff0c\u5f53\u5230\u8fbe\u7f51\u7edc\u8fb9\u754c\u65f6\u53ea\u5269\u4e0b\u4e00\u79cd\u9009\u62e9\uff0c\u56e0\u6b64\u5b9e\u9645\u7684\u8def\u5f84\u6570\u91cf\u4f1a\u5c11\u4e00\u4e9b\u3002
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"\u6211\u4eec\u5f15\u5165\u4e00\u4e2a\u548c\u7f51\u683c
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.pygrid\u76f8\u540c\u5c3a\u5bf8\u7684\u8bb0\u5fc6\u5217\u8868mem\uff0c\u7528\u4e8e\u8bb0\u5f55\u5404\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u8fdb\u884c\u526a\u679d\uff1a
min_path_sum.cppdef min_path_sum_dfs_mem(\n grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -> int:\n \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 and j == 0:\n return grid[0][0]\n # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 or j < 0:\n return inf\n # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][j] != -1:\n return mem[i][j]\n # \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n # \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = min(left, up) + grid[i][j]\n return mem[i][j]\n
min_path_sum.java/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(vector<vector<int>> &grid, vector<vector<int>> &mem, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return INT_MAX;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFSMem(grid, mem, i - 1, j);\n int left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n return mem[i][j];\n}\n
min_path_sum.cs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Integer.MAX_VALUE;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFSMem(grid, mem, i - 1, j);\n int left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = Math.min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.go/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return int.MaxValue;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = Math.Min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.swift/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 && j == 0 {\n return grid[0][0]\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return math.MaxInt\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][j] != -1 {\n return mem[i][j]\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n up := minPathSumDFSMem(grid, mem, i-1, j)\n left := minPathSumDFSMem(grid, mem, i, j-1)\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n return mem[i][j]\n}\n
min_path_sum.js/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -> Int {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0, j == 0 {\n return grid[0][0]\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return .max\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][j] != -1 {\n return mem[i][j]\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n let up = minPathSumDFSMem(grid: grid, mem: &mem, i: i - 1, j: j)\n let left = minPathSumDFSMem(grid: grid, mem: &mem, i: i, j: j - 1)\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = min(left, up) + grid[i][j]\n return mem[i][j]\n}\n
min_path_sum.ts/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i === 0 && j === 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Infinity;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] !== -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n const up = minPathSumDFSMem(grid, mem, i - 1, j);\n const left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = Math.min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.dart/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(\n grid: Array<Array<number>>,\n mem: Array<Array<number>>,\n i: number,\n j: number\n): number {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i === 0 && j === 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return Infinity;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n const up = minPathSumDFSMem(grid, mem, i - 1, j);\n const left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = Math.min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.rs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(List<List<int>> grid, List<List<int>> mem, int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n return BigInt.from(2).pow(31).toInt();\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFSMem(grid, mem, i - 1, j);\n int left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = min(left, up) + grid[i][j];\n return mem[i][j];\n}\n
min_path_sum.c/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn min_path_sum_dfs_mem(grid: &Vec<Vec<i32>>, mem: &mut Vec<Vec<i32>>, i: i32, j: i32) -> i32 {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if i == 0 && j == 0 {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if i < 0 || j < 0 {\n return i32::MAX;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i as usize][j as usize] != -1 {\n return mem[i as usize][j as usize];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n mem[i as usize][j as usize]\n}\n
min_path_sum.zig/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 && j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 || j < 0) {\n return INT_MAX;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][j] != -1) {\n return mem[i][j];\n }\n // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n int up = minPathSumDFSMem(grid, mem, i - 1, j);\n int left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n return mem[i][j];\n}\n// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {\n // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n if (i == 0 and j == 0) {\n return grid[0][0];\n }\n // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n if (i < 0 or j < 0) {\n return std.math.maxInt(i32);\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {\n return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n }\n // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n var up = minPathSumDFSMem(grid, mem, i - 1, j);\n var left = minPathSumDFSMem(grid, mem, i, j - 1);\n // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n\u5982\u56fe 14-15 \u6240\u793a\uff0c\u5728\u5f15\u5165\u8bb0\u5fc6\u5316\u540e\uff0c\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u72b6\u6001\u603b\u6570\uff0c\u5373\u7f51\u683c\u5c3a\u5bf8 \\(O(nm)\\) \u3002
\u56fe 14-15 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py
min_path_sum.cppdef min_path_sum_dp(grid: list[list[int]]) -> int:\n \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n, m = len(grid), len(grid[0])\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * m for _ in range(n)]\n dp[0][0] = grid[0][0]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in range(1, m):\n dp[0][j] = dp[0][j - 1] + grid[0][j]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for i in range(1, n):\n dp[i][0] = dp[i - 1][0] + grid[i][0]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in range(1, n):\n for j in range(1, m):\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n return dp[n - 1][m - 1]\n
min_path_sum.java/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(vector<vector<int>> &grid) {\n int n = grid.size(), m = grid[0].size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n, vector<int>(m));\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.cs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int[][] grid) {\n int n = grid.length, m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n][m];\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.go/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDP(int[][] grid) {\n int n = grid.Length, m = grid[0].Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n, m];\n dp[0, 0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0, j] = dp[0, j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n }\n }\n return dp[n - 1, m - 1];\n}\n
min_path_sum.swift/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid [][]int) int {\n n, m := len(grid), len(grid[0])\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n)\n for i := 0; i < n; i++ {\n dp[i] = make([]int, m)\n }\n dp[0][0] = grid[0][0]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j := 1; j < m; j++ {\n dp[0][j] = dp[0][j-1] + grid[0][j]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for i := 1; i < n; i++ {\n dp[i][0] = dp[i-1][0] + grid[i][0]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i < n; i++ {\n for j := 1; j < m; j++ {\n dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n }\n }\n return dp[n-1][m-1]\n}\n
min_path_sum.js/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid: [[Int]]) -> Int {\n let n = grid.count\n let m = grid[0].count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n dp[0][0] = grid[0][0]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in stride(from: 1, to: m, by: 1) {\n dp[0][j] = dp[0][j - 1] + grid[0][j]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for i in stride(from: 1, to: n, by: 1) {\n dp[i][0] = dp[i - 1][0] + grid[i][0]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in stride(from: 1, to: n, by: 1) {\n for j in stride(from: 1, to: m, by: 1) {\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n }\n }\n return dp[n - 1][m - 1]\n}\n
min_path_sum.ts/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid) {\n const n = grid.length,\n m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n }, () =>\n Array.from({ length: m }, () => 0)\n );\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (let j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (let i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i < n; i++) {\n for (let j = 1; j < m; j++) {\n dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.dart/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid: Array<Array<number>>): number {\n const n = grid.length,\n m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n }, () =>\n Array.from({ length: m }, () => 0)\n );\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (let j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (let i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i < n; i++) {\n for (let j: number = 1; j < m; j++) {\n dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.rs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(List<List<int>> grid) {\n int n = grid.length, m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n, (i) => List.filled(m, 0));\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n
min_path_sum.c/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp(grid: &Vec<Vec<i32>>) -> i32 {\n let (n, m) = (grid.len(), grid[0].len());\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; m]; n];\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in 1..m {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for i in 1..n {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in 1..n {\n for j in 1..m {\n dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n dp[n - 1][m - 1]\n}\n
min_path_sum.zig/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc(n * sizeof(int *));\n for (int i = 0; i < n; i++) {\n dp[i] = calloc(m, sizeof(int));\n }\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (int i = 1; i < n; i++) {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i < n; i++) {\n for (int j = 1; j < m; j++) {\n dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n int res = dp[n - 1][m - 1];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i < n; i++) {\n free(dp[i]);\n }\n return res;\n}\n// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\nfn minPathSumDP(comptime grid: anytype) i32 {\n comptime var n = grid.len;\n comptime var m = grid[0].len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][m]i32{[_]i32{0} ** m} ** n;\n dp[0][0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (1..m) |j| {\n dp[0][j] = dp[0][j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n for (1..n) |i| {\n dp[i][0] = dp[i - 1][0] + grid[i][0];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (1..n) |i| {\n for (1..m) |j| {\n dp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n }\n }\n return dp[n - 1][m - 1];\n}\n\u56fe 14-16 \u5c55\u793a\u4e86\u6700\u5c0f\u8def\u5f84\u548c\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\uff0c\u5176\u904d\u5386\u4e86\u6574\u4e2a\u7f51\u683c\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002
\u6570\u7ec4
<1><2><3><4><5><6><7><8><9><10><11><12>dp\u5927\u5c0f\u4e3a \\(n \\times m\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002\u56fe 14-16 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u53ea\u4e0e\u5176\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u683c\u5b50\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u53ea\u7528\u4e00\u4e2a\u5355\u884c\u6570\u7ec4\u6765\u5b9e\u73b0 \\(dp\\) \u8868\u3002
\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.pydp\u53ea\u80fd\u8868\u793a\u4e00\u884c\u7684\u72b6\u6001\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u6cd5\u63d0\u524d\u521d\u59cb\u5316\u9996\u5217\u72b6\u6001\uff0c\u800c\u662f\u5728\u904d\u5386\u6bcf\u884c\u65f6\u66f4\u65b0\u5b83\uff1a
min_path_sum.cppdef min_path_sum_dp_comp(grid: list[list[int]]) -> int:\n \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n, m = len(grid), len(grid[0])\n # \u521d\u59cb\u5316 dp \u8868\n dp = [0] * m\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0]\n for j in range(1, m):\n dp[j] = dp[j - 1] + grid[0][j]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in range(1, n):\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in range(1, m):\n dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n return dp[m - 1]\n
min_path_sum.java/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(vector<vector<int>> &grid) {\n int n = grid.size(), m = grid[0].size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(m);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.cs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int[][] grid) {\n int n = grid.length, m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[m];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.go/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDPComp(int[][] grid) {\n int n = grid.Length, m = grid[0].Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[m];\n dp[0] = grid[0][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.swift/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid [][]int) int {\n n, m := len(grid), len(grid[0])\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, m)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0]\n for j := 1; j < m; j++ {\n dp[j] = dp[j-1] + grid[0][j]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i < n; i++ {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j := 1; j < m; j++ {\n dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n }\n }\n return dp[m-1]\n}\n
min_path_sum.js/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid: [[Int]]) -> Int {\n let n = grid.count\n let m = grid[0].count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: 0, count: m)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0]\n for j in stride(from: 1, to: m, by: 1) {\n dp[j] = dp[j - 1] + grid[0][j]\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in stride(from: 1, to: n, by: 1) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0]\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in stride(from: 1, to: m, by: 1) {\n dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n }\n }\n return dp[m - 1]\n}\n
min_path_sum.ts/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid) {\n const n = grid.length,\n m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = new Array(m);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (let j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (let i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (let j = 1; j < m; j++) {\n dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.dart/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid: Array<Array<number>>): number {\n const n = grid.length,\n m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = new Array(m);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (let j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (let i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (let j = 1; j < m; j++) {\n dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.rs/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(List<List<int>> grid) {\n int n = grid.length, m = grid[0].length;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(m, 0);\n dp[0] = grid[0][0];\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n
min_path_sum.c/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp_comp(grid: &Vec<Vec<i32>>) -> i32 {\n let (n, m) = (grid.len(), grid[0].len());\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; m];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for j in 1..m {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in 1..n {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in 1..m {\n dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n dp[m - 1]\n}\n
min_path_sum.zig/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(m, sizeof(int));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (int j = 1; j < m; j++) {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i < n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j < m; j++) {\n dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n int res = dp[m - 1];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/edit_distance_problem/","title":"14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898","text":"// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minPathSumDPComp(comptime grid: anytype) i32 {\n comptime var n = grid.len;\n comptime var m = grid[0].len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** m;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n dp[0] = grid[0][0];\n for (1..m) |j| {\n dp[j] = dp[j - 1] + grid[0][j];\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (1..n) |i| {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n dp[0] = dp[0] + grid[i][0];\n for (1..m) |j| {\n dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];\n }\n }\n return dp[m - 1];\n}\n\u7f16\u8f91\u8ddd\u79bb\uff0c\u4e5f\u79f0 Levenshtein \u8ddd\u79bb\uff0c\u6307\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u4e92\u76f8\u8f6c\u6362\u7684\u6700\u5c11\u4fee\u6539\u6b21\u6570\uff0c\u901a\u5e38\u7528\u4e8e\u5728\u4fe1\u606f\u68c0\u7d22\u548c\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\u5ea6\u91cf\u4e24\u4e2a\u5e8f\u5217\u7684\u76f8\u4f3c\u5ea6\u3002
Question
\u8f93\u5165\u4e24\u4e2a\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \uff0c\u8fd4\u56de\u5c06 \\(s\\) \u8f6c\u6362\u4e3a \\(t\\) \u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002
\u4f60\u53ef\u4ee5\u5728\u4e00\u4e2a\u5b57\u7b26\u4e32\u4e2d\u8fdb\u884c\u4e09\u79cd\u7f16\u8f91\u64cd\u4f5c\uff1a\u63d2\u5165\u4e00\u4e2a\u5b57\u7b26\u3001\u5220\u9664\u4e00\u4e2a\u5b57\u7b26\u3001\u5c06\u5b57\u7b26\u66ff\u6362\u4e3a\u4efb\u610f\u4e00\u4e2a\u5b57\u7b26\u3002
\u5982\u56fe 14-27 \u6240\u793a\uff0c\u5c06
kitten\u8f6c\u6362\u4e3asitting\u9700\u8981\u7f16\u8f91 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u4e0e 1 \u6b21\u6dfb\u52a0\u64cd\u4f5c\uff1b\u5c06hello\u8f6c\u6362\u4e3aalgo\u9700\u8981 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u548c 1 \u6b21\u5220\u9664\u64cd\u4f5c\u3002\u56fe 14-27 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u793a\u4f8b\u6570\u636e
\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u53ef\u4ee5\u5f88\u81ea\u7136\u5730\u7528\u51b3\u7b56\u6811\u6a21\u578b\u6765\u89e3\u91ca\u3002\u5b57\u7b26\u4e32\u5bf9\u5e94\u6811\u8282\u70b9\uff0c\u4e00\u8f6e\u51b3\u7b56\uff08\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\uff09\u5bf9\u5e94\u6811\u7684\u4e00\u6761\u8fb9\u3002
\u5982\u56fe 14-28 \u6240\u793a\uff0c\u5728\u4e0d\u9650\u5236\u64cd\u4f5c\u7684\u60c5\u51b5\u4e0b\uff0c\u6bcf\u4e2a\u8282\u70b9\u90fd\u53ef\u4ee5\u6d3e\u751f\u51fa\u8bb8\u591a\u6761\u8fb9\uff0c\u6bcf\u6761\u8fb9\u5bf9\u5e94\u4e00\u79cd\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u4ece
hello\u8f6c\u6362\u5230algo\u6709\u8bb8\u591a\u79cd\u53ef\u80fd\u7684\u8def\u5f84\u3002\u4ece\u51b3\u7b56\u6811\u7684\u89d2\u5ea6\u770b\uff0c\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u8282\u70b9
hello\u548c\u8282\u70b9algo\u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u3002\u56fe 14-28 \u00a0 \u57fa\u4e8e\u51b3\u7b56\u6811\u6a21\u578b\u8868\u793a\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898
"},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868
\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u662f\u5bf9\u5b57\u7b26\u4e32 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\u3002
\u6211\u4eec\u5e0c\u671b\u5728\u7f16\u8f91\u64cd\u4f5c\u7684\u8fc7\u7a0b\u4e2d\uff0c\u95ee\u9898\u7684\u89c4\u6a21\u9010\u6e10\u7f29\u5c0f\uff0c\u8fd9\u6837\u624d\u80fd\u6784\u5efa\u5b50\u95ee\u9898\u3002\u8bbe\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \u7684\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \uff0c\u6211\u4eec\u5148\u8003\u8651\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u3002
- \u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u76f8\u540c\uff0c\u6211\u4eec\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u76f4\u63a5\u8003\u8651 \\(s[n-2]\\) \u548c \\(t[m-2]\\) \u3002
- \u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u4e0d\u540c\uff0c\u6211\u4eec\u9700\u8981\u5bf9 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\uff08\u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\uff09\uff0c\u4f7f\u5f97\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26\u76f8\u540c\uff0c\u4ece\u800c\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u8003\u8651\u89c4\u6a21\u66f4\u5c0f\u7684\u95ee\u9898\u3002
\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u5728\u5b57\u7b26\u4e32 \\(s\\) \u4e2d\u8fdb\u884c\u7684\u6bcf\u4e00\u8f6e\u51b3\u7b56\uff08\u7f16\u8f91\u64cd\u4f5c\uff09\uff0c\u90fd\u4f1a\u4f7f\u5f97 \\(s\\) \u548c \\(t\\) \u4e2d\u5269\u4f59\u7684\u5f85\u5339\u914d\u5b57\u7b26\u53d1\u751f\u53d8\u5316\u3002\u56e0\u6b64\uff0c\u72b6\u6001\u4e3a\u5f53\u524d\u5728 \\(s\\) \u548c \\(t\\) \u4e2d\u8003\u8651\u7684\u7b2c \\(i\\) \u548c\u7b2c \\(j\\) \u4e2a\u5b57\u7b26\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002
\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\uff1a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002
\u81f3\u6b64\uff0c\u5f97\u5230\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((i+1) \\times (j+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002
\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
\u8003\u8651\u5b50\u95ee\u9898 \\(dp[i, j]\\) \uff0c\u5176\u5bf9\u5e94\u7684\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5c3e\u90e8\u5b57\u7b26\u4e3a \\(s[i-1]\\) \u548c \\(t[j-1]\\) \uff0c\u53ef\u6839\u636e\u4e0d\u540c\u7f16\u8f91\u64cd\u4f5c\u5206\u4e3a\u56fe 14-29 \u6240\u793a\u7684\u4e09\u79cd\u60c5\u51b5\u3002
- \u5728 \\(s[i-1]\\) \u4e4b\u540e\u6dfb\u52a0 \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i, j-1]\\) \u3002
- \u5220\u9664 \\(s[i-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j]\\) \u3002
- \u5c06 \\(s[i-1]\\) \u66ff\u6362\u4e3a \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j-1]\\) \u3002
\u56fe 14-29 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u72b6\u6001\u8f6c\u79fb
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u5f97\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\\(dp[i, j]\\) \u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(dp[i, j-1]\\)\u3001\\(dp[i-1, j]\\)\u3001\\(dp[i-1, j-1]\\) \u4e09\u8005\u4e2d\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u518d\u52a0\u4e0a\u672c\u6b21\u7684\u7f16\u8f91\u6b65\u6570 \\(1\\) \u3002\u5bf9\u5e94\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a
\\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\]\u8bf7\u6ce8\u610f\uff0c\u5f53 \\(s[i-1]\\) \u548c \\(t[j-1]\\) \u76f8\u540c\u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a
\\[ dp[i, j] = dp[i-1, j-1] \\]\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
\u5f53\u4e24\u5b57\u7b26\u4e32\u90fd\u4e3a\u7a7a\u65f6\uff0c\u7f16\u8f91\u6b65\u6570\u4e3a \\(0\\) \uff0c\u5373 \\(dp[0, 0] = 0\\) \u3002\u5f53 \\(s\\) \u4e3a\u7a7a\u4f46 \\(t\\) \u4e0d\u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(t\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u884c \\(dp[0, j] = j\\) \u3002\u5f53 \\(s\\) \u4e0d\u4e3a\u7a7a\u4f46 \\(t\\) \u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(s\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u5217 \\(dp[i, 0] = i\\) \u3002
\u89c2\u5bdf\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u89e3 \\(dp[i, j]\\) \u4f9d\u8d56\u5de6\u65b9\u3001\u4e0a\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u89e3\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002
"},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig edit_distance.py
edit_distance.cppdef edit_distance_dp(s: str, t: str) -> int:\n \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n, m = len(s), len(t)\n dp = [[0] * (m + 1) for _ in range(n + 1)]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for i in range(1, n + 1):\n dp[i][0] = i\n for j in range(1, m + 1):\n dp[0][j] = j\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in range(1, n + 1):\n for j in range(1, m + 1):\n if s[i - 1] == t[j - 1]:\n # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1]\n else:\n # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n return dp[n][m]\n
edit_distance.java/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(string s, string t) {\n int n = s.length(), m = t.length();\n vector<vector<int>> dp(n + 1, vector<int>(m + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.cs/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n int n = s.length(), m = t.length();\n int[][] dp = new int[n + 1][m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s.charAt(i - 1) == t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.go/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDP(string s, string t) {\n int n = s.Length, m = t.Length;\n int[,] dp = new int[n + 1, m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i, 0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0, j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i, j] = dp[i - 1, j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n }\n }\n }\n return dp[n, m];\n}\n
edit_distance.swift/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s string, t string) int {\n n := len(s)\n m := len(t)\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, m+1)\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for i := 1; i <= n; i++ {\n dp[i][0] = i\n }\n for j := 1; j <= m; j++ {\n dp[0][j] = j\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i <= n; i++ {\n for j := 1; j <= m; j++ {\n if s[i-1] == t[j-1] {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i-1][j-1]\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n }\n }\n }\n return dp[n][m]\n}\n
edit_distance.js/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s: String, t: String) -> Int {\n let n = s.utf8CString.count\n let m = t.utf8CString.count\n var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for i in stride(from: 1, through: n, by: 1) {\n dp[i][0] = i\n }\n for j in stride(from: 1, through: m, by: 1) {\n dp[0][j] = j\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in stride(from: 1, through: n, by: 1) {\n for j in stride(from: 1, through: m, by: 1) {\n if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1]\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n }\n }\n }\n return dp[n][m]\n}\n
edit_distance.ts/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s, t) {\n const n = s.length,\n m = t.length;\n const dp = Array.from({ length: n + 1 }, () => new Array(m + 1).fill(0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (let i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (let j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i <= n; i++) {\n for (let j = 1; j <= m; j++) {\n if (s.charAt(i - 1) === t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] =\n Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.dart/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s: string, t: string): number {\n const n = s.length,\n m = t.length;\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: m + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (let i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (let j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i <= n; i++) {\n for (let j = 1; j <= m; j++) {\n if (s.charAt(i - 1) === t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] =\n Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.rs/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n int n = s.length, m = t.length;\n List<List<int>> dp = List.generate(n + 1, (_) => List.filled(m + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n
edit_distance.c/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp(s: &str, t: &str) -> i32 {\n let (n, m) = (s.len(), t.len());\n let mut dp = vec![vec![0; m + 1]; n + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for i in 1..= n {\n dp[i][0] = i as i32;\n }\n for j in 1..m {\n dp[0][j] = j as i32;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in 1..=n {\n for j in 1..=m {\n if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n dp[n][m]\n}\n
edit_distance.zig/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(char *s, char *t, int n, int m) {\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(m + 1, sizeof(int));\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int i = 1; i <= n; i++) {\n dp[i][0] = i;\n }\n for (int j = 1; j <= m; j++) {\n dp[0][j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int j = 1; j <= m; j++) {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n int res = dp[n][m];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n return res;\n}\n// \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\nfn editDistanceDP(comptime s: []const u8, comptime t: []const u8) i32 {\n comptime var n = s.len;\n comptime var m = t.len;\n var dp = [_][m + 1]i32{[_]i32{0} ** (m + 1)} ** (n + 1);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (1..n + 1) |i| {\n dp[i][0] = @intCast(i);\n }\n for (1..m + 1) |j| {\n dp[0][j] = @intCast(j);\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (1..n + 1) |i| {\n for (1..m + 1) |j| {\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[i][j] = dp[i - 1][j - 1];\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[i][j] = @min(@min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n }\n }\n }\n return dp[n][m];\n}\n\u5982\u56fe 14-30 \u6240\u793a\uff0c\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\u4e0e\u80cc\u5305\u95ee\u9898\u975e\u5e38\u7c7b\u4f3c\uff0c\u90fd\u53ef\u4ee5\u770b\u4f5c\u586b\u5199\u4e00\u4e2a\u4e8c\u7ef4\u7f51\u683c\u7684\u8fc7\u7a0b\u3002
<1><2><3><4><5><6><7><8><9><10><11><12><13><14><15>\u56fe 14-30 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
"},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"\u7531\u4e8e \\(dp[i,j]\\) \u662f\u7531\u4e0a\u65b9 \\(dp[i-1, j]\\)\u3001\u5de6\u65b9 \\(dp[i, j-1]\\)\u3001\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \u8f6c\u79fb\u800c\u6765\u7684\uff0c\u800c\u6b63\u5e8f\u904d\u5386\u4f1a\u4e22\u5931\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \uff0c\u5012\u5e8f\u904d\u5386\u65e0\u6cd5\u63d0\u524d\u6784\u5efa \\(dp[i, j-1]\\) \uff0c\u56e0\u6b64\u4e24\u79cd\u904d\u5386\u987a\u5e8f\u90fd\u4e0d\u53ef\u53d6\u3002
\u4e3a\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf
PythonC++JavaC#GoSwiftJSTSDartRustCZig edit_distance.pyleftup\u6765\u6682\u5b58\u5de6\u4e0a\u65b9\u7684\u89e3 \\(dp[i-1, j-1]\\) \uff0c\u4ece\u800c\u53ea\u9700\u8003\u8651\u5de6\u65b9\u548c\u4e0a\u65b9\u7684\u89e3\u3002\u6b64\u65f6\u7684\u60c5\u51b5\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u53ef\u4f7f\u7528\u6b63\u5e8f\u904d\u5386\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
edit_distance.cppdef edit_distance_dp_comp(s: str, t: str) -> int:\n \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n, m = len(s), len(t)\n dp = [0] * (m + 1)\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in range(1, m + 1):\n dp[j] = j\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in range(1, n + 1):\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n leftup = dp[0] # \u6682\u5b58 dp[i-1, j-1]\n dp[0] += 1\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in range(1, m + 1):\n temp = dp[j]\n if s[i - 1] == t[j - 1]:\n # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup\n else:\n # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n leftup = temp # \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n return dp[m]\n
edit_distance.java/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(string s, string t) {\n int n = s.length(), m = t.length();\n vector<int> dp(m + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.cs/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n int n = s.length(), m = t.length();\n int[] dp = new int[m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s.charAt(i - 1) == t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.go/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDPComp(string s, string t) {\n int n = s.Length, m = t.Length;\n int[] dp = new int[m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.swift/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s string, t string) int {\n n := len(s)\n m := len(t)\n dp := make([]int, m+1)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j := 1; j <= m; j++ {\n dp[j] = j\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i := 1; i <= n; i++ {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n leftUp := dp[0] // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j := 1; j <= m; j++ {\n temp := dp[j]\n if s[i-1] == t[j-1] {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftUp\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n }\n leftUp = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m]\n}\n
edit_distance.js/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s: String, t: String) -> Int {\n let n = s.utf8CString.count\n let m = t.utf8CString.count\n var dp = Array(repeating: 0, count: m + 1)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in stride(from: 1, through: m, by: 1) {\n dp[j] = j\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in stride(from: 1, through: n, by: 1) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n var leftup = dp[0] // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in stride(from: 1, through: m, by: 1) {\n let temp = dp[j]\n if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n }\n leftup = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m]\n}\n
edit_distance.ts/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s, t) {\n const n = s.length,\n m = t.length;\n const dp = new Array(m + 1).fill(0);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (let j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (let i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (let j = 1; j <= m; j++) {\n const temp = dp[j];\n if (s.charAt(i - 1) === t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.dart/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s: string, t: string): number {\n const n = s.length,\n m = t.length;\n const dp = new Array(m + 1).fill(0);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (let j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (let i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (let j = 1; j <= m; j++) {\n const temp = dp[j];\n if (s.charAt(i - 1) === t.charAt(j - 1)) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.rs/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n int n = s.length, m = t.length;\n List<int> dp = List.filled(m + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n
edit_distance.c/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp_comp(s: &str, t: &str) -> i32 {\n let (n, m) = (s.len(), t.len());\n let mut dp = vec![0; m + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for j in 1..m {\n dp[j] = j as i32;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for i in 1..=n {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n let mut leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i as i32;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for j in 1..=m {\n let temp = dp[j];\n if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n dp[m]\n}\n
edit_distance.zig/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n int *dp = calloc(m + 1, sizeof(int));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (int j = 1; j <= m; j++) {\n dp[j] = j;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (int i = 1; i <= n; i++) {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = i;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (int j = 1; j <= m; j++) {\n int temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n int res = dp[m];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212","text":"// \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn editDistanceDPComp(comptime s: []const u8, comptime t: []const u8) i32 {\n comptime var n = s.len;\n comptime var m = t.len;\n var dp = [_]i32{0} ** (m + 1);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n for (1..m + 1) |j| {\n dp[j] = @intCast(j);\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n for (1..n + 1) |i| {\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n var leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n dp[0] = @intCast(i);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n for (1..m + 1) |j| {\n var temp = dp[j];\n if (s[i - 1] == t[j - 1]) {\n // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n dp[j] = leftup;\n } else {\n // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n dp[j] = @min(@min(dp[j - 1], dp[j]), leftup) + 1;\n }\n leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n }\n }\n return dp[m];\n}\n\u300c\u52a8\u6001\u89c4\u5212 dynamic programming\u300d\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u7b97\u6cd5\u8303\u5f0f\uff0c\u5b83\u5c06\u4e00\u4e2a\u95ee\u9898\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\uff0c\u4ece\u800c\u5927\u5e45\u63d0\u5347\u65f6\u95f4\u6548\u7387\u3002
\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u4ece\u4e00\u4e2a\u7ecf\u5178\u4f8b\u9898\u5165\u624b\uff0c\u5148\u7ed9\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c2\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u518d\u9010\u6b65\u5bfc\u51fa\u66f4\u9ad8\u6548\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002
\u722c\u697c\u68af
\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f
\u5982\u56fe 14-1 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e00\u4e2a \\(3\\) \u9636\u697c\u68af\uff0c\u5171\u6709 \\(3\\) \u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002
\u56fe 14-1 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf
\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u56de\u6eaf\u6765\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c06\u722c\u697c\u68af\u60f3\u8c61\u4e3a\u4e00\u4e2a\u591a\u8f6e\u9009\u62e9\u7684\u8fc7\u7a0b\uff1a\u4ece\u5730\u9762\u51fa\u53d1\uff0c\u6bcf\u8f6e\u9009\u62e9\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u6bcf\u5f53\u5230\u8fbe\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u65b9\u6848\u6570\u91cf\u52a0 \\(1\\) \uff0c\u5f53\u8d8a\u8fc7\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u5176\u526a\u679d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_backtrack.py
climbing_stairs_backtrack.cppdef backtrack(choices: list[int], state: int, n: int, res: list[int]) -> int:\n \"\"\"\u56de\u6eaf\"\"\"\n # \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if state == n:\n res[0] += 1\n # \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices:\n # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if state + choice > n:\n continue\n # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res)\n # \u56de\u9000\n\ndef climbing_stairs_backtrack(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u56de\u6eaf\"\"\"\n choices = [1, 2] # \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n state = 0 # \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n res = [0] # \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res)\n return res[0]\n
climbing_stairs_backtrack.java/* \u56de\u6eaf */\nvoid backtrack(vector<int> &choices, int state, int n, vector<int> &res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n)\n res[0]++;\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (auto &choice : choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n)\n continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n vector<int> choices = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n vector<int> res = {0}; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res[0];\n}\n
climbing_stairs_backtrack.cs/* \u56de\u6eaf */\nvoid backtrack(List<Integer> choices, int state, int n, List<Integer> res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n)\n res.set(0, res.get(0) + 1);\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (Integer choice : choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n)\n continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n List<Integer> choices = Arrays.asList(1, 2); // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n List<Integer> res = new ArrayList<>();\n res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res.get(0);\n}\n
climbing_stairs_backtrack.go/* \u56de\u6eaf */\nvoid Backtrack(List<int> choices, int state, int n, List<int> res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n)\n res[0]++;\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n foreach (int choice in choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n)\n continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n Backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint ClimbingStairsBacktrack(int n) {\n List<int> choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n List<int> res = [0]; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n Backtrack(choices, state, n, res);\n return res[0];\n}\n
climbing_stairs_backtrack.swift/* \u56de\u6eaf */\nfunc backtrack(choices []int, state, n int, res []int) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if state == n {\n res[0] = res[0] + 1\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for _, choice := range choices {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if state+choice > n {\n continue\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state+choice, n, res)\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n int) int {\n // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n choices := []int{1, 2}\n // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n state := 0\n res := make([]int, 1)\n // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n res[0] = 0\n backtrack(choices, state, n, res)\n return res[0]\n}\n
climbing_stairs_backtrack.js/* \u56de\u6eaf */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if state == n {\n res[0] += 1\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for choice in choices {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if state + choice > n {\n continue\n }\n backtrack(choices: choices, state: state + choice, n: n, res: &res)\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n: Int) -> Int {\n let choices = [1, 2] // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n let state = 0 // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n var res: [Int] = []\n res.append(0) // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices: choices, state: state, n: n, res: &res)\n return res[0]\n}\n
climbing_stairs_backtrack.ts/* \u56de\u6eaf */\nfunction backtrack(choices, state, n, res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state === n) res.set(0, res.get(0) + 1);\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (const choice of choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n) continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n) {\n const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n const res = new Map();\n res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res.get(0);\n}\n
climbing_stairs_backtrack.dart/* \u56de\u6eaf */\nfunction backtrack(\n choices: number[],\n state: number,\n n: number,\n res: Map<0, any>\n): void {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state === n) res.set(0, res.get(0) + 1);\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (const choice of choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n) continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n: number): number {\n const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n const res = new Map();\n res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res.get(0);\n}\n
climbing_stairs_backtrack.rs/* \u56de\u6eaf */\nvoid backtrack(List<int> choices, int state, int n, List<int> res) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n) {\n res[0]++;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int choice in choices) {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n) continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n List<int> choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n List<int> res = [];\n res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(choices, state, n, res);\n return res[0];\n}\n
climbing_stairs_backtrack.c/* \u56de\u6eaf */\nfn backtrack(choices: &[i32], state: i32, n: i32, res: &mut [i32]) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if state == n { res[0] = res[0] + 1; }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for &choice in choices {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if state + choice > n { continue; }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfn climbing_stairs_backtrack(n: usize) -> i32 {\n let choices = vec![ 1, 2 ]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n let state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n let mut res = Vec::new();\n res.push(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(&choices, state, n as i32, &mut res);\n res[0]\n}\n
climbing_stairs_backtrack.zig/* \u56de\u6eaf */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n)\n res[0]++;\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (int i = 0; i < len; i++) {\n int choice = choices[i];\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n)\n continue;\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res, len);\n // \u56de\u9000\n }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n int choices[2] = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n int *res = (int *)malloc(sizeof(int));\n *res = 0; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n int len = sizeof(choices) / sizeof(int);\n backtrack(choices, state, n, res, len);\n int result = *res;\n free(res);\n return result;\n}\n
"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411","title":"14.1.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"// \u56de\u6eaf\nfn backtrack(choices: []i32, state: i32, n: i32, res: std.ArrayList(i32)) void {\n // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n if (state == n) {\n res.items[0] = res.items[0] + 1;\n }\n // \u904d\u5386\u6240\u6709\u9009\u62e9\n for (choices) |choice| {\n // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n if (state + choice > n) {\n continue;\n }\n // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n backtrack(choices, state + choice, n, res);\n // \u56de\u9000\n }\n}\n\n// \u722c\u697c\u68af\uff1a\u56de\u6eaf\nfn climbingStairsBacktrack(n: usize) !i32 {\n var choices = [_]i32{ 1, 2 }; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n var state: i32 = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n var res = std.ArrayList(i32).init(std.heap.page_allocator);\n defer res.deinit();\n try res.append(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n backtrack(&choices, state, @intCast(n), res);\n return res.items[0];\n}\n\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u5e76\u4e0d\u663e\u5f0f\u5730\u5bf9\u95ee\u9898\u8fdb\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c06\u6c42\u89e3\u95ee\u9898\u770b\u4f5c\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\uff0c\u901a\u8fc7\u8bd5\u63a2\u548c\u526a\u679d\uff0c\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002
\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u4ece\u95ee\u9898\u5206\u89e3\u7684\u89d2\u5ea6\u5206\u6790\u8fd9\u9053\u9898\u3002\u8bbe\u722c\u5230\u7b2c \\(i\\) \u9636\u5171\u6709 \\(dp[i]\\) \u79cd\u65b9\u6848\uff0c\u90a3\u4e48 \\(dp[i]\\) \u5c31\u662f\u539f\u95ee\u9898\uff0c\u5176\u5b50\u95ee\u9898\u5305\u62ec\uff1a
\\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\]\u7531\u4e8e\u6bcf\u8f6e\u53ea\u80fd\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u56e0\u6b64\u5f53\u6211\u4eec\u7ad9\u5728\u7b2c \\(i\\) \u9636\u697c\u68af\u4e0a\u65f6\uff0c\u4e0a\u4e00\u8f6e\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u4e0a\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ea\u80fd\u4ece\u7b2c \\(i -1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u8fc8\u5411\u7b2c \\(i\\) \u9636\u3002
\u7531\u6b64\u4fbf\u53ef\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u63a8\u8bba\uff1a\u722c\u5230\u7b2c \\(i - 1\\) \u9636\u7684\u65b9\u6848\u6570\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u9636\u7684\u65b9\u6848\u6570\u5c31\u7b49\u4e8e\u722c\u5230\u7b2c \\(i\\) \u9636\u7684\u65b9\u6848\u6570\u3002\u516c\u5f0f\u5982\u4e0b\uff1a
\\[ dp[i] = dp[i-1] + dp[i-2] \\]\u8fd9\u610f\u5473\u7740\u5728\u722c\u697c\u68af\u95ee\u9898\u4e2d\uff0c\u5404\u4e2a\u5b50\u95ee\u9898\u4e4b\u95f4\u5b58\u5728\u9012\u63a8\u5173\u7cfb\uff0c\u539f\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u7684\u89e3\u6784\u5efa\u5f97\u6765\u3002\u56fe 14-2 \u5c55\u793a\u4e86\u8be5\u9012\u63a8\u5173\u7cfb\u3002
\u56fe 14-2 \u00a0 \u65b9\u6848\u6570\u91cf\u9012\u63a8\u5173\u7cfb
\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9012\u63a8\u516c\u5f0f\u5f97\u5230\u66b4\u529b\u641c\u7d22\u89e3\u6cd5\u3002\u4ee5 \\(dp[n]\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u9012\u5f52\u5730\u5c06\u4e00\u4e2a\u8f83\u5927\u95ee\u9898\u62c6\u89e3\u4e3a\u4e24\u4e2a\u8f83\u5c0f\u95ee\u9898\u7684\u548c\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898 \\(dp[1]\\) \u548c \\(dp[2]\\) \u65f6\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373 \\(dp[1] = 1\\)\u3001\\(dp[2] = 2\\) \uff0c\u8868\u793a\u722c\u5230\u7b2c \\(1\\)\u3001\\(2\\) \u9636\u5206\u522b\u6709 \\(1\\)\u3001\\(2\\) \u79cd\u65b9\u6848\u3002
\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5b83\u548c\u6807\u51c6\u56de\u6eaf\u4ee3\u7801\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f46\u66f4\u52a0\u7b80\u6d01\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dfs.py
climbing_stairs_dfs.cppdef dfs(i: int) -> int:\n \"\"\"\u641c\u7d22\"\"\"\n # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 or i == 2:\n return i\n # dp[i] = dp[i-1] + dp[i-2]\n count = dfs(i - 1) + dfs(i - 2)\n return count\n\ndef climbing_stairs_dfs(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u641c\u7d22\"\"\"\n return dfs(n)\n
climbing_stairs_dfs.java/* \u641c\u7d22 */\nint dfs(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n return dfs(n);\n}\n
climbing_stairs_dfs.cs/* \u641c\u7d22 */\nint dfs(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n return dfs(n);\n}\n
climbing_stairs_dfs.go/* \u641c\u7d22 */\nint DFS(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = DFS(i - 1) + DFS(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint ClimbingStairsDFS(int n) {\n return DFS(n);\n}\n
climbing_stairs_dfs.swift/* \u641c\u7d22 */\nfunc dfs(i int) int {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 {\n return i\n }\n // dp[i] = dp[i-1] + dp[i-2]\n count := dfs(i-1) + dfs(i-2)\n return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n int) int {\n return dfs(n)\n}\n
climbing_stairs_dfs.js/* \u641c\u7d22 */\nfunc dfs(i: Int) -> Int {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 {\n return i\n }\n // dp[i] = dp[i-1] + dp[i-2]\n let count = dfs(i: i - 1) + dfs(i: i - 2)\n return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n: Int) -> Int {\n dfs(i: n)\n}\n
climbing_stairs_dfs.ts/* \u641c\u7d22 */\nfunction dfs(i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i === 1 || i === 2) return i;\n // dp[i] = dp[i-1] + dp[i-2]\n const count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n) {\n return dfs(n);\n}\n
climbing_stairs_dfs.dart/* \u641c\u7d22 */\nfunction dfs(i: number): number {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i === 1 || i === 2) return i;\n // dp[i] = dp[i-1] + dp[i-2]\n const count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n: number): number {\n return dfs(n);\n}\n
climbing_stairs_dfs.rs/* \u641c\u7d22 */\nint dfs(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2) return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n return dfs(n);\n}\n
climbing_stairs_dfs.c/* \u641c\u7d22 */\nfn dfs(i: usize) -> i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 { return i as i32; }\n // dp[i] = dp[i-1] + dp[i-2]\n let count = dfs(i - 1) + dfs(i - 2);\n count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfn climbing_stairs_dfs(n: usize) -> i32 {\n dfs(n) \n}\n
climbing_stairs_dfs.zig/* \u641c\u7d22 */\nint dfs(int i) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n return dfs(n);\n}\n// \u641c\u7d22\nfn dfs(i: usize) i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 or i == 2) {\n return @intCast(i);\n }\n // dp[i] = dp[i-1] + dp[i-2]\n var count = dfs(i - 1) + dfs(i - 2);\n return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u641c\u7d22\nfn climbingStairsDFS(comptime n: usize) i32 {\n return dfs(n);\n}\n\u56fe 14-3 \u5c55\u793a\u4e86\u66b4\u529b\u641c\u7d22\u5f62\u6210\u7684\u9012\u5f52\u6811\u3002\u5bf9\u4e8e\u95ee\u9898 \\(dp[n]\\) \uff0c\u5176\u9012\u5f52\u6811\u7684\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002\u6307\u6570\u9636\u5c5e\u4e8e\u7206\u70b8\u5f0f\u589e\u957f\uff0c\u5982\u679c\u6211\u4eec\u8f93\u5165\u4e00\u4e2a\u6bd4\u8f83\u5927\u7684 \\(n\\) \uff0c\u5219\u4f1a\u9677\u5165\u6f2b\u957f\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002
\u56fe 14-3 \u00a0 \u722c\u697c\u68af\u5bf9\u5e94\u9012\u5f52\u6811
\u89c2\u5bdf\u56fe 14-3 \uff0c\u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u201c\u91cd\u53e0\u5b50\u95ee\u9898\u201d\u5bfc\u81f4\u7684\u3002\u4f8b\u5982 \\(dp[9]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\\(dp[8]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u4e24\u8005\u90fd\u5305\u542b\u5b50\u95ee\u9898 \\(dp[7]\\) \u3002
\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5b50\u95ee\u9898\u4e2d\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5b50\u5b50\u5b59\u5b59\u65e0\u7a77\u5c3d\u4e5f\u3002\u7edd\u5927\u90e8\u5206\u8ba1\u7b97\u8d44\u6e90\u90fd\u6d6a\u8d39\u5728\u8fd9\u4e9b\u91cd\u53e0\u7684\u95ee\u9898\u4e0a\u3002
"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412","title":"14.1.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"\u4e3a\u4e86\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u6240\u6709\u7684\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u6570\u7ec4
mem\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u526a\u679d\u3002- \u5f53\u9996\u6b21\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u5c06\u5176\u8bb0\u5f55\u81f3
mem[i]\uff0c\u4ee5\u4fbf\u4e4b\u540e\u4f7f\u7528\u3002 - \u5f53\u518d\u6b21\u9700\u8981\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u4fbf\u53ef\u76f4\u63a5\u4ece
mem[i]\u4e2d\u83b7\u53d6\u7ed3\u679c\uff0c\u4ece\u800c\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u8be5\u5b50\u95ee\u9898\u3002
\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dfs_mem.py
climbing_stairs_dfs_mem.cppdef dfs(i: int, mem: list[int]) -> int:\n \"\"\"\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 or i == 2:\n return i\n # \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if mem[i] != -1:\n return mem[i]\n # dp[i] = dp[i-1] + dp[i-2]\n count = dfs(i - 1, mem) + dfs(i - 2, mem)\n # \u8bb0\u5f55 dp[i]\n mem[i] = count\n return count\n\ndef climbing_stairs_dfs_mem(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n # mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n mem = [-1] * (n + 1)\n return dfs(n, mem)\n
climbing_stairs_dfs_mem.java/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, vector<int> &mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1)\n return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n vector<int> mem(n + 1, -1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.cs/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1)\n return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n int[] mem = new int[n + 1];\n Arrays.fill(mem, -1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.go/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint DFS(int i, int[] mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1)\n return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint ClimbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n int[] mem = new int[n + 1];\n Array.Fill(mem, -1);\n return DFS(n, mem);\n}\n
climbing_stairs_dfs_mem.swift/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfsMem(i int, mem []int) int {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 {\n return i\n }\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if mem[i] != -1 {\n return mem[i]\n }\n // dp[i] = dp[i-1] + dp[i-2]\n count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n // \u8bb0\u5f55 dp[i]\n mem[i] = count\n return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n int) int {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n mem := make([]int, n+1)\n for i := range mem {\n mem[i] = -1\n }\n return dfsMem(n, mem)\n}\n
climbing_stairs_dfs_mem.js/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfs(i: Int, mem: inout [Int]) -> Int {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 {\n return i\n }\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if mem[i] != -1 {\n return mem[i]\n }\n // dp[i] = dp[i-1] + dp[i-2]\n let count = dfs(i: i - 1, mem: &mem) + dfs(i: i - 2, mem: &mem)\n // \u8bb0\u5f55 dp[i]\n mem[i] = count\n return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n: Int) -> Int {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n var mem = Array(repeating: -1, count: n + 1)\n return dfs(i: n, mem: &mem)\n}\n
climbing_stairs_dfs_mem.ts/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i, mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i === 1 || i === 2) return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1) return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n const mem = new Array(n + 1).fill(-1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.dart/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i: number, mem: number[]): number {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i === 1 || i === 2) return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1) return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n: number): number {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n const mem = new Array(n + 1).fill(-1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.rs/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, List<int> mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2) return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1) return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n List<int> mem = List.filled(n + 1, -1);\n return dfs(n, mem);\n}\n
climbing_stairs_dfs_mem.c/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn dfs(i: usize, mem: &mut [i32]) -> i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if i == 1 || i == 2 { return i as i32; }\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if mem[i] != -1 { return mem[i]; }\n // dp[i] = dp[i-1] + dp[i-2]\n let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn climbing_stairs_dfs_mem(n: usize) -> i32 {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n let mut mem = vec![-1; n + 1];\n dfs(n, &mut mem)\n}\n
climbing_stairs_dfs_mem.zig/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int *mem) {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 || i == 2)\n return i;\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1)\n return mem[i];\n // dp[i] = dp[i-1] + dp[i-2]\n int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n int *mem = (int *)malloc((n + 1) * sizeof(int));\n for (int i = 0; i <= n; i++) {\n mem[i] = -1;\n }\n int result = dfs(n, mem);\n free(mem);\n return result;\n}\n// \u8bb0\u5fc6\u5316\u641c\u7d22\nfn dfs(i: usize, mem: []i32) i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (i == 1 or i == 2) {\n return @intCast(i);\n }\n // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n if (mem[i] != -1) {\n return mem[i];\n }\n // dp[i] = dp[i-1] + dp[i-2]\n var count = dfs(i - 1, mem) + dfs(i - 2, mem);\n // \u8bb0\u5f55 dp[i]\n mem[i] = count;\n return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn climbingStairsDFSMem(comptime n: usize) i32 {\n // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n var mem = [_]i32{ -1 } ** (n + 1);\n return dfs(n, &mem);\n}\n\u89c2\u5bdf\u56fe 14-4 \uff0c\u7ecf\u8fc7\u8bb0\u5fc6\u5316\u5904\u7406\u540e\uff0c\u6240\u6709\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u8fd9\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u98de\u8dc3\u3002
\u56fe 14-4 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u5bf9\u5e94\u9012\u5f52\u6811
"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413","title":"14.1.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"\u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u201c\u4ece\u9876\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u4eec\u4ece\u539f\u95ee\u9898\uff08\u6839\u8282\u70b9\uff09\u5f00\u59cb\uff0c\u9012\u5f52\u5730\u5c06\u8f83\u5927\u5b50\u95ee\u9898\u5206\u89e3\u4e3a\u8f83\u5c0f\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u53f6\u8282\u70b9\uff09\u3002\u4e4b\u540e\uff0c\u901a\u8fc7\u56de\u6eaf\u9010\u5c42\u6536\u96c6\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002
\u4e0e\u4e4b\u76f8\u53cd\uff0c\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u201c\u4ece\u5e95\u81f3\u9876\u201d\u7684\u65b9\u6cd5\uff1a\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u5f00\u59cb\uff0c\u8fed\u4ee3\u5730\u6784\u5efa\u66f4\u5927\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002
\u7531\u4e8e\u52a8\u6001\u89c4\u5212\u4e0d\u5305\u542b\u56de\u6eaf\u8fc7\u7a0b\uff0c\u56e0\u6b64\u53ea\u9700\u4f7f\u7528\u5faa\u73af\u8fed\u4ee3\u5b9e\u73b0\uff0c\u65e0\u987b\u4f7f\u7528\u9012\u5f52\u3002\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u4e00\u4e2a\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dp.pydp\u6765\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5b83\u8d77\u5230\u4e86\u4e0e\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u6570\u7ec4mem\u76f8\u540c\u7684\u8bb0\u5f55\u4f5c\u7528\uff1a
climbing_stairs_dp.cppdef climbing_stairs_dp(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n if n == 1 or n == 2:\n return n\n # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp = [0] * (n + 1)\n # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1], dp[2] = 1, 2\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in range(3, n + 1):\n dp[i] = dp[i - 1] + dp[i - 2]\n return dp[n]\n
climbing_stairs_dp.java/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n if (n == 1 || n == 2)\n return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n vector<int> dp(n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.cs/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n if (n == 1 || n == 2)\n return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[] dp = new int[n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.go/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDP(int n) {\n if (n == 1 || n == 2)\n return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int[] dp = new int[n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.swift/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n int) int {\n if n == 1 || n == 2 {\n return n\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n dp := make([]int, n+1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1\n dp[2] = 2\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n dp[i] = dp[i-1] + dp[i-2]\n }\n return dp[n]\n}\n
climbing_stairs_dp.js/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n: Int) -> Int {\n if n == 1 || n == 2 {\n return n\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = Array(repeating: 0, count: n + 1)\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1\n dp[2] = 2\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in stride(from: 3, through: n, by: 1) {\n dp[i] = dp[i - 1] + dp[i - 2]\n }\n return dp[n]\n}\n
climbing_stairs_dp.ts/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n) {\n if (n === 1 || n === 2) return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = new Array(n + 1).fill(-1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.dart/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n: number): number {\n if (n === 1 || n === 2) return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n const dp = new Array(n + 1).fill(-1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (let i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.rs/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n if (n == 1 || n == 2) return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n List<int> dp = List.filled(n + 1, 0);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n
climbing_stairs_dp.c/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp(n: usize) -> i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if n == 1 || n == 2 { return n as i32; }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n let mut dp = vec![-1; n + 1];\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i in 3..=n {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n dp[n]\n}\n
climbing_stairs_dp.zig/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n if (n == 1 || n == 2)\n return n;\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n int *dp = (int *)malloc((n + 1) * sizeof(int));\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (int i = 3; i <= n; i++) {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n int result = dp[n];\n free(dp);\n return result;\n}\n// \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsDP(comptime n: usize) i32 {\n // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n if (n == 1 or n == 2) {\n return @intCast(n);\n }\n // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n var dp = [_]i32{-1} ** (n + 1);\n // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n dp[1] = 1;\n dp[2] = 2;\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for (3..n + 1) |i| {\n dp[i] = dp[i - 1] + dp[i - 2];\n }\n return dp[n];\n}\n\u56fe 14-5 \u6a21\u62df\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u6267\u884c\u8fc7\u7a0b\u3002
\u56fe 14-5 \u00a0 \u722c\u697c\u68af\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
\u4e0e\u56de\u6eaf\u7b97\u6cd5\u4e00\u6837\uff0c\u52a8\u6001\u89c4\u5212\u4e5f\u4f7f\u7528\u201c\u72b6\u6001\u201d\u6982\u5ff5\u6765\u8868\u793a\u95ee\u9898\u6c42\u89e3\u7684\u7279\u5b9a\u9636\u6bb5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\u4ee5\u53ca\u76f8\u5e94\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u4f8b\u5982\uff0c\u722c\u697c\u68af\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570 \\(i\\) \u3002
\u6839\u636e\u4ee5\u4e0a\u5185\u5bb9\uff0c\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u52a8\u6001\u89c4\u5212\u7684\u5e38\u7528\u672f\u8bed\u3002
- \u5c06\u6570\u7ec4
dp\u79f0\u4e3a\u300c\\(dp\\) \u8868\u300d\uff0c\\(dp[i]\\) \u8868\u793a\u72b6\u6001 \\(i\\) \u5bf9\u5e94\u5b50\u95ee\u9898\u7684\u89e3\u3002 - \u5c06\u6700\u5c0f\u5b50\u95ee\u9898\u5bf9\u5e94\u7684\u72b6\u6001\uff08\u7b2c \\(1\\) \u9636\u548c\u7b2c \\(2\\) \u9636\u697c\u68af\uff09\u79f0\u4e3a\u300c\u521d\u59cb\u72b6\u6001\u300d\u3002
- \u5c06\u9012\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u79f0\u4e3a\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\u3002
\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u7531\u4e8e \\(dp[i]\\) \u53ea\u4e0e \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u987b\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dp.pydp\u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u800c\u53ea\u9700\u4e24\u4e2a\u53d8\u91cf\u6eda\u52a8\u524d\u8fdb\u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
climbing_stairs_dp.cppdef climbing_stairs_dp_comp(n: int) -> int:\n \"\"\"\u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n if n == 1 or n == 2:\n return n\n a, b = 1, 2\n for _ in range(3, n + 1):\n a, b = b, a + b\n return b\n
climbing_stairs_dp.java/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n if (n == 1 || n == 2)\n return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.cs/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n if (n == 1 || n == 2)\n return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.go/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDPComp(int n) {\n if (n == 1 || n == 2)\n return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.swift/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n int) int {\n if n == 1 || n == 2 {\n return n\n }\n a, b := 1, 2\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n for i := 3; i <= n; i++ {\n a, b = b, a+b\n }\n return b\n}\n
climbing_stairs_dp.js/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n: Int) -> Int {\n if n == 1 || n == 2 {\n return n\n }\n var a = 1\n var b = 2\n for _ in stride(from: 3, through: n, by: 1) {\n (a, b) = (b, a + b)\n }\n return b\n}\n
climbing_stairs_dp.ts/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n) {\n if (n === 1 || n === 2) return n;\n let a = 1,\n b = 2;\n for (let i = 3; i <= n; i++) {\n const tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.dart/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n: number): number {\n if (n === 1 || n === 2) return n;\n let a = 1,\n b = 2;\n for (let i = 3; i <= n; i++) {\n const tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.rs/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n if (n == 1 || n == 2) return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n
climbing_stairs_dp.c/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp_comp(n: usize) -> i32 {\n if n == 1 || n == 2 { return n as i32; }\n let (mut a, mut b) = (1, 2);\n for _ in 3..=n {\n let tmp = b;\n b = a + b;\n a = tmp;\n }\n b\n}\n
climbing_stairs_dp.zig/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n if (n == 1 || n == 2)\n return n;\n int a = 1, b = 2;\n for (int i = 3; i <= n; i++) {\n int tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n// \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn climbingStairsDPComp(comptime n: usize) i32 {\n if (n == 1 or n == 2) {\n return @intCast(n);\n }\n var a: i32 = 1;\n var b: i32 = 2;\n for (3..n + 1) |_| {\n var tmp = b;\n b = a + b;\n a = tmp;\n }\n return b;\n}\n\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u7531\u4e8e\u7701\u53bb\u4e86\u6570\u7ec4
dp\u5360\u7528\u7684\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002\u5728\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e2d\uff0c\u5f53\u524d\u72b6\u6001\u5f80\u5f80\u4ec5\u4e0e\u524d\u9762\u6709\u9650\u4e2a\u72b6\u6001\u6709\u5173\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72b6\u6001\uff0c\u901a\u8fc7\u201c\u964d\u7ef4\u201d\u6765\u8282\u7701\u5185\u5b58\u7a7a\u95f4\u3002\u8fd9\u79cd\u7a7a\u95f4\u4f18\u5316\u6280\u5de7\u88ab\u79f0\u4e3a\u201c\u6eda\u52a8\u53d8\u91cf\u201d\u6216\u201c\u6eda\u52a8\u6570\u7ec4\u201d\u3002
"},{"location":"chapter_dynamic_programming/knapsack_problem/","title":"14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898","text":"\u80cc\u5305\u95ee\u9898\u662f\u4e00\u4e2a\u975e\u5e38\u597d\u7684\u52a8\u6001\u89c4\u5212\u5165\u95e8\u9898\u76ee\uff0c\u662f\u52a8\u6001\u89c4\u5212\u4e2d\u6700\u5e38\u89c1\u7684\u95ee\u9898\u5f62\u5f0f\u3002\u5176\u5177\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982 0-1 \u80cc\u5305\u95ee\u9898\u3001\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u3001\u591a\u91cd\u80cc\u5305\u95ee\u9898\u7b49\u3002
\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6765\u6c42\u89e3\u6700\u5e38\u89c1\u7684 0-1 \u80cc\u5305\u95ee\u9898\u3002
Question
\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002
\u89c2\u5bdf\u56fe 14-17 \uff0c\u7531\u4e8e\u7269\u54c1\u7f16\u53f7 \\(i\\) \u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u6570\u7ec4\u7d22\u5f15\u4ece \\(0\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u56e0\u6b64\u7269\u54c1 \\(i\\) \u5bf9\u5e94\u91cd\u91cf \\(wgt[i-1]\\) \u548c\u4ef7\u503c \\(val[i-1]\\) \u3002
\u56fe 14-17 \u00a0 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u6570\u636e
\u6211\u4eec\u53ef\u4ee5\u5c06 0-1 \u80cc\u5305\u95ee\u9898\u770b\u4f5c\u4e00\u4e2a\u7531 \\(n\\) \u8f6e\u51b3\u7b56\u7ec4\u6210\u7684\u8fc7\u7a0b\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u4f53\u90fd\u6709\u4e0d\u653e\u5165\u548c\u653e\u5165\u4e24\u79cd\u51b3\u7b56\uff0c\u56e0\u6b64\u8be5\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\u3002
\u8be5\u95ee\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u201c\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u201d\uff0c\u56e0\u6b64\u8f83\u5927\u6982\u7387\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\u3002
\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868
\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u54c1\u6765\u8bf4\uff0c\u4e0d\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff1b\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u51cf\u5c0f\u3002\u7531\u6b64\u53ef\u5f97\u72b6\u6001\u5b9a\u4e49\uff1a\u5f53\u524d\u7269\u54c1\u7f16\u53f7 \\(i\\) \u548c\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u8bb0\u4e3a \\([i, c]\\) \u3002
\u72b6\u6001 \\([i, c]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\uff0c\u8bb0\u4e3a \\(dp[i, c]\\) \u3002
\u5f85\u6c42\u89e3\u7684\u662f \\(dp[n, cap]\\) \uff0c\u56e0\u6b64\u9700\u8981\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((n+1) \\times (cap+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002
\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
\u5f53\u6211\u4eec\u505a\u51fa\u7269\u54c1 \\(i\\) \u7684\u51b3\u7b56\u540e\uff0c\u5269\u4f59\u7684\u662f\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u7684\u51b3\u7b56\uff0c\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002
- \u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c]\\) \u3002
- \u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u51cf\u5c11 \\(wgt[i-1]\\) \uff0c\u4ef7\u503c\u589e\u52a0 \\(val[i-1]\\) \uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c-wgt[i-1]]\\) \u3002
\u4e0a\u8ff0\u5206\u6790\u5411\u6211\u4eec\u63ed\u793a\u4e86\u672c\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6700\u5927\u4ef7\u503c \\(dp[i, c]\\) \u7b49\u4e8e\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \u548c\u653e\u5165\u7269\u54c1 \\(i\\) \u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\u3002\u7531\u6b64\u53ef\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a
\\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\]\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf \\(wgt[i - 1]\\) \u8d85\u51fa\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002
\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
\u5f53\u65e0\u7269\u54c1\u6216\u65e0\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u65f6\u6700\u5927\u4ef7\u503c\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217 \\(dp[i, 0]\\) \u548c\u9996\u884c \\(dp[0, c]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002
\u5f53\u524d\u72b6\u6001 \\([i, c]\\) \u4ece\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c]\\) \u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c-wgt[i-1]]\\) \u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002
\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u6309\u987a\u5e8f\u5b9e\u73b0\u66b4\u529b\u641c\u7d22\u3001\u8bb0\u5fc6\u5316\u641c\u7d22\u3001\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002
"},{"location":"chapter_dynamic_programming/knapsack_problem/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"\u641c\u7d22\u4ee3\u7801\u5305\u542b\u4ee5\u4e0b\u8981\u7d20\u3002
- \u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, c]\\) \u3002
- \u8fd4\u56de\u503c\uff1a\u5b50\u95ee\u9898\u7684\u89e3 \\(dp[i, c]\\) \u3002
- \u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53\u7269\u54c1\u7f16\u53f7\u8d8a\u754c \\(i = 0\\) \u6216\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\u4e3a \\(0\\) \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5e76\u8fd4\u56de\u4ef7\u503c \\(0\\) \u3002
- \u526a\u679d\uff1a\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf\u8d85\u51fa\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002
knapsack.cppdef knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -> int:\n \"\"\"0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 or c == 0:\n return 0\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c:\n return knapsack_dfs(wgt, val, i - 1, c)\n # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n no = knapsack_dfs(wgt, val, i - 1, c)\n yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n # \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return max(no, yes)\n
knapsack.java/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(vector<int> &wgt, vector<int> &val, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFS(wgt, val, i - 1, c);\n int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return max(no, yes);\n}\n
knapsack.cs/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFS(wgt, val, i - 1, c);\n int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return Math.max(no, yes);\n}\n
knapsack.go/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (weight[i - 1] > c) {\n return KnapsackDFS(weight, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = KnapsackDFS(weight, val, i - 1, c);\n int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return Math.Max(no, yes);\n}\n
knapsack.swift/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i-1] > c {\n return knapsackDFS(wgt, val, i-1, c)\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n no := knapsackDFS(wgt, val, i-1, c)\n yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return int(math.Max(float64(no), float64(yes)))\n}\n
knapsack.js/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -> Int {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c {\n return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return max(no, yes)\n}\n
knapsack.ts/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(wgt, val, i, c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i === 0 || c === 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n const no = knapsackDFS(wgt, val, i - 1, c);\n const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return Math.max(no, yes);\n}\n
knapsack.dart/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(\n wgt: Array<number>,\n val: Array<number>,\n i: number,\n c: number\n): number {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i === 0 || c === 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n const no = knapsackDFS(wgt, val, i - 1, c);\n const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return Math.max(no, yes);\n}\n
knapsack.rs/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(List<int> wgt, List<int> val, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFS(wgt, val, i - 1, c);\n int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return max(no, yes);\n}\n
knapsack.c/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfn knapsack_dfs(wgt: &[i32], val: &[i32], i: usize, c: usize) -> i32 {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c as i32 {\n return knapsack_dfs(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n let no = knapsack_dfs(wgt, val, i - 1, c);\n let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n std::cmp::max(no, yes)\n}\n
knapsack.zig/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFS(wgt, val, i - 1, c);\n int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return myMax(no, yes);\n}\n// 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\nfn knapsackDFS(wgt: []i32, val: []i32, i: usize, c: usize) i32 {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 or c == 0) {\n return 0;\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFS(wgt, val, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n var no = knapsackDFS(wgt, val, i - 1, c);\n var yes = knapsackDFS(wgt, val, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n return @max(no, yes);\n}\n\u5982\u56fe 14-18 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u7269\u54c1\u90fd\u4f1a\u4ea7\u751f\u4e0d\u9009\u548c\u9009\u4e24\u6761\u641c\u7d22\u5206\u652f\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002
\u89c2\u5bdf\u9012\u5f52\u6811\uff0c\u5bb9\u6613\u53d1\u73b0\u5176\u4e2d\u5b58\u5728\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u4f8b\u5982 \\(dp[1, 10]\\) \u7b49\u3002\u800c\u5f53\u7269\u54c1\u8f83\u591a\u3001\u80cc\u5305\u5bb9\u91cf\u8f83\u5927\uff0c\u5c24\u5176\u662f\u76f8\u540c\u91cd\u91cf\u7684\u7269\u54c1\u8f83\u591a\u65f6\uff0c\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u6570\u91cf\u5c06\u4f1a\u5927\u5e45\u589e\u591a\u3002
\u56fe 14-18 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811
"},{"location":"chapter_dynamic_programming/knapsack_problem/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"\u4e3a\u4e86\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\uff0c\u6211\u4eec\u501f\u52a9\u8bb0\u5fc6\u5217\u8868
mem\u6765\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5176\u4e2dmem[i][c]\u5bf9\u5e94 \\(dp[i, c]\\) \u3002\u5f15\u5165\u8bb0\u5fc6\u5316\u4e4b\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5b50\u95ee\u9898\u6570\u91cf\uff0c\u4e5f\u5c31\u662f \\(O(n \\times cap)\\) \u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py
knapsack.cppdef knapsack_dfs_mem(\n wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -> int:\n \"\"\"0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 or c == 0:\n return 0\n # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][c] != -1:\n return mem[i][c]\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c:\n return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n # \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = max(no, yes)\n return mem[i][c]\n
knapsack.java/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(vector<int> &wgt, vector<int> &val, vector<vector<int>> &mem, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = max(no, yes);\n return mem[i][c];\n}\n
knapsack.cs/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = Math.max(no, yes);\n return mem[i][c];\n}\n
knapsack.go/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (weight[i - 1] > c) {\n return KnapsackDFSMem(weight, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = Math.Max(no, yes);\n return mem[i][c];\n}\n
knapsack.swift/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][c] != -1 {\n return mem[i][c]\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i-1] > c {\n return knapsackDFSMem(wgt, val, mem, i-1, c)\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n no := knapsackDFSMem(wgt, val, mem, i-1, c)\n yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = int(math.Max(float64(no), float64(yes)))\n return mem[i][c]\n}\n
knapsack.js/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -> Int {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][c] != -1 {\n return mem[i][c]\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c {\n return knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: i - 1, c: c)\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n let no = knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: i - 1, c: c)\n let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = max(no, yes)\n return mem[i][c]\n}\n
knapsack.ts/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i === 0 || c === 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] !== -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n const yes =\n knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = Math.max(no, yes);\n return mem[i][c];\n}\n
knapsack.dart/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(\n wgt: Array<number>,\n val: Array<number>,\n mem: Array<Array<number>>,\n i: number,\n c: number\n): number {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i === 0 || c === 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] !== -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n const yes =\n knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = Math.max(no, yes);\n return mem[i][c];\n}\n
knapsack.rs/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(\n List<int> wgt,\n List<int> val,\n List<List<int>> mem,\n int i,\n int c,\n) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = max(no, yes);\n return mem[i][c];\n}\n
knapsack.c/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn knapsack_dfs_mem(wgt: &[i32], val: &[i32], mem: &mut Vec<Vec<i32>>, i: usize, c: usize) -> i32 {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if i == 0 || c == 0 {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if mem[i][c] != -1 {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if wgt[i - 1] > c as i32 {\n return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = std::cmp::max(no, yes);\n mem[i][c]\n}\n
knapsack.zig/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 || c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = myMax(no, yes);\n return mem[i][c];\n}\n// 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn knapsackDFSMem(wgt: []i32, val: []i32, mem: anytype, i: usize, c: usize) i32 {\n // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n if (i == 0 or c == 0) {\n return 0;\n }\n // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (mem[i][c] != -1) {\n return mem[i][c];\n }\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n if (wgt[i - 1] > c) {\n return knapsackDFSMem(wgt, val, mem, i - 1, c);\n }\n // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n var no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n var yes = knapsackDFSMem(wgt, val, mem, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n mem[i][c] = @max(no, yes);\n return mem[i][c];\n}\n\u56fe 14-19 \u5c55\u793a\u4e86\u5728\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u88ab\u526a\u6389\u7684\u641c\u7d22\u5206\u652f\u3002
\u56fe 14-19 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811
"},{"location":"chapter_dynamic_programming/knapsack_problem/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"\u52a8\u6001\u89c4\u5212\u5b9e\u8d28\u4e0a\u5c31\u662f\u5728\u72b6\u6001\u8f6c\u79fb\u4e2d\u586b\u5145 \\(dp\\) \u8868\u7684\u8fc7\u7a0b\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py
knapsack.cppdef knapsack_dp(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(wgt)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * (cap + 1) for _ in range(n + 1)]\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n for c in range(1, cap + 1):\n if wgt[i - 1] > c:\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c]\n else:\n # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n return dp[n][cap]\n
knapsack.java/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(vector<int> &wgt, vector<int> &val, int cap) {\n int n = wgt.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n + 1, vector<int>(cap + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.cs/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n + 1][cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.go/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n int n = weight.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n + 1, cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (weight[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i, c] = dp[i - 1, c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n }\n }\n }\n return dp[n, cap];\n}\n
knapsack.swift/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt, val []int, cap int) int {\n n := len(wgt)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, cap+1)\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n for c := 1; c <= cap; c++ {\n if wgt[i-1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i-1][c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n }\n }\n }\n return dp[n][cap]\n}\n
knapsack.js/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -> Int {\n let n = wgt.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for c in stride(from: 1, through: cap, by: 1) {\n if wgt[i - 1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n }\n }\n }\n return dp[n][cap]\n}\n
knapsack.ts/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(wgt, val, cap) {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array(n + 1)\n .fill(0)\n .map(() => Array(cap + 1).fill(0));\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(\n dp[i - 1][c],\n dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n );\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.dart/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(\n wgt: Array<number>,\n val: Array<number>,\n cap: number\n): number {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: cap + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(\n dp[i - 1][c],\n dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n );\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.rs/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(List<int> wgt, List<int> val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(cap + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
knapsack.c/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {\n let n = wgt.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; cap + 1]; n + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for c in 1..=cap {\n if wgt[i - 1] > c as i32 {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1]);\n }\n }\n }\n dp[n][cap]\n}\n
knapsack.zig/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n int n = wgtSize;\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(cap + 1, sizeof(int));\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n int res = dp[n][cap];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n return res;\n}\n// 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn knapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n comptime var n = wgt.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..cap + 1) |c| {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = @max(dp[i - 1][c], dp[i - 1][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n\u5982\u56fe 14-20 \u6240\u793a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u7531\u6570\u7ec4
<1><2><3><4><5><6><7><8><9><10><11><12><13><14>dp\u5927\u5c0f\u51b3\u5b9a\uff0c\u5373 \\(O(n \\times cap)\\) \u3002\u56fe 14-20 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
"},{"location":"chapter_dynamic_programming/knapsack_problem/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u90fd\u53ea\u4e0e\u5176\u4e0a\u4e00\u884c\u7684\u72b6\u6001\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6570\u7ec4\u6eda\u52a8\u524d\u8fdb\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \u3002
\u8fdb\u4e00\u6b65\u601d\u8003\uff0c\u6211\u4eec\u80fd\u5426\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u5b9e\u73b0\u7a7a\u95f4\u4f18\u5316\u5462\uff1f\u89c2\u5bdf\u53ef\u77e5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u662f\u7531\u6b63\u4e0a\u65b9\u6216\u5de6\u4e0a\u65b9\u7684\u683c\u5b50\u8f6c\u79fb\u8fc7\u6765\u7684\u3002\u5047\u8bbe\u53ea\u6709\u4e00\u4e2a\u6570\u7ec4\uff0c\u5f53\u5f00\u59cb\u904d\u5386\u7b2c \\(i\\) \u884c\u65f6\uff0c\u8be5\u6570\u7ec4\u5b58\u50a8\u7684\u4ecd\u7136\u662f\u7b2c \\(i-1\\) \u884c\u7684\u72b6\u6001\u3002
- \u5982\u679c\u91c7\u53d6\u6b63\u5e8f\u904d\u5386\uff0c\u90a3\u4e48\u904d\u5386\u5230 \\(dp[i, j]\\) \u65f6\uff0c\u5de6\u4e0a\u65b9 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) \u503c\u53ef\u80fd\u5df2\u7ecf\u88ab\u8986\u76d6\uff0c\u6b64\u65f6\u5c31\u65e0\u6cd5\u5f97\u5230\u6b63\u786e\u7684\u72b6\u6001\u8f6c\u79fb\u7ed3\u679c\u3002
- \u5982\u679c\u91c7\u53d6\u5012\u5e8f\u904d\u5386\uff0c\u5219\u4e0d\u4f1a\u53d1\u751f\u8986\u76d6\u95ee\u9898\uff0c\u72b6\u6001\u8f6c\u79fb\u53ef\u4ee5\u6b63\u786e\u8fdb\u884c\u3002
\u56fe 14-21 \u5c55\u793a\u4e86\u5728\u5355\u4e2a\u6570\u7ec4\u4e0b\u4ece\u7b2c \\(i = 1\\) \u884c\u8f6c\u6362\u81f3\u7b2c \\(i = 2\\) \u884c\u7684\u8fc7\u7a0b\u3002\u8bf7\u601d\u8003\u6b63\u5e8f\u904d\u5386\u548c\u5012\u5e8f\u904d\u5386\u7684\u533a\u522b\u3002
<1><2><3><4><5><6>\u56fe 14-21 \u00a0 0-1 \u80cc\u5305\u7684\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4ec5\u9700\u5c06\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.pydp\u7684\u7b2c\u4e00\u7ef4 \\(i\\) \u76f4\u63a5\u5220\u9664\uff0c\u5e76\u4e14\u628a\u5185\u5faa\u73af\u66f4\u6539\u4e3a\u5012\u5e8f\u904d\u5386\u5373\u53ef\uff1a
knapsack.cppdef knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(wgt)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [0] * (cap + 1)\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n # \u5012\u5e8f\u904d\u5386\n for c in range(cap, 0, -1):\n if wgt[i - 1] > c:\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c]\n else:\n # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n return dp[cap]\n
knapsack.java/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(vector<int> &wgt, vector<int> &val, int cap) {\n int n = wgt.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(cap + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.cs/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.go/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n int n = weight.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c > 0; c--) {\n if (weight[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.swift/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n n := len(wgt)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, cap+1)\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n // \u5012\u5e8f\u904d\u5386\n for c := cap; c >= 1; c-- {\n if wgt[i-1] <= c {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n }\n }\n }\n return dp[cap]\n}\n
knapsack.js/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -> Int {\n let n = wgt.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: 0, count: cap + 1)\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n // \u5012\u5e8f\u904d\u5386\n for c in stride(from: cap, through: 1, by: -1) {\n if wgt[i - 1] <= c {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n }\n }\n }\n return dp[cap]\n}\n
knapsack.ts/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(wgt, val, cap) {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array(cap + 1).fill(0);\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (let c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.dart/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(\n wgt: Array<number>,\n val: Array<number>,\n cap: number\n): number {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array(cap + 1).fill(0);\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (let c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.rs/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(List<int> wgt, List<int> val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(cap + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
knapsack.c/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp_comp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {\n let n = wgt.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n // \u5012\u5e8f\u904d\u5386\n for c in (1..=cap).rev() {\n if wgt[i - 1] <= c as i32 {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n }\n }\n }\n dp[cap]\n}\n
knapsack.zig/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n int n = wgtSize;\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(cap + 1, sizeof(int));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n // \u5012\u5e8f\u904d\u5386\n for (int c = cap; c >= 1; c--) {\n if (wgt[i - 1] <= c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n int res = dp[cap];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/summary/","title":"14.7 \u00a0 \u5c0f\u7ed3","text":"// 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn knapsackDPComp(wgt: []i32, val: []i32, comptime cap: usize) i32 {\n var n = wgt.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** (cap + 1);\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n // \u5012\u5e8f\u904d\u5386\n var c = cap;\n while (c > 0) : (c -= 1) {\n if (wgt[i - 1] < c) {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n- \u52a8\u6001\u89c4\u5212\u5bf9\u95ee\u9898\u8fdb\u884c\u5206\u89e3\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u89c4\u907f\u91cd\u590d\u8ba1\u7b97\uff0c\u63d0\u9ad8 \u8ba1\u7b97\u6548\u7387\u3002
- \u4e0d\u8003\u8651\u65f6\u95f4\u7684\u524d\u63d0\u4e0b\uff0c\u6240\u6709\u52a8\u6001\u89c4\u5212\u95ee\u9898\u90fd\u53ef\u4ee5\u7528\u56de\u6eaf\uff08\u66b4\u529b\u641c\u7d22\uff09\u8fdb\u884c\u6c42\u89e3\uff0c\u4f46\u9012\u5f52\u6811\u4e2d\u5b58\u5728\u5927\u91cf\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u6548\u7387\u6781\u4f4e\u3002\u901a\u8fc7\u5f15\u5165\u8bb0\u5fc6\u5316\u5217\u8868\uff0c\u53ef\u4ee5\u5b58\u50a8\u6240\u6709\u8ba1\u7b97\u8fc7\u7684\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u4ece\u800c\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002
- \u8bb0\u5fc6\u5316\u9012\u5f52\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u9012\u5f52\u5f0f\u89e3\u6cd5\uff0c\u800c\u4e0e\u4e4b\u5bf9\u5e94\u7684\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u4ece\u5e95\u81f3\u9876\u7684\u9012\u63a8\u5f0f\u89e3\u6cd5\uff0c\u5176\u5982\u540c\u201c\u586b\u5199\u8868\u683c\u201d\u4e00\u6837\u3002\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u4ec5\u4f9d\u8d56\u67d0\u4e9b\u5c40\u90e8\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u6d88\u9664 \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\uff0c\u4ece\u800c\u964d\u4f4e\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
- \u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u5177\u6709\u4e0d\u540c\u7684\u6027\u8d28\u3002
- \u52a8\u6001\u89c4\u5212\u95ee\u9898\u6709\u4e09\u5927\u7279\u6027\uff1a\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002
- \u5982\u679c\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\uff0c\u5219\u5b83\u5c31\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u3002
- \u65e0\u540e\u6548\u6027\u6307\u5bf9\u4e8e\u4e00\u4e2a\u72b6\u6001\uff0c\u5176\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u8be5\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u4e0d\u5177\u6709\u65e0\u540e\u6548\u6027\uff0c\u65e0\u6cd5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u5feb\u901f\u6c42\u89e3\u3002
\u80cc\u5305\u95ee\u9898
- \u80cc\u5305\u95ee\u9898\u662f\u6700\u5178\u578b\u7684\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e4b\u4e00\uff0c\u5177\u6709 0-1 \u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305\u3001\u591a\u91cd\u80cc\u5305\u7b49\u53d8\u79cd\u3002
- 0-1 \u80cc\u5305\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\u3002\u6839\u636e\u4e0d\u653e\u5165\u80cc\u5305\u548c\u653e\u5165\u80cc\u5305\u4e24\u79cd\u51b3\u7b56\uff0c\u53ef\u5f97\u5230\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\uff0c\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u9700\u8981\u5012\u5e8f\u904d\u5386\u5217\u8868\uff0c\u907f\u514d\u5de6\u4e0a\u65b9\u72b6\u6001\u88ab\u8986\u76d6\u3002
- \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u6bcf\u79cd\u7269\u54c1\u7684\u9009\u53d6\u6570\u91cf\u65e0\u9650\u5236\uff0c\u56e0\u6b64\u9009\u62e9\u653e\u5165\u7269\u54c1\u7684\u72b6\u6001\u8f6c\u79fb\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\u3002\u7531\u4e8e\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u6b63\u5de6\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\u5e94\u5f53\u6b63\u5e8f\u904d\u5386\u3002
- \u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u4e2a\u53d8\u79cd\u3002\u5b83\u4ece\u6c42\u201c\u6700\u5927\u201d\u4ef7\u503c\u53d8\u4e3a\u6c42\u201c\u6700\u5c0f\u201d\u786c\u5e01\u6570\u91cf\uff0c\u56e0\u6b64\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\max()\\) \u5e94\u6539\u4e3a \\(\\min()\\) \u3002\u4ece\u8ffd\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u5230\u8ffd\u6c42\u201c\u6070\u597d\u201d\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u4f7f\u7528 \\(amt + 1\\) \u6765\u8868\u793a\u201c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u201d\u7684\u65e0\u6548\u89e3\u3002
- \u96f6\u94b1\u5151\u6362 II \u95ee\u9898\u4ece\u6c42\u201c\u6700\u5c11\u786c\u5e01\u6570\u91cf\u201d\u6539\u4e3a\u6c42\u201c\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u201d\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u76f8\u5e94\u5730\u4ece \\(\\min()\\) \u6539\u4e3a\u6c42\u548c\u8fd0\u7b97\u7b26\u3002
\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898
- \u7f16\u8f91\u8ddd\u79bb\uff08Levenshtein \u8ddd\u79bb\uff09\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\uff0c\u5176\u5b9a\u4e49\u4e3a\u4ece\u4e00\u4e2a\u5b57\u7b26\u4e32\u5230\u53e6\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u7f16\u8f91\u64cd\u4f5c\u5305\u62ec\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\u3002
- \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002\u5f53 \\(s[i] \\ne t[j]\\) \u65f6\uff0c\u5177\u6709\u4e09\u79cd\u51b3\u7b56\uff1a\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\uff0c\u5b83\u4eec\u90fd\u6709\u76f8\u5e94\u7684\u5269\u4f59\u5b50\u95ee\u9898\u3002\u636e\u6b64\u4fbf\u53ef\u4ee5\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u6784\u5efa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u800c\u5f53 \\(s[i] = t[j]\\) \u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\u3002
- \u5728\u7f16\u8f91\u8ddd\u79bb\u4e2d\uff0c\u72b6\u6001\u4f9d\u8d56\u5176\u6b63\u4e0a\u65b9\u3001\u6b63\u5de6\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u6b63\u5e8f\u6216\u5012\u5e8f\u904d\u5386\u90fd\u65e0\u6cd5\u6b63\u786e\u5730\u8fdb\u884c\u72b6\u6001\u8f6c\u79fb\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u53d8\u91cf\u6682\u5b58\u5de6\u4e0a\u65b9\u72b6\u6001\uff0c\u4ece\u800c\u8f6c\u5316\u5230\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7b49\u4ef7\u7684\u60c5\u51b5\uff0c\u53ef\u4ee5\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002
\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6c42\u89e3\u53e6\u4e00\u4e2a\u5e38\u89c1\u7684\u80cc\u5305\u95ee\u9898\uff1a\u5b8c\u5168\u80cc\u5305\uff0c\u518d\u4e86\u89e3\u5b83\u7684\u4e00\u79cd\u7279\u4f8b\uff1a\u96f6\u94b1\u5151\u6362\u3002
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","title":"14.5.1 \u00a0 \u5b8c\u5168\u80cc\u5305","text":"Question
\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 14-22 \u6240\u793a\u3002
\u56fe 14-22 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\uff0c\u533a\u522b\u4ec5\u5728\u4e8e\u4e0d\u9650\u5236\u7269\u54c1\u7684\u9009\u62e9\u6b21\u6570\u3002
- \u5728 0-1 \u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u53ea\u6709\u4e00\u4e2a\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u53ea\u80fd\u4ece\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002
- \u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u7684\u6570\u91cf\u662f\u65e0\u9650\u7684\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u4ecd\u53ef\u4ee5\u4ece\u524d \\(i\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002
\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u89c4\u5b9a\u4e0b\uff0c\u72b6\u6001 \\([i, c]\\) \u7684\u53d8\u5316\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5\u3002
- \u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i-1, c]\\) \u3002
- \u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i, c-wgt[i-1]]\\) \u3002
\u4ece\u800c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u53d8\u4e3a\uff1a
\\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5bf9\u6bd4\u4e24\u9053\u9898\u76ee\u7684\u4ee3\u7801\uff0c\u72b6\u6001\u8f6c\u79fb\u4e2d\u6709\u4e00\u5904\u4ece \\(i-1\\) \u53d8\u4e3a \\(i\\) \uff0c\u5176\u4f59\u5b8c\u5168\u4e00\u81f4\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig unbounded_knapsack.py
unbounded_knapsack.cppdef unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(wgt)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * (cap + 1) for _ in range(n + 1)]\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n for c in range(1, cap + 1):\n if wgt[i - 1] > c:\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c]\n else:\n # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n return dp[n][cap]\n
unbounded_knapsack.java/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(vector<int> &wgt, vector<int> &val, int cap) {\n int n = wgt.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n + 1, vector<int>(cap + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.cs/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n + 1][cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.go/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n int n = wgt.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n + 1, cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i, c] = dp[i - 1, c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n, cap];\n}\n
unbounded_knapsack.swift/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n n := len(wgt)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, cap+1)\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n for c := 1; c <= cap; c++ {\n if wgt[i-1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i-1][c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n }\n }\n }\n return dp[n][cap]\n}\n
unbounded_knapsack.js/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -> Int {\n let n = wgt.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for c in stride(from: 1, through: cap, by: 1) {\n if wgt[i - 1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n }\n }\n }\n return dp[n][cap]\n}\n
unbounded_knapsack.ts/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: cap + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(\n dp[i - 1][c],\n dp[i][c - wgt[i - 1]] + val[i - 1]\n );\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.dart/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(\n wgt: Array<number>,\n val: Array<number>,\n cap: number\n): number {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: cap + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = Math.max(\n dp[i - 1][c],\n dp[i][c - wgt[i - 1]] + val[i - 1]\n );\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.rs/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(List<int> wgt, List<int> val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(cap + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.c/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {\n let n = wgt.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; cap + 1]; n + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for c in 1..=cap {\n if wgt[i - 1] > c as i32 {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n
unbounded_knapsack.zig/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n int n = wgtSize;\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(cap + 1, sizeof(int));\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n int res = dp[n][cap];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"// \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n comptime var n = wgt.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..cap + 1) |c| {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[i][c] = dp[i - 1][c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[i][c] = @max(dp[i - 1][c], dp[i][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n }\n }\n }\n return dp[n][cap];\n}\n\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u662f\u4ece\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u72b6\u6001\u8f6c\u79fb\u800c\u6765\u7684\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u5e94\u8be5\u5bf9 \\(dp\\) \u8868\u4e2d\u7684\u6bcf\u4e00\u884c\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002
\u8fd9\u4e2a\u904d\u5386\u987a\u5e8f\u4e0e 0-1 \u80cc\u5305\u6b63\u597d\u76f8\u53cd\u3002\u8bf7\u501f\u52a9\u56fe 14-23 \u6765\u7406\u89e3\u4e24\u8005\u7684\u533a\u522b\u3002
<1><2><3><4><5><6>\u56fe 14-23 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
\u4ee3\u7801\u5b9e\u73b0\u6bd4\u8f83\u7b80\u5355\uff0c\u4ec5\u9700\u5c06\u6570\u7ec4
PythonC++JavaC#GoSwiftJSTSDartRustCZig unbounded_knapsack.pydp\u7684\u7b2c\u4e00\u7ef4\u5220\u9664\uff1a
unbounded_knapsack.cppdef unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(wgt)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [0] * (cap + 1)\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n # \u6b63\u5e8f\u904d\u5386\n for c in range(1, cap + 1):\n if wgt[i - 1] > c:\n # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c]\n else:\n # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n return dp[cap]\n
unbounded_knapsack.java/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(vector<int> &wgt, vector<int> &val, int cap) {\n int n = wgt.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(cap + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.cs/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.go/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n int n = wgt.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.swift/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n n := len(wgt)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, cap+1)\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n for c := 1; c <= cap; c++ {\n if wgt[i-1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n }\n }\n }\n return dp[cap]\n}\n
unbounded_knapsack.js/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -> Int {\n let n = wgt.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: 0, count: cap + 1)\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for c in stride(from: 1, through: cap, by: 1) {\n if wgt[i - 1] > c {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c]\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n }\n }\n }\n return dp[cap]\n}\n
unbounded_knapsack.ts/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: cap + 1 }, () => 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.dart/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(\n wgt: Array<number>,\n val: Array<number>,\n cap: number\n): number {\n const n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: cap + 1 }, () => 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.rs/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(List<int> wgt, List<int> val, int cap) {\n int n = wgt.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(cap + 1, 0);\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n
unbounded_knapsack.c/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp_comp(wgt: &[i32], val: &[i32], cap: usize) -> i32 {\n let n = wgt.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; cap + 1];\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for c in 1..=cap {\n if wgt[i - 1] > c as i32 {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n }\n }\n }\n dp[cap]\n}\n
unbounded_knapsack.zig/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n int n = wgtSize;\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(cap + 1, sizeof(int));\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int c = 1; c <= cap; c++) {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n }\n }\n }\n int res = dp[cap];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","title":"14.5.2 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898","text":"// \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDPComp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n comptime var n = wgt.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** (cap + 1);\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..cap + 1) |c| {\n if (wgt[i - 1] > c) {\n // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n dp[c] = dp[c];\n } else {\n // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n }\n }\n }\n return dp[cap];\n}\n\u80cc\u5305\u95ee\u9898\u662f\u4e00\u5927\u7c7b\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4ee3\u8868\uff0c\u5176\u62e5\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002
Question
\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 14-24 \u6240\u793a\u3002
\u56fe 14-24 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"\u96f6\u94b1\u5151\u6362\u53ef\u4ee5\u770b\u4f5c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u79cd\u7279\u6b8a\u60c5\u51b5\uff0c\u4e24\u8005\u5177\u6709\u4ee5\u4e0b\u8054\u7cfb\u4e0e\u4e0d\u540c\u70b9\u3002
- \u4e24\u9053\u9898\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\uff0c\u201c\u7269\u54c1\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u201d\u3001\u201c\u7269\u54c1\u91cd\u91cf\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u9762\u503c\u201d\u3001\u201c\u80cc\u5305\u5bb9\u91cf\u201d\u5bf9\u5e94\u201c\u76ee\u6807\u91d1\u989d\u201d\u3002
- \u4f18\u5316\u76ee\u6807\u76f8\u53cd\uff0c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u8981\u6700\u5927\u5316\u7269\u54c1\u4ef7\u503c\uff0c\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u8981\u6700\u5c0f\u5316\u786c\u5e01\u6570\u91cf\u3002
- \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u89e3\uff0c\u96f6\u94b1\u5151\u6362\u662f\u6c42\u201c\u6070\u597d\u201d\u51d1\u5230\u76ee\u6807\u91d1\u989d\u7684\u89e3\u3002
\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868
\u72b6\u6001 \\([i, a]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\uff0c\u8bb0\u4e3a \\(dp[i, a]\\) \u3002
\u4e8c\u7ef4 \\(dp\\) \u8868\u7684\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt+1)\\) \u3002
\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b
\u672c\u9898\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5b58\u5728\u4ee5\u4e0b\u4e24\u70b9\u5dee\u5f02\u3002
- \u672c\u9898\u8981\u6c42\u6700\u5c0f\u503c\uff0c\u56e0\u6b64\u9700\u5c06\u8fd0\u7b97\u7b26 \\(\\max()\\) \u66f4\u6539\u4e3a \\(\\min()\\) \u3002
- \u4f18\u5316\u4e3b\u4f53\u662f\u786c\u5e01\u6570\u91cf\u800c\u975e\u5546\u54c1\u4ef7\u503c\uff0c\u56e0\u6b64\u5728\u9009\u4e2d\u786c\u5e01\u65f6\u6267\u884c \\(+1\\) \u5373\u53ef\u3002
\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f
\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u51d1\u51fa\u5b83\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002
\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u610f \\(> 0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u5373\u662f\u65e0\u6548\u89e3\u3002\u4e3a\u4f7f\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\min()\\) \u51fd\u6570\u80fd\u591f\u8bc6\u522b\u5e76\u8fc7\u6ee4\u65e0\u6548\u89e3\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528 \\(+ \\infty\\) \u6765\u8868\u793a\u5b83\u4eec\uff0c\u5373\u4ee4\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(+ \\infty\\) \u3002
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u5e76\u672a\u63d0\u4f9b \\(+ \\infty\\) \u53d8\u91cf\uff0c\u53ea\u80fd\u4f7f\u7528\u6574\u578b
int\u7684\u6700\u5927\u503c\u6765\u4ee3\u66ff\u3002\u800c\u8fd9\u53c8\u4f1a\u5bfc\u81f4\u5927\u6570\u8d8a\u754c\uff1a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(+ 1\\) \u64cd\u4f5c\u53ef\u80fd\u53d1\u751f\u6ea2\u51fa\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u6570\u5b57 \\(amt + 1\\) \u6765\u8868\u793a\u65e0\u6548\u89e3\uff0c\u56e0\u4e3a\u51d1\u51fa \\(amt\\) \u7684\u786c\u5e01\u6570\u91cf\u6700\u591a\u4e3a \\(amt\\) \u3002\u6700\u540e\u8fd4\u56de\u524d\uff0c\u5224\u65ad \\(dp[n, amt]\\) \u662f\u5426\u7b49\u4e8e \\(amt + 1\\) \uff0c\u82e5\u662f\u5219\u8fd4\u56de \\(-1\\) \uff0c\u4ee3\u8868\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change.py
coin_change.cppdef coin_change_dp(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(coins)\n MAX = amt + 1\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * (amt + 1) for _ in range(n + 1)]\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for a in range(1, amt + 1):\n dp[0][a] = MAX\n # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in range(1, n + 1):\n for a in range(1, amt + 1):\n if coins[i - 1] > a:\n # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a]\n else:\n # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n return dp[n][amt] if dp[n][amt] != MAX else -1\n
coin_change.java/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(vector<int> &coins, int amt) {\n int n = coins.size();\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n + 1, vector<int>(amt + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n
coin_change.cs/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int[] coins, int amt) {\n int n = coins.length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n + 1][amt + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n
coin_change.go/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDP(int[] coins, int amt) {\n int n = coins.Length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n + 1, amt + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0, a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i, a] = dp[i - 1, a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n
coin_change.swift/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins []int, amt int) int {\n n := len(coins)\n max := amt + 1\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, amt+1)\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for a := 1; a <= amt; a++ {\n dp[0][a] = max\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i <= n; i++ {\n for a := 1; a <= amt; a++ {\n if coins[i-1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i-1][a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n }\n }\n }\n if dp[n][amt] != max {\n return dp[n][amt]\n }\n return -1\n}\n
coin_change.js/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins: [Int], amt: Int) -> Int {\n let n = coins.count\n let MAX = amt + 1\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for a in stride(from: 1, through: amt, by: 1) {\n dp[0][a] = MAX\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in stride(from: 1, through: n, by: 1) {\n for a in stride(from: 1, through: amt, by: 1) {\n if coins[i - 1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n }\n }\n }\n return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n
coin_change.ts/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins, amt) {\n const n = coins.length;\n const MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: amt + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (let a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n
coin_change.dart/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins: Array<number>, amt: number): number {\n const n = coins.length;\n const MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: amt + 1 }, () => 0)\n );\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (let a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n
coin_change.rs/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(List<int> coins, int amt) {\n int n = coins.length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(amt + 1, 0));\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n
coin_change.c/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp(coins: &[i32], amt: usize) -> i32 {\n let n = coins.len();\n let max = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; amt + 1]; n + 1];\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for a in 1..= amt {\n dp[0][a] = max;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i in 1..=n {\n for a in 1..=amt {\n if coins[i - 1] > a as i32 {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n }\n }\n }\n if dp[n][amt] != max { return dp[n][amt] as i32; } else { -1 }\n}\n
coin_change.zig/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n int n = coinsSize;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(amt + 1, sizeof(int));\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (int a = 1; a <= amt; a++) {\n dp[0][a] = MAX;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n }\n }\n }\n int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n free(dp);\n return res;\n}\n// \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeDP(comptime coins: []i32, comptime amt: usize) i32 {\n comptime var n = coins.len;\n comptime var max = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n for (1..amt + 1) |a| {\n dp[0][a] = max;\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for (1..n + 1) |i| {\n for (1..amt + 1) |a| {\n if (coins[i - 1] > @as(i32, @intCast(a))) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = @min(dp[i - 1][a], dp[i][a - @as(usize, @intCast(coins[i - 1]))] + 1);\n }\n }\n }\n if (dp[n][amt] != max) {\n return @intCast(dp[n][amt]);\n } else {\n return -1;\n }\n}\n\u56fe 14-25 \u5c55\u793a\u4e86\u96f6\u94b1\u5151\u6362\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\uff0c\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\u3002
<1><2><3><4><5><6><7><8><9><10><11><12><13><14><15>\u56fe 14-25 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"\u96f6\u94b1\u5151\u6362\u7684\u7a7a\u95f4\u4f18\u5316\u7684\u5904\u7406\u65b9\u5f0f\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e00\u81f4\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change.py
coin_change.cppdef coin_change_dp_comp(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(coins)\n MAX = amt + 1\n # \u521d\u59cb\u5316 dp \u8868\n dp = [MAX] * (amt + 1)\n dp[0] = 0\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n # \u6b63\u5e8f\u904d\u5386\n for a in range(1, amt + 1):\n if coins[i - 1] > a:\n # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n else:\n # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n return dp[amt] if dp[amt] != MAX else -1\n
coin_change.java/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(vector<int> &coins, int amt) {\n int n = coins.size();\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(amt + 1, MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1;\n}\n
coin_change.cs/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int[] coins, int amt) {\n int n = coins.length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[amt + 1];\n Arrays.fill(dp, MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1;\n}\n
coin_change.go/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDPComp(int[] coins, int amt) {\n int n = coins.Length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[amt + 1];\n Array.Fill(dp, MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1;\n}\n
coin_change.swift/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins []int, amt int) int {\n n := len(coins)\n max := amt + 1\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, amt+1)\n for i := 1; i <= amt; i++ {\n dp[i] = max\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n // \u5012\u5e8f\u904d\u5386\n for a := 1; a <= amt; a++ {\n if coins[i-1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n }\n }\n }\n if dp[amt] != max {\n return dp[amt]\n }\n return -1\n}\n
coin_change.js/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -> Int {\n let n = coins.count\n let MAX = amt + 1\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: MAX, count: amt + 1)\n dp[0] = 0\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for a in stride(from: 1, through: amt, by: 1) {\n if coins[i - 1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1\n}\n
coin_change.ts/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins, amt) {\n const n = coins.length;\n const MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: amt + 1 }, () => MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] !== MAX ? dp[amt] : -1;\n}\n
coin_change.dart/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins: Array<number>, amt: number): number {\n const n = coins.length;\n const MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: amt + 1 }, () => MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] !== MAX ? dp[amt] : -1;\n}\n
coin_change.rs/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(List<int> coins, int amt) {\n int n = coins.length;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(amt + 1, MAX);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n return dp[amt] != MAX ? dp[amt] : -1;\n}\n
coin_change.c/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp_comp(coins: &[i32], amt: usize) -> i32 {\n let n = coins.len();\n let max = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; amt + 1];\n dp.fill(max);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for a in 1..=amt {\n if coins[i - 1] > a as i32 {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n }\n }\n }\n if dp[amt] != max { return dp[amt] as i32; } else { -1 }\n}\n
coin_change.zig/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n int n = coinsSize;\n int MAX = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(amt + 1, sizeof(int));\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n }\n }\n }\n int res = dp[amt] != MAX ? dp[amt] : -1;\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","title":"14.5.3 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II","text":"// \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n comptime var n = coins.len;\n comptime var max = amt + 1;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** (amt + 1);\n @memset(&dp, max);\n dp[0] = 0;\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..amt + 1) |a| {\n if (coins[i - 1] > @as(i32, @intCast(a))) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = @min(dp[a], dp[a - @as(usize, @intCast(coins[i - 1]))] + 1);\n }\n }\n }\n if (dp[amt] != max) {\n return @intCast(dp[amt]);\n } else {\n return -1;\n }\n}\nQuestion
\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u3002\u793a\u4f8b\u5982\u56fe 14-26 \u6240\u793a\u3002
\u56fe 14-26 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II \u7684\u793a\u4f8b\u6570\u636e
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"\u76f8\u6bd4\u4e8e\u4e0a\u4e00\u9898\uff0c\u672c\u9898\u76ee\u6807\u662f\u6c42\u7ec4\u5408\u6570\u91cf\uff0c\u56e0\u6b64\u5b50\u95ee\u9898\u53d8\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u7ec4\u5408\u6570\u91cf\u3002\u800c \\(dp\\) \u8868\u4ecd\u7136\u662f\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt + 1)\\) \u7684\u4e8c\u7ef4\u77e9\u9635\u3002
\u5f53\u524d\u72b6\u6001\u7684\u7ec4\u5408\u6570\u91cf\u7b49\u4e8e\u4e0d\u9009\u5f53\u524d\u786c\u5e01\u4e0e\u9009\u5f53\u524d\u786c\u5e01\u8fd9\u4e24\u79cd\u51b3\u7b56\u7684\u7ec4\u5408\u6570\u91cf\u4e4b\u548c\u3002\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a
\\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\]\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u65e0\u987b\u9009\u62e9\u4efb\u4f55\u786c\u5e01\u5373\u53ef\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u5e94\u5c06\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u521d\u59cb\u5316\u4e3a \\(1\\) \u3002\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u4f55 \\(>0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_ii.py
coin_change_ii.cppdef coin_change_ii_dp(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(coins)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [[0] * (amt + 1) for _ in range(n + 1)]\n # \u521d\u59cb\u5316\u9996\u5217\n for i in range(n + 1):\n dp[i][0] = 1\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n for a in range(1, amt + 1):\n if coins[i - 1] > a:\n # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a]\n else:\n # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n return dp[n][amt]\n
coin_change_ii.java/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(vector<int> &coins, int amt) {\n int n = coins.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<vector<int>> dp(n + 1, vector<int>(amt + 1, 0));\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.cs/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int[] coins, int amt) {\n int n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[][] dp = new int[n + 1][amt + 1];\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.go/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDP(int[] coins, int amt) {\n int n = coins.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[,] dp = new int[n + 1, amt + 1];\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i, 0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i, a] = dp[i - 1, a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n }\n }\n }\n return dp[n, amt];\n}\n
coin_change_ii.swift/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins []int, amt int) int {\n n := len(coins)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([][]int, n+1)\n for i := 0; i <= n; i++ {\n dp[i] = make([]int, amt+1)\n }\n // \u521d\u59cb\u5316\u9996\u5217\n for i := 0; i <= n; i++ {\n dp[i][0] = 1\n }\n // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n for i := 1; i <= n; i++ {\n for a := 1; a <= amt; a++ {\n if coins[i-1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i-1][a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n }\n }\n }\n return dp[n][amt]\n}\n
coin_change_ii.js/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -> Int {\n let n = coins.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n // \u521d\u59cb\u5316\u9996\u5217\n for i in stride(from: 0, through: n, by: 1) {\n dp[i][0] = 1\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for a in stride(from: 1, through: amt, by: 1) {\n if coins[i - 1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n }\n }\n }\n return dp[n][amt]\n}\n
coin_change_ii.ts/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins, amt) {\n const n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: amt + 1 }, () => 0)\n );\n // \u521d\u59cb\u5316\u9996\u5217\n for (let i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.dart/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins: Array<number>, amt: number): number {\n const n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: n + 1 }, () =>\n Array.from({ length: amt + 1 }, () => 0)\n );\n // \u521d\u59cb\u5316\u9996\u5217\n for (let i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.rs/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(List<int> coins, int amt) {\n int n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<List<int>> dp = List.generate(n + 1, (index) => List.filled(amt + 1, 0));\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n return dp[n][amt];\n}\n
coin_change_ii.c/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp(coins: &[i32], amt: usize) -> i32 {\n let n = coins.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![vec![0; amt + 1]; n + 1];\n // \u521d\u59cb\u5316\u9996\u5217\n for i in 0..= n {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for a in 1..=amt {\n if coins[i - 1] > a as i32 {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n }\n }\n }\n dp[n][amt]\n}\n
coin_change_ii.zig/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n int n = coinsSize;\n // \u521d\u59cb\u5316 dp \u8868\n int **dp = malloc((n + 1) * sizeof(int *));\n for (int i = 0; i <= n; i++) {\n dp[i] = calloc(amt + 1, sizeof(int));\n }\n // \u521d\u59cb\u5316\u9996\u5217\n for (int i = 0; i <= n; i++) {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n }\n }\n }\n int res = dp[n][amt];\n // \u91ca\u653e\u5185\u5b58\n for (int i = 0; i <= n; i++) {\n free(dp[i]);\n }\n free(dp);\n return res;\n}\n
"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"// \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDP(comptime coins: []i32, comptime amt: usize) i32 {\n comptime var n = coins.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n // \u521d\u59cb\u5316\u9996\u5217\n for (0..n + 1) |i| {\n dp[i][0] = 1;\n }\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..amt + 1) |a| {\n if (coins[i - 1] > @as(i32, @intCast(a))) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[i][a] = dp[i - 1][a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[i][a] = dp[i - 1][a] + dp[i][a - @as(usize, @intCast(coins[i - 1]))];\n }\n }\n }\n return dp[n][amt];\n}\n\u7a7a\u95f4\u4f18\u5316\u5904\u7406\u65b9\u5f0f\u76f8\u540c\uff0c\u5220\u9664\u786c\u5e01\u7ef4\u5ea6\u5373\u53ef\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_ii.py
coin_change_ii.cppdef coin_change_ii_dp_comp(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n n = len(coins)\n # \u521d\u59cb\u5316 dp \u8868\n dp = [0] * (amt + 1)\n dp[0] = 1\n # \u72b6\u6001\u8f6c\u79fb\n for i in range(1, n + 1):\n # \u6b63\u5e8f\u904d\u5386\n for a in range(1, amt + 1):\n if coins[i - 1] > a:\n # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n else:\n # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]]\n return dp[amt]\n
coin_change_ii.java/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(vector<int> &coins, int amt) {\n int n = coins.size();\n // \u521d\u59cb\u5316 dp \u8868\n vector<int> dp(amt + 1, 0);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.cs/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int[] coins, int amt) {\n int n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[amt + 1];\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.go/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n int n = coins.Length;\n // \u521d\u59cb\u5316 dp \u8868\n int[] dp = new int[amt + 1];\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.swift/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n n := len(coins)\n // \u521d\u59cb\u5316 dp \u8868\n dp := make([]int, amt+1)\n dp[0] = 1\n // \u72b6\u6001\u8f6c\u79fb\n for i := 1; i <= n; i++ {\n // \u5012\u5e8f\u904d\u5386\n for a := 1; a <= amt; a++ {\n if coins[i-1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a-coins[i-1]]\n }\n }\n }\n return dp[amt]\n}\n
coin_change_ii.js/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -> Int {\n let n = coins.count\n // \u521d\u59cb\u5316 dp \u8868\n var dp = Array(repeating: 0, count: amt + 1)\n dp[0] = 1\n // \u72b6\u6001\u8f6c\u79fb\n for i in stride(from: 1, through: n, by: 1) {\n for a in stride(from: 1, through: amt, by: 1) {\n if coins[i - 1] > a {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a]\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]]\n }\n }\n }\n return dp[amt]\n}\n
coin_change_ii.ts/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins, amt) {\n const n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: amt + 1 }, () => 0);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.dart/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins: Array<number>, amt: number): number {\n const n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n const dp = Array.from({ length: amt + 1 }, () => 0);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (let i = 1; i <= n; i++) {\n for (let a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.rs/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(List<int> coins, int amt) {\n int n = coins.length;\n // \u521d\u59cb\u5316 dp \u8868\n List<int> dp = List.filled(amt + 1, 0);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n return dp[amt];\n}\n
coin_change_ii.c/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp_comp(coins: &[i32], amt: usize) -> i32 {\n let n = coins.len();\n // \u521d\u59cb\u5316 dp \u8868\n let mut dp = vec![0; amt + 1];\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for i in 1..=n {\n for a in 1..=amt {\n if coins[i - 1] > a as i32 {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n }\n }\n }\n dp[amt]\n}\n
coin_change_ii.zig/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n int n = coinsSize;\n // \u521d\u59cb\u5316 dp \u8868\n int *dp = calloc(amt + 1, sizeof(int));\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (int i = 1; i <= n; i++) {\n for (int a = 1; a <= amt; a++) {\n if (coins[i - 1] > a) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n dp[a] = dp[a] + dp[a - coins[i - 1]];\n }\n }\n }\n int res = dp[amt];\n // \u91ca\u653e\u5185\u5b58\n free(dp);\n return res;\n}\n
"},{"location":"chapter_graph/","title":"\u7b2c 9 \u7ae0 \u00a0 \u56fe","text":"// \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n comptime var n = coins.len;\n // \u521d\u59cb\u5316 dp \u8868\n var dp = [_]i32{0} ** (amt + 1);\n dp[0] = 1;\n // \u72b6\u6001\u8f6c\u79fb\n for (1..n + 1) |i| {\n for (1..amt + 1) |a| {\n if (coins[i - 1] > @as(i32, @intCast(a))) {\n // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n dp[a] = dp[a];\n } else {\n // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n dp[a] = dp[a] + dp[a - @as(usize, @intCast(coins[i - 1]))];\n }\n }\n }\n return dp[amt];\n}\nAbstract
\u5728\u751f\u547d\u65c5\u9014\u4e2d\uff0c\u6211\u4eec\u5c31\u50cf\u662f\u4e00\u4e2a\u4e2a\u8282\u70b9\uff0c\u88ab\u65e0\u6570\u770b\u4e0d\u89c1\u7684\u8fb9\u76f8\u8fde\u3002
\u6bcf\u4e00\u6b21\u7684\u76f8\u8bc6\u4e0e\u76f8\u79bb\uff0c\u90fd\u5728\u8fd9\u5f20\u5de8\u5927\u7684\u7f51\u7edc\u56fe\u4e2d\u7559\u4e0b\u72ec\u7279\u7684\u5370\u8bb0\u3002
"},{"location":"chapter_graph/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 9.1 \u00a0 \u56fe
- 9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c
- 9.3 \u00a0 \u56fe\u7684\u904d\u5386
- 9.4 \u00a0 \u5c0f\u7ed3
\u300c\u56fe graph\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u300c\u9876\u70b9 vertex\u300d\u548c\u300c\u8fb9 edge\u300d\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe\u3002
\\[ \\begin{aligned} V & = \\{ 1, 2, 3, 4, 5 \\} \\newline E & = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G & = \\{ V, E \\} \\newline \\end{aligned} \\]\u5982\u679c\u5c06\u9876\u70b9\u770b\u4f5c\u8282\u70b9\uff0c\u5c06\u8fb9\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u56fe\u770b\u4f5c\u4e00\u79cd\u4ece\u94fe\u8868\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u5982\u56fe 9-1 \u6240\u793a\uff0c\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002
\u56fe 9-1 \u00a0 \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb
"},{"location":"chapter_graph/graph/#911","title":"9.1.1 \u00a0 \u56fe\u5e38\u89c1\u7c7b\u578b\u4e0e\u672f\u8bed","text":"\u6839\u636e\u8fb9\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u4e3a\u300c\u65e0\u5411\u56fe undirected graph\u300d\u548c\u300c\u6709\u5411\u56fe directed graph\u300d\uff0c\u5982\u56fe 9-2 \u6240\u793a\u3002
- \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u7684\u201c\u53cc\u5411\u201d\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\u3002
- \u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\u3002
\u56fe 9-2 \u00a0 \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe
\u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u53ef\u5206\u4e3a\u300c\u8fde\u901a\u56fe connected graph\u300d\u548c\u300c\u975e\u8fde\u901a\u56fe disconnected graph\u300d\uff0c\u5982\u56fe 9-3 \u6240\u793a\u3002
- \u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\u3002
- \u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002
\u56fe 9-3 \u00a0 \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe
\u6211\u4eec\u8fd8\u53ef\u4ee5\u4e3a\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u4ece\u800c\u5f97\u5230\u5982\u56fe 9-4 \u6240\u793a\u7684\u300c\u6709\u6743\u56fe weighted graph\u300d\u3002\u4f8b\u5982\u5728\u201c\u738b\u8005\u8363\u8000\u201d\u7b49\u624b\u6e38\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002
\u56fe 9-4 \u00a0 \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe
\u56fe\u6570\u636e\u7ed3\u6784\u5305\u542b\u4ee5\u4e0b\u5e38\u7528\u672f\u8bed\u3002
- \u300c\u90bb\u63a5 adjacency\u300d\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u5b58\u5728\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u8fd9\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2\u30013\u30015\u3002
- \u300c\u8def\u5f84 path\u300d\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u7ecf\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u6761\u8def\u5f84\u3002
- \u300c\u5ea6 degree\u300d\uff1a\u4e00\u4e2a\u9876\u70b9\u62e5\u6709\u7684\u8fb9\u6570\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u300c\u5165\u5ea6 in-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u300c\u51fa\u5ea6 out-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002
\u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u5f0f\u5305\u62ec\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u3002\u4ee5\u4e0b\u4f7f\u7528\u65e0\u5411\u56fe\u8fdb\u884c\u4e3e\u4f8b\u3002
"},{"location":"chapter_graph/graph/#1","title":"1. \u00a0 \u90bb\u63a5\u77e9\u9635","text":"\u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u300c\u90bb\u63a5\u77e9\u9635 adjacency matrix\u300d\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u662f\u5426\u5b58\u5728\u8fb9\u3002
\u5982\u56fe 9-5 \u6240\u793a\uff0c\u8bbe\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\)\u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u90a3\u4e48\u77e9\u9635\u5143\u7d20 \\(M[i, j] = 1\\) \u8868\u793a\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u5b58\u5728\u8fb9\uff0c\u53cd\u4e4b \\(M[i, j] = 0\\) \u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002
\u56fe 9-5 \u00a0 \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a
\u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002
- \u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u6b64\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002
- \u5bf9\u4e8e\u65e0\u5411\u56fe\uff0c\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002
- \u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) \u548c \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u53ef\u8868\u793a\u6709\u6743\u56fe\u3002
\u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u4ee5\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u591a\u3002
"},{"location":"chapter_graph/graph/#2","title":"2. \u00a0 \u90bb\u63a5\u8868","text":"\u300c\u90bb\u63a5\u8868 adjacency list\u300d\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u8282\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002\u56fe 9-6 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f7f\u7528\u90bb\u63a5\u8868\u5b58\u50a8\u7684\u56fe\u7684\u793a\u4f8b\u3002
\u56fe 9-6 \u00a0 \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a
\u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b9e\u9645\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u901a\u5e38\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u7136\u800c\uff0c\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002
\u89c2\u5bdf\u56fe 9-6 \uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u4e2d\u7684\u201c\u94fe\u5f0f\u5730\u5740\u201d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u91c7\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u6bd4\u5982\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff1b\u8fd8\u53ef\u4ee5\u628a\u94fe\u8868\u8f6c\u6362\u4e3a\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u81f3 \\(O(1)\\) \u3002
"},{"location":"chapter_graph/graph/#913","title":"9.1.3 \u00a0 \u56fe\u5e38\u89c1\u5e94\u7528","text":"\u5982\u8868 9-1 \u6240\u793a\uff0c\u8bb8\u591a\u73b0\u5b9e\u7cfb\u7edf\u53ef\u4ee5\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u76f8\u5e94\u7684\u95ee\u9898\u4e5f\u53ef\u4ee5\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002
\u8868 9-1 \u00a0 \u73b0\u5b9e\u751f\u6d3b\u4e2d\u5e38\u89c1\u7684\u56fe
\u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c","text":"\u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u53ef\u5206\u4e3a\u5bf9\u201c\u8fb9\u201d\u7684\u64cd\u4f5c\u548c\u5bf9\u201c\u9876\u70b9\u201d\u7684\u64cd\u4f5c\u3002\u5728\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u4e24\u79cd\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5b9e\u73b0\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002
"},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"\u7ed9\u5b9a\u4e00\u4e2a\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \u7684\u65e0\u5411\u56fe\uff0c\u5219\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u5f0f\u5982\u56fe 9-7 \u6240\u793a\u3002
- \u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
- \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
- \u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
- \u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868
vertices\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635adjMat\uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
\u56fe 9-7 \u00a0 \u90bb\u63a5\u77e9\u9635\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9
\u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_adjacency_matrix.py
graph_adjacency_matrix.cppclass GraphAdjMat:\n \"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n def __init__(self, vertices: list[int], edges: list[list[int]]):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n # \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n self.vertices: list[int] = []\n # \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n self.adj_mat: list[list[int]] = []\n # \u6dfb\u52a0\u9876\u70b9\n for val in vertices:\n self.add_vertex(val)\n # \u6dfb\u52a0\u8fb9\n # \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for e in edges:\n self.add_edge(e[0], e[1])\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n return len(self.vertices)\n\n def add_vertex(self, val: int):\n \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n n = self.size()\n # \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n self.vertices.append(val)\n # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n new_row = [0] * n\n self.adj_mat.append(new_row)\n # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for row in self.adj_mat:\n row.append(0)\n\n def remove_vertex(self, index: int):\n \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n if index >= self.size():\n raise IndexError()\n # \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n self.vertices.pop(index)\n # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n self.adj_mat.pop(index)\n # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for row in self.adj_mat:\n row.pop(index)\n\n def add_edge(self, i: int, j: int):\n \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\n raise IndexError()\n # \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n self.adj_mat[i][j] = 1\n self.adj_mat[j][i] = 1\n\n def remove_edge(self, i: int, j: int):\n \"\"\"\u5220\u9664\u8fb9\"\"\"\n # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\n raise IndexError()\n self.adj_mat[i][j] = 0\n self.adj_mat[j][i] = 0\n\n def print(self):\n \"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\n print(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\n print(\"\u90bb\u63a5\u77e9\u9635 =\")\n print_matrix(self.adj_mat)\n
graph_adjacency_matrix.java/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n vector<int> vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n vector<vector<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n GraphAdjMat(const vector<int> &vertices, const vector<vector<int>> &edges) {\n // \u6dfb\u52a0\u9876\u70b9\n for (int val : vertices) {\n addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (const vector<int> &edge : edges) {\n addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int size() const {\n return vertices.size();\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n void addVertex(int val) {\n int n = size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.push_back(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n adjMat.emplace_back(vector<int>(n, 0));\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (vector<int> &row : adjMat) {\n row.push_back(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n void removeVertex(int index) {\n if (index >= size()) {\n throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.erase(vertices.begin() + index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.erase(adjMat.begin() + index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (vector<int> &row : adjMat) {\n row.erase(row.begin() + index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n void addEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\n throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat[i][j] = 1;\n adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n void removeEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\n throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n }\n adjMat[i][j] = 0;\n adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n void print() {\n cout << \"\u9876\u70b9\u5217\u8868 = \";\n printVector(vertices);\n cout << \"\u90bb\u63a5\u77e9\u9635 =\" << endl;\n printVectorMatrix(adjMat);\n }\n};\n
graph_adjacency_matrix.cs/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n List<Integer> vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n List<List<Integer>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public GraphAdjMat(int[] vertices, int[][] edges) {\n this.vertices = new ArrayList<>();\n this.adjMat = new ArrayList<>();\n // \u6dfb\u52a0\u9876\u70b9\n for (int val : vertices) {\n addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (int[] e : edges) {\n addEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n public int size() {\n return vertices.size();\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public void addVertex(int val) {\n int n = size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.add(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n List<Integer> newRow = new ArrayList<>(n);\n for (int j = 0; j < n; j++) {\n newRow.add(0);\n }\n adjMat.add(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (List<Integer> row : adjMat) {\n row.add(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public void removeVertex(int index) {\n if (index >= size())\n throw new IndexOutOfBoundsException();\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.remove(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.remove(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (List<Integer> row : adjMat) {\n row.remove(index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n public void addEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\n throw new IndexOutOfBoundsException();\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat.get(i).set(j, 1);\n adjMat.get(j).set(i, 1);\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n public void removeEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\n throw new IndexOutOfBoundsException();\n adjMat.get(i).set(j, 0);\n adjMat.get(j).set(i, 0);\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n public void print() {\n System.out.print(\"\u9876\u70b9\u5217\u8868 = \");\n System.out.println(vertices);\n System.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\n PrintUtil.printMatrix(adjMat);\n }\n}\n
graph_adjacency_matrix.go/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n List<int> vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n List<List<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u51fd\u6570 */\n public GraphAdjMat(int[] vertices, int[][] edges) {\n this.vertices = [];\n this.adjMat = [];\n // \u6dfb\u52a0\u9876\u70b9\n foreach (int val in vertices) {\n AddVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n foreach (int[] e in edges) {\n AddEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int Size() {\n return vertices.Count;\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public void AddVertex(int val) {\n int n = Size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.Add(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n List<int> newRow = new(n);\n for (int j = 0; j < n; j++) {\n newRow.Add(0);\n }\n adjMat.Add(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n foreach (List<int> row in adjMat) {\n row.Add(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public void RemoveVertex(int index) {\n if (index >= Size())\n throw new IndexOutOfRangeException();\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.RemoveAt(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.RemoveAt(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n foreach (List<int> row in adjMat) {\n row.RemoveAt(index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n public void AddEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= Size() || j >= Size() || i == j)\n throw new IndexOutOfRangeException();\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat[i][j] = 1;\n adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n public void RemoveEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= Size() || j >= Size() || i == j)\n throw new IndexOutOfRangeException();\n adjMat[i][j] = 0;\n adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n public void Print() {\n Console.Write(\"\u9876\u70b9\u5217\u8868 = \");\n PrintUtil.PrintList(vertices);\n Console.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\n PrintUtil.PrintMatrix(adjMat);\n }\n}\n
graph_adjacency_matrix.swift/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n vertices []int\n // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n adjMat [][]int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n // \u6dfb\u52a0\u9876\u70b9\n n := len(vertices)\n adjMat := make([][]int, n)\n for i := range adjMat {\n adjMat[i] = make([]int, n)\n }\n // \u521d\u59cb\u5316\u56fe\n g := &graphAdjMat{\n vertices: vertices,\n adjMat: adjMat,\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for i := range edges {\n g.addEdge(edges[i][0], edges[i][1])\n }\n return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\n return len(g.vertices)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\n n := g.size()\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n g.vertices = append(g.vertices, val)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n newRow := make([]int, n)\n g.adjMat = append(g.adjMat, newRow)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for i := range g.adjMat {\n g.adjMat[i] = append(g.adjMat[i], 0)\n }\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\n if index >= g.size() {\n return\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for i := range g.adjMat {\n g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n }\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\n fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n g.adjMat[i][j] = 1\n g.adjMat[j][i] = 1\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\n fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n }\n g.adjMat[i][j] = 0\n g.adjMat[j][i] = 0\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\n fmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\n fmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\n for i := range g.adjMat {\n fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n }\n}\n
graph_adjacency_matrix.js/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n private var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n private var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init(vertices: [Int], edges: [[Int]]) {\n self.vertices = []\n adjMat = []\n // \u6dfb\u52a0\u9876\u70b9\n for val in vertices {\n addVertex(val: val)\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for e in edges {\n addEdge(i: e[0], j: e[1])\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n func size() -> Int {\n vertices.count\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n func addVertex(val: Int) {\n let n = size()\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.append(val)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n let newRow = Array(repeating: 0, count: n)\n adjMat.append(newRow)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for i in adjMat.indices {\n adjMat[i].append(0)\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n func removeVertex(index: Int) {\n if index >= size() {\n fatalError(\"\u8d8a\u754c\")\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.remove(at: index)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.remove(at: index)\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for i in adjMat.indices {\n adjMat[i].remove(at: index)\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n func addEdge(i: Int, j: Int) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 || j < 0 || i >= size() || j >= size() || i == j {\n fatalError(\"\u8d8a\u754c\")\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat[i][j] = 1\n adjMat[j][i] = 1\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n func removeEdge(i: Int, j: Int) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i < 0 || j < 0 || i >= size() || j >= size() || i == j {\n fatalError(\"\u8d8a\u754c\")\n }\n adjMat[i][j] = 0\n adjMat[j][i] = 0\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n func print() {\n Swift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\n Swift.print(vertices)\n Swift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\n PrintUtil.printMatrix(matrix: adjMat)\n }\n}\n
graph_adjacency_matrix.ts/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u51fd\u6570 */\n constructor(vertices, edges) {\n this.vertices = [];\n this.adjMat = [];\n // \u6dfb\u52a0\u9876\u70b9\n for (const val of vertices) {\n this.addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (const e of edges) {\n this.addEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n size() {\n return this.vertices.length;\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n addVertex(val) {\n const n = this.size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n this.vertices.push(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n const newRow = [];\n for (let j = 0; j < n; j++) {\n newRow.push(0);\n }\n this.adjMat.push(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (const row of this.adjMat) {\n row.push(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n removeVertex(index) {\n if (index >= this.size()) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n this.vertices.splice(index, 1);\n\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n this.adjMat.splice(index, 1);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (const row of this.adjMat) {\n row.splice(index, 1);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n addEdge(i, j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n this.adjMat[i][j] = 1;\n this.adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n removeEdge(i, j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n this.adjMat[i][j] = 0;\n this.adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n print() {\n console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n }\n}\n
graph_adjacency_matrix.dart/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n vertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n adjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u51fd\u6570 */\n constructor(vertices: number[], edges: number[][]) {\n this.vertices = [];\n this.adjMat = [];\n // \u6dfb\u52a0\u9876\u70b9\n for (const val of vertices) {\n this.addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (const e of edges) {\n this.addEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n size(): number {\n return this.vertices.length;\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n addVertex(val: number): void {\n const n: number = this.size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n this.vertices.push(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n const newRow: number[] = [];\n for (let j: number = 0; j < n; j++) {\n newRow.push(0);\n }\n this.adjMat.push(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (const row of this.adjMat) {\n row.push(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n removeVertex(index: number): void {\n if (index >= this.size()) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n this.vertices.splice(index, 1);\n\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n this.adjMat.splice(index, 1);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (const row of this.adjMat) {\n row.splice(index, 1);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n addEdge(i: number, j: number): void {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n this.adjMat[i][j] = 1;\n this.adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n removeEdge(i: number, j: number): void {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\n throw new RangeError('Index Out Of Bounds Exception');\n }\n this.adjMat[i][j] = 0;\n this.adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n print(): void {\n console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n }\n}\n
graph_adjacency_matrix.rs/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n List<int> vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n List<List<int>> adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n /* \u6784\u9020\u65b9\u6cd5 */\n GraphAdjMat(List<int> vertices, List<List<int>> edges) {\n this.vertices = [];\n this.adjMat = [];\n // \u6dfb\u52a0\u9876\u70b9\n for (int val in vertices) {\n addVertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for (List<int> e in edges) {\n addEdge(e[0], e[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int size() {\n return vertices.length;\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n void addVertex(int val) {\n int n = size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n vertices.add(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n List<int> newRow = List.filled(n, 0, growable: true);\n adjMat.add(newRow);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for (List<int> row in adjMat) {\n row.add(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n void removeVertex(int index) {\n if (index >= size()) {\n throw IndexError;\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n vertices.removeAt(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n adjMat.removeAt(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (List<int> row in adjMat) {\n row.removeAt(index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n void addEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\n throw IndexError;\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n adjMat[i][j] = 1;\n adjMat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n void removeEdge(int i, int j) {\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\n throw IndexError;\n }\n adjMat[i][j] = 0;\n adjMat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n void printAdjMat() {\n print(\"\u9876\u70b9\u5217\u8868 = $vertices\");\n print(\"\u90bb\u63a5\u77e9\u9635 = \");\n printMatrix(adjMat);\n }\n}\n
graph_adjacency_matrix.c/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjMat {\n // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n pub vertices: Vec<i32>,\n // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n pub adj_mat: Vec<Vec<i32>>,\n}\n\nimpl GraphAdjMat {\n /* \u6784\u9020\u65b9\u6cd5 */\n pub fn new(vertices: Vec<i32>, edges: Vec<[usize; 2]>) -> Self {\n let mut graph = GraphAdjMat {\n vertices: vec![],\n adj_mat: vec![],\n };\n // \u6dfb\u52a0\u9876\u70b9\n for val in vertices {\n graph.add_vertex(val);\n }\n // \u6dfb\u52a0\u8fb9\n // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n for edge in edges {\n graph.add_edge(edge[0], edge[1])\n }\n\n graph\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n pub fn size(&self) -> usize {\n self.vertices.len()\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n pub fn add_vertex(&mut self, val: i32) {\n let n = self.size();\n // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n self.vertices.push(val);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n self.adj_mat.push(vec![0; n]);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n for row in &mut self.adj_mat {\n row.push(0);\n }\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n pub fn remove_vertex(&mut self, index: usize) {\n if index >= self.size() {\n panic!(\"index error\")\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n self.vertices.remove(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n self.adj_mat.remove(index);\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for row in &mut self.adj_mat {\n row.remove(index);\n }\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n pub fn add_edge(&mut self, i: usize, j: usize) {\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i >= self.size() || j >= self.size() || i == j {\n panic!(\"index error\")\n }\n // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n self.adj_mat[i][j] = 1;\n self.adj_mat[j][i] = 1;\n }\n\n /* \u5220\u9664\u8fb9 */\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n pub fn remove_edge(&mut self, i: usize, j: usize) {\n // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n if i >= self.size() || j >= self.size() || i == j {\n panic!(\"index error\")\n }\n self.adj_mat[i][j] = 0;\n self.adj_mat[j][i] = 0;\n }\n\n /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n pub fn print(&self) {\n println!(\"\u9876\u70b9\u5217\u8868 = {:?}\", self.vertices);\n println!(\"\u90bb\u63a5\u77e9\u9635 =\");\n println!(\"[\");\n for row in &self.adj_mat {\n println!(\" {:?},\", row);\n }\n println!(\"]\")\n }\n}\n
graph_adjacency_matrix.zig/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7ed3\u6784\u4f53 */\ntypedef struct {\n int vertices[MAX_SIZE];\n int adjMat[MAX_SIZE][MAX_SIZE];\n int size;\n} GraphAdjMat;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjMat *newGraphAdjMat() {\n GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n graph->size = 0;\n for (int i = 0; i < MAX_SIZE; i++) {\n for (int j = 0; j < MAX_SIZE; j++) {\n graph->adjMat[i][j] = 0;\n }\n }\n return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n free(graph);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjMat *graph, int val) {\n if (graph->size == MAX_SIZE) {\n fprintf(stderr, \"\u56fe\u7684\u9876\u70b9\u6570\u91cf\u5df2\u8fbe\u6700\u5927\u503c\\n\");\n return;\n }\n // \u6dfb\u52a0\u7b2c n \u4e2a\u9876\u70b9\uff0c\u5e76\u5c06\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\n int n = graph->size;\n graph->vertices[n] = val;\n for (int i = 0; i <= n; i++) {\n graph->adjMat[n][i] = graph->adjMat[i][n] = 0;\n }\n graph->size++;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n if (index < 0 || index >= graph->size) {\n fprintf(stderr, \"\u9876\u70b9\u7d22\u5f15\u8d8a\u754c\\n\");\n return;\n }\n // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n for (int i = index; i < graph->size - 1; i++) {\n graph->vertices[i] = graph->vertices[i + 1];\n }\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n for (int i = index; i < graph->size - 1; i++) {\n for (int j = 0; j < graph->size; j++) {\n graph->adjMat[i][j] = graph->adjMat[i + 1][j];\n }\n }\n // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n for (int i = 0; i < graph->size; i++) {\n for (int j = index; j < graph->size - 1; j++) {\n graph->adjMat[i][j] = graph->adjMat[i][j + 1];\n }\n }\n graph->size--;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n if (i < 0 || j < 0 || i >= graph->size || j >= graph->size || i == j) {\n fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n return;\n }\n graph->adjMat[i][j] = 1;\n graph->adjMat[j][i] = 1;\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n if (i < 0 || j < 0 || i >= graph->size || j >= graph->size || i == j) {\n fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n return;\n }\n graph->adjMat[i][j] = 0;\n graph->adjMat[j][i] = 0;\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n printf(\"\u9876\u70b9\u5217\u8868 = \");\n printArray(graph->vertices, graph->size);\n printf(\"\u90bb\u63a5\u77e9\u9635 =\\n\");\n for (int i = 0; i < graph->size; i++) {\n printArray(graph->adjMat[i], graph->size);\n }\n}\n
"},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"[class]{GraphAdjMat}-[func]{}\n\u8bbe\u65e0\u5411\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\)\u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u53ef\u6839\u636e\u56fe 9-8 \u6240\u793a\u7684\u65b9\u6cd5\u5b9e\u73b0\u5404\u79cd\u64cd\u4f5c\u3002
- \u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u672b\u5c3e\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
- \u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u627e\u5e76\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
- \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\uff0c\u5e76\u5c06\u65b0\u589e\u9876\u70b9\u4f5c\u4e3a\u94fe\u8868\u5934\u8282\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
- \u5220\u9664\u9876\u70b9\uff1a\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
- \u521d\u59cb\u5316\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u521b\u5efa \\(n\\) \u4e2a\u9876\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
\u56fe 9-8 \u00a0 \u90bb\u63a5\u8868\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9
\u4ee5\u4e0b\u662f\u90bb\u63a5\u8868\u7684\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u6bd4\u56fe 9-8 \uff0c\u5b9e\u9645\u4ee3\u7801\u6709\u4ee5\u4e0b\u4e0d\u540c\u3002
- \u4e3a\u4e86\u65b9\u4fbf\u6dfb\u52a0\u4e0e\u5220\u9664\u9876\u70b9\uff0c\u4ee5\u53ca\u7b80\u5316\u4ee3\u7801\uff0c\u6211\u4eec\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u6765\u4ee3\u66ff\u94fe\u8868\u3002
- \u4f7f\u7528\u54c8\u5e0c\u8868\u6765\u5b58\u50a8\u90bb\u63a5\u8868\uff0c
key\u4e3a\u9876\u70b9\u5b9e\u4f8b\uff0cvalue\u4e3a\u8be5\u9876\u70b9\u7684\u90bb\u63a5\u9876\u70b9\u5217\u8868\uff08\u94fe\u8868\uff09\u3002
\u53e6\u5916\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528
PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_adjacency_list.pyVertex\u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u662f\uff1a\u5982\u679c\u4e0e\u90bb\u63a5\u77e9\u9635\u4e00\u6837\uff0c\u7528\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u5047\u8bbe\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u6240\u6709\u5927\u4e8e \\(i\\) \u7684\u7d22\u5f15\u5168\u90e8\u51cf \\(1\\) \uff0c\u6548\u7387\u5f88\u4f4e\u3002\u800c\u5982\u679c\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684Vertex\u5b9e\u4f8b\uff0c\u5220\u9664\u67d0\u4e00\u9876\u70b9\u4e4b\u540e\u5c31\u65e0\u987b\u6539\u52a8\u5176\u4ed6\u9876\u70b9\u4e86\u3002
graph_adjacency_list.cppclass GraphAdjList:\n \"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n def __init__(self, edges: list[list[Vertex]]):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n # \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n self.adj_list = dict[Vertex, list[Vertex]]()\n # \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for edge in edges:\n self.add_vertex(edge[0])\n self.add_vertex(edge[1])\n self.add_edge(edge[0], edge[1])\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n return len(self.adj_list)\n\n def add_edge(self, vet1: Vertex, vet2: Vertex):\n \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n raise ValueError()\n # \u6dfb\u52a0\u8fb9 vet1 - vet2\n self.adj_list[vet1].append(vet2)\n self.adj_list[vet2].append(vet1)\n\n def remove_edge(self, vet1: Vertex, vet2: Vertex):\n \"\"\"\u5220\u9664\u8fb9\"\"\"\n if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n raise ValueError()\n # \u5220\u9664\u8fb9 vet1 - vet2\n self.adj_list[vet1].remove(vet2)\n self.adj_list[vet2].remove(vet1)\n\n def add_vertex(self, vet: Vertex):\n \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n if vet in self.adj_list:\n return\n # \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n self.adj_list[vet] = []\n\n def remove_vertex(self, vet: Vertex):\n \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n if vet not in self.adj_list:\n raise ValueError()\n # \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n self.adj_list.pop(vet)\n # \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for vertex in self.adj_list:\n if vet in self.adj_list[vertex]:\n self.adj_list[vertex].remove(vet)\n\n def print(self):\n \"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\n print(\"\u90bb\u63a5\u8868 =\")\n for vertex in self.adj_list:\n tmp = [v.val for v in self.adj_list[vertex]]\n print(f\"{vertex.val}: {tmp},\")\n
graph_adjacency_list.java/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n public:\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n unordered_map<Vertex *, vector<Vertex *>> adjList;\n\n /* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\n void remove(vector<Vertex *> &vec, Vertex *vet) {\n for (int i = 0; i < vec.size(); i++) {\n if (vec[i] == vet) {\n vec.erase(vec.begin() + i);\n break;\n }\n }\n }\n\n /* \u6784\u9020\u65b9\u6cd5 */\n GraphAdjList(const vector<vector<Vertex *>> &edges) {\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for (const vector<Vertex *> &edge : edges) {\n addVertex(edge[0]);\n addVertex(edge[1]);\n addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int size() {\n return adjList.size();\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n void addEdge(Vertex *vet1, Vertex *vet2) {\n if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList[vet1].push_back(vet2);\n adjList[vet2].push_back(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n void removeEdge(Vertex *vet1, Vertex *vet2) {\n if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n // \u5220\u9664\u8fb9 vet1 - vet2\n remove(adjList[vet1], vet2);\n remove(adjList[vet2], vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n void addVertex(Vertex *vet) {\n if (adjList.count(vet))\n return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList[vet] = vector<Vertex *>();\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n void removeVertex(Vertex *vet) {\n if (!adjList.count(vet))\n throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.erase(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (auto &adj : adjList) {\n remove(adj.second, vet);\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n void print() {\n cout << \"\u90bb\u63a5\u8868 =\" << endl;\n for (auto &adj : adjList) {\n const auto &key = adj.first;\n const auto &vec = adj.second;\n cout << key->val << \": \";\n printVector(vetsToVals(vec));\n }\n }\n};\n
graph_adjacency_list.cs/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n Map<Vertex, List<Vertex>> adjList;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public GraphAdjList(Vertex[][] edges) {\n this.adjList = new HashMap<>();\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for (Vertex[] edge : edges) {\n addVertex(edge[0]);\n addVertex(edge[1]);\n addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n public int size() {\n return adjList.size();\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n public void addEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n throw new IllegalArgumentException();\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList.get(vet1).add(vet2);\n adjList.get(vet2).add(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n public void removeEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n throw new IllegalArgumentException();\n // \u5220\u9664\u8fb9 vet1 - vet2\n adjList.get(vet1).remove(vet2);\n adjList.get(vet2).remove(vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public void addVertex(Vertex vet) {\n if (adjList.containsKey(vet))\n return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList.put(vet, new ArrayList<>());\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public void removeVertex(Vertex vet) {\n if (!adjList.containsKey(vet))\n throw new IllegalArgumentException();\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.remove(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (List<Vertex> list : adjList.values()) {\n list.remove(vet);\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n public void print() {\n System.out.println(\"\u90bb\u63a5\u8868 =\");\n for (Map.Entry<Vertex, List<Vertex>> pair : adjList.entrySet()) {\n List<Integer> tmp = new ArrayList<>();\n for (Vertex vertex : pair.getValue())\n tmp.add(vertex.val);\n System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n }\n }\n}\n
graph_adjacency_list.go/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n public Dictionary<Vertex, List<Vertex>> adjList;\n\n /* \u6784\u9020\u51fd\u6570 */\n public GraphAdjList(Vertex[][] edges) {\n adjList = [];\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n foreach (Vertex[] edge in edges) {\n AddVertex(edge[0]);\n AddVertex(edge[1]);\n AddEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int Size() {\n return adjList.Count;\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n public void AddEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n throw new InvalidOperationException();\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList[vet1].Add(vet2);\n adjList[vet2].Add(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n public void RemoveEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n throw new InvalidOperationException();\n // \u5220\u9664\u8fb9 vet1 - vet2\n adjList[vet1].Remove(vet2);\n adjList[vet2].Remove(vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public void AddVertex(Vertex vet) {\n if (adjList.ContainsKey(vet))\n return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList.Add(vet, []);\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public void RemoveVertex(Vertex vet) {\n if (!adjList.ContainsKey(vet))\n throw new InvalidOperationException();\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.Remove(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n foreach (List<Vertex> list in adjList.Values) {\n list.Remove(vet);\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n public void Print() {\n Console.WriteLine(\"\u90bb\u63a5\u8868 =\");\n foreach (KeyValuePair<Vertex, List<Vertex>> pair in adjList) {\n List<int> tmp = [];\n foreach (Vertex vertex in pair.Value)\n tmp.Add(vertex.val);\n Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n }\n }\n}\n
graph_adjacency_list.swift/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n adjList map[Vertex][]Vertex\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n g := &graphAdjList{\n adjList: make(map[Vertex][]Vertex),\n }\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for _, edge := range edges {\n g.addVertex(edge[0])\n g.addVertex(edge[1])\n g.addEdge(edge[0], edge[1])\n }\n return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\n return len(g.adjList)\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n _, ok1 := g.adjList[vet1]\n _, ok2 := g.adjList[vet2]\n if !ok1 || !ok2 || vet1 == vet2 {\n panic(\"error\")\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\n g.adjList[vet1] = append(g.adjList[vet1], vet2)\n g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n _, ok1 := g.adjList[vet1]\n _, ok2 := g.adjList[vet2]\n if !ok1 || !ok2 || vet1 == vet2 {\n panic(\"error\")\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n _, ok := g.adjList[vet]\n if ok {\n return\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n _, ok := g.adjList[vet]\n if !ok {\n panic(\"error\")\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n delete(g.adjList, vet)\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for v, list := range g.adjList {\n g.adjList[v] = DeleteSliceElms(list, vet)\n }\n}\n\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\n var builder strings.Builder\n fmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\n for k, v := range g.adjList {\n builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n for _, vet := range v {\n builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n }\n fmt.Println(builder.String())\n builder.Reset()\n }\n}\n
graph_adjacency_list.js/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n public private(set) var adjList: [Vertex: [Vertex]]\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public init(edges: [[Vertex]]) {\n adjList = [:]\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for edge in edges {\n addVertex(vet: edge[0])\n addVertex(vet: edge[1])\n addEdge(vet1: edge[0], vet2: edge[1])\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n public func size() -> Int {\n adjList.count\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n public func addEdge(vet1: Vertex, vet2: Vertex) {\n if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n fatalError(\"\u53c2\u6570\u9519\u8bef\")\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList[vet1]?.append(vet2)\n adjList[vet2]?.append(vet1)\n }\n\n /* \u5220\u9664\u8fb9 */\n public func removeEdge(vet1: Vertex, vet2: Vertex) {\n if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n fatalError(\"\u53c2\u6570\u9519\u8bef\")\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n adjList[vet1]?.removeAll(where: { $0 == vet2 })\n adjList[vet2]?.removeAll(where: { $0 == vet1 })\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n public func addVertex(vet: Vertex) {\n if adjList[vet] != nil {\n return\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList[vet] = []\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n public func removeVertex(vet: Vertex) {\n if adjList[vet] == nil {\n fatalError(\"\u53c2\u6570\u9519\u8bef\")\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.removeValue(forKey: vet)\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for key in adjList.keys {\n adjList[key]?.removeAll(where: { $0 == vet })\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n public func print() {\n Swift.print(\"\u90bb\u63a5\u8868 =\")\n for pair in adjList {\n var tmp: [Int] = []\n for vertex in pair.value {\n tmp.append(vertex.val)\n }\n Swift.print(\"\\(pair.key.val): \\(tmp),\")\n }\n }\n}\n
graph_adjacency_list.ts/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n adjList;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(edges) {\n this.adjList = new Map();\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for (const edge of edges) {\n this.addVertex(edge[0]);\n this.addVertex(edge[1]);\n this.addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n size() {\n return this.adjList.size;\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n addEdge(vet1, vet2) {\n if (\n !this.adjList.has(vet1) ||\n !this.adjList.has(vet2) ||\n vet1 === vet2\n ) {\n throw new Error('Illegal Argument Exception');\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n this.adjList.get(vet1).push(vet2);\n this.adjList.get(vet2).push(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n removeEdge(vet1, vet2) {\n if (\n !this.adjList.has(vet1) ||\n !this.adjList.has(vet2) ||\n vet1 === vet2\n ) {\n throw new Error('Illegal Argument Exception');\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n addVertex(vet) {\n if (this.adjList.has(vet)) return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n this.adjList.set(vet, []);\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n removeVertex(vet) {\n if (!this.adjList.has(vet)) {\n throw new Error('Illegal Argument Exception');\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n this.adjList.delete(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (const set of this.adjList.values()) {\n const index = set.indexOf(vet);\n if (index > -1) {\n set.splice(index, 1);\n }\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n print() {\n console.log('\u90bb\u63a5\u8868 =');\n for (const [key, value] of this.adjList) {\n const tmp = [];\n for (const vertex of value) {\n tmp.push(vertex.val);\n }\n console.log(key.val + ': ' + tmp.join());\n }\n }\n}\n
graph_adjacency_list.dart/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n adjList: Map<Vertex, Vertex[]>;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(edges: Vertex[][]) {\n this.adjList = new Map();\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for (const edge of edges) {\n this.addVertex(edge[0]);\n this.addVertex(edge[1]);\n this.addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n size(): number {\n return this.adjList.size;\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n addEdge(vet1: Vertex, vet2: Vertex): void {\n if (\n !this.adjList.has(vet1) ||\n !this.adjList.has(vet2) ||\n vet1 === vet2\n ) {\n throw new Error('Illegal Argument Exception');\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n this.adjList.get(vet1).push(vet2);\n this.adjList.get(vet2).push(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n removeEdge(vet1: Vertex, vet2: Vertex): void {\n if (\n !this.adjList.has(vet1) ||\n !this.adjList.has(vet2) ||\n vet1 === vet2\n ) {\n throw new Error('Illegal Argument Exception');\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n addVertex(vet: Vertex): void {\n if (this.adjList.has(vet)) return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n this.adjList.set(vet, []);\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n removeVertex(vet: Vertex): void {\n if (!this.adjList.has(vet)) {\n throw new Error('Illegal Argument Exception');\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n this.adjList.delete(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (const set of this.adjList.values()) {\n const index: number = set.indexOf(vet);\n if (index > -1) {\n set.splice(index, 1);\n }\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n print(): void {\n console.log('\u90bb\u63a5\u8868 =');\n for (const [key, value] of this.adjList.entries()) {\n const tmp = [];\n for (const vertex of value) {\n tmp.push(vertex.val);\n }\n console.log(key.val + ': ' + tmp.join());\n }\n }\n}\n
graph_adjacency_list.rs/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n Map<Vertex, List<Vertex>> adjList = {};\n\n /* \u6784\u9020\u65b9\u6cd5 */\n GraphAdjList(List<List<Vertex>> edges) {\n for (List<Vertex> edge in edges) {\n addVertex(edge[0]);\n addVertex(edge[1]);\n addEdge(edge[0], edge[1]);\n }\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n int size() {\n return adjList.length;\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n void addEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.containsKey(vet1) ||\n !adjList.containsKey(vet2) ||\n vet1 == vet2) {\n throw ArgumentError;\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n adjList[vet1]!.add(vet2);\n adjList[vet2]!.add(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n void removeEdge(Vertex vet1, Vertex vet2) {\n if (!adjList.containsKey(vet1) ||\n !adjList.containsKey(vet2) ||\n vet1 == vet2) {\n throw ArgumentError;\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n adjList[vet1]!.remove(vet2);\n adjList[vet2]!.remove(vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n void addVertex(Vertex vet) {\n if (adjList.containsKey(vet)) return;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n adjList[vet] = [];\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n void removeVertex(Vertex vet) {\n if (!adjList.containsKey(vet)) {\n throw ArgumentError;\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n adjList.remove(vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n adjList.forEach((key, value) {\n value.remove(vet);\n });\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n void printAdjList() {\n print(\"\u90bb\u63a5\u8868 =\");\n adjList.forEach((key, value) {\n List<int> tmp = [];\n for (Vertex vertex in value) {\n tmp.add(vertex.val);\n }\n print(\"${key.val}: $tmp,\");\n });\n }\n}\n
graph_adjacency_list.c/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjList {\n // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n pub adj_list: HashMap<Vertex, Vec<Vertex>>,\n}\n\nimpl GraphAdjList {\n /* \u6784\u9020\u65b9\u6cd5 */\n pub fn new(edges: Vec<[Vertex; 2]>) -> Self {\n let mut graph = GraphAdjList {\n adj_list: HashMap::new(),\n };\n // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n for edge in edges {\n graph.add_vertex(edge[0]);\n graph.add_vertex(edge[1]);\n graph.add_edge(edge[0], edge[1]);\n }\n\n graph\n }\n\n /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n #[allow(unused)]\n pub fn size(&self) -> usize {\n self.adj_list.len()\n }\n\n /* \u6dfb\u52a0\u8fb9 */\n pub fn add_edge(&mut self, vet1: Vertex, vet2: Vertex) {\n if !self.adj_list.contains_key(&vet1) || !self.adj_list.contains_key(&vet2) || vet1 == vet2\n {\n panic!(\"value error\");\n }\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n self.adj_list.get_mut(&vet1).unwrap().push(vet2);\n self.adj_list.get_mut(&vet2).unwrap().push(vet1);\n }\n\n /* \u5220\u9664\u8fb9 */\n #[allow(unused)]\n pub fn remove_edge(&mut self, vet1: Vertex, vet2: Vertex) {\n if !self.adj_list.contains_key(&vet1) || !self.adj_list.contains_key(&vet2) || vet1 == vet2\n {\n panic!(\"value error\");\n }\n // \u5220\u9664\u8fb9 vet1 - vet2\n self.adj_list\n .get_mut(&vet1)\n .unwrap()\n .retain(|&vet| vet != vet2);\n self.adj_list\n .get_mut(&vet2)\n .unwrap()\n .retain(|&vet| vet != vet1);\n }\n\n /* \u6dfb\u52a0\u9876\u70b9 */\n pub fn add_vertex(&mut self, vet: Vertex) {\n if self.adj_list.contains_key(&vet) {\n return;\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n self.adj_list.insert(vet, vec![]);\n }\n\n /* \u5220\u9664\u9876\u70b9 */\n #[allow(unused)]\n pub fn remove_vertex(&mut self, vet: Vertex) {\n if !self.adj_list.contains_key(&vet) {\n panic!(\"value error\");\n }\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n self.adj_list.remove(&vet);\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for list in self.adj_list.values_mut() {\n list.retain(|&v| v != vet);\n }\n }\n\n /* \u6253\u5370\u90bb\u63a5\u8868 */\n pub fn print(&self) {\n println!(\"\u90bb\u63a5\u8868 =\");\n for (vertex, list) in &self.adj_list {\n let list = list.iter().map(|vertex| vertex.val).collect::<Vec<i32>>();\n println!(\"{}: {:?},\", vertex.val, list);\n }\n }\n}\n
graph_adjacency_list.zig/* \u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct AdjListNode {\n Vertex *vertex; // \u9876\u70b9\n struct AdjListNode *next; // \u540e\u7ee7\u8282\u70b9\n} AdjListNode;\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n for (int i = 0; i < graph->size; i++) {\n if (graph->heads[i]->vertex == vet) {\n return graph->heads[i];\n }\n }\n return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n node->vertex = vet;\n // \u5934\u63d2\u6cd5\n node->next = head->next;\n head->next = node;\n}\n\n/* \u5220\u9664\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n AdjListNode *pre = head;\n AdjListNode *cur = head->next;\n // \u5728\u94fe\u8868\u4e2d\u641c\u7d22 vet \u5bf9\u5e94\u8282\u70b9\n while (cur != NULL && cur->vertex != vet) {\n pre = cur;\n cur = cur->next;\n }\n if (cur == NULL)\n return;\n // \u5c06 vet \u5bf9\u5e94\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\n pre->next = cur->next;\n // \u91ca\u653e\u5185\u5b58\n free(cur);\n}\n\n/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntypedef struct {\n AdjListNode *heads[MAX_SIZE]; // \u8282\u70b9\u6570\u7ec4\n int size; // \u8282\u70b9\u6570\u91cf\n} GraphAdjList;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjList *newGraphAdjList() {\n GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n if (!graph) {\n return NULL;\n }\n graph->size = 0;\n for (int i = 0; i < MAX_SIZE; i++) {\n graph->heads[i] = NULL;\n }\n return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjList(GraphAdjList *graph) {\n for (int i = 0; i < graph->size; i++) {\n AdjListNode *cur = graph->heads[i];\n while (cur != NULL) {\n AdjListNode *next = cur->next;\n if (cur != graph->heads[i]) {\n free(cur);\n }\n cur = next;\n }\n free(graph->heads[i]->vertex);\n free(graph->heads[i]);\n }\n free(graph);\n}\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n for (int i = 0; i < graph->size; i++) {\n if (graph->heads[i]->vertex == vet) {\n return graph->heads[i];\n }\n }\n return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n AdjListNode *head1 = findNode(graph, vet1);\n AdjListNode *head2 = findNode(graph, vet2);\n assert(head1 != NULL && head2 != NULL && head1 != head2);\n // \u6dfb\u52a0\u8fb9 vet1 - vet2\n addEdgeHelper(head1, vet2);\n addEdgeHelper(head2, vet1);\n}\n\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n AdjListNode *head1 = findNode(graph, vet1);\n AdjListNode *head2 = findNode(graph, vet2);\n assert(head1 != NULL && head2 != NULL);\n // \u5220\u9664\u8fb9 vet1 - vet2\n removeEdgeHelper(head1, head2->vertex);\n removeEdgeHelper(head2, head1->vertex);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n assert(graph != NULL && graph->size < MAX_SIZE);\n AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n head->vertex = vet;\n head->next = NULL;\n // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n graph->heads[graph->size++] = head;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n AdjListNode *node = findNode(graph, vet);\n assert(node != NULL);\n // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n AdjListNode *cur = node, *pre = NULL;\n while (cur) {\n pre = cur;\n cur = cur->next;\n free(pre);\n }\n // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n for (int i = 0; i < graph->size; i++) {\n cur = graph->heads[i];\n pre = NULL;\n while (cur) {\n pre = cur;\n cur = cur->next;\n if (cur && cur->vertex == vet) {\n pre->next = cur->next;\n free(cur);\n break;\n }\n }\n }\n // \u5c06\u8be5\u9876\u70b9\u4e4b\u540e\u7684\u9876\u70b9\u5411\u524d\u79fb\u52a8\uff0c\u4ee5\u586b\u8865\u7a7a\u7f3a\n int i;\n for (i = 0; i < graph->size; i++) {\n if (graph->heads[i] == node)\n break;\n }\n for (int j = i; j < graph->size - 1; j++) {\n graph->heads[j] = graph->heads[j + 1];\n }\n graph->size--;\n free(vet);\n}\n
"},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3 \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"[class]{GraphAdjList}-[func]{}\n\u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u8868 9-2 \u5bf9\u6bd4\u4e86\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u3002
\u8868 9-2 \u00a0 \u90bb\u63a5\u77e9\u9635\u4e0e\u90bb\u63a5\u8868\u5bf9\u6bd4
\u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\)\u89c2\u5bdf\u8868 9-2 \uff0c\u4f3c\u4e4e\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u6548\u7387\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7efc\u5408\u6765\u770b\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u539f\u5219\uff0c\u800c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u7684\u539f\u5219\u3002
"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 \u56fe\u7684\u904d\u5386","text":"\u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\u3002\u663e\u7136\uff0c\u6811\u7684\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u7684\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u79cd\u7279\u4f8b\u3002
\u56fe\u548c\u6811\u90fd\u9700\u8981\u5e94\u7528\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u53ef\u5206\u4e3a\u4e24\u79cd\uff1a\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal\u300d\u3002\u5b83\u4eec\u4e5f\u5e38\u88ab\u79f0\u4e3a\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 breadth-first search\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22 depth-first search\u300d\uff0c\u7b80\u79f0 BFS \u548c DFS \u3002
"},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1 \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u67d0\u4e2a\u8282\u70b9\u51fa\u53d1\uff0c\u59cb\u7ec8\u4f18\u5148\u8bbf\u95ee\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5982\u56fe 9-9 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u9996\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u7136\u540e\u904d\u5386\u4e0b\u4e00\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6bd5\u3002
\u56fe 9-9 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386
"},{"location":"chapter_graph/graph_traversal/#1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"BFS \u901a\u5e38\u501f\u52a9\u961f\u5217\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u7684\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u5f02\u66f2\u540c\u5de5\u3002
- \u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9
startVet\u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\u3002 - \u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u7136\u540e\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\u3002
- \u5faa\u73af\u6b65\u9aa4
2.\uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u88ab\u8bbf\u95ee\u5b8c\u6bd5\u540e\u7ed3\u675f\u3002
\u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_bfs.pyvisited\u6765\u8bb0\u5f55\u54ea\u4e9b\u8282\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002
graph_bfs.cppdef graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]:\n \"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS\"\"\"\n # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n res = []\n # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n visited = set[Vertex]([start_vet])\n # \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n que = deque[Vertex]([start_vet])\n # \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while len(que) > 0:\n vet = que.popleft() # \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.append(vet) # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for adj_vet in graph.adj_list[vet]:\n if adj_vet in visited:\n continue # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n que.append(adj_vet) # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.add(adj_vet) # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n # \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res\n
graph_bfs.java/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex *> graphBFS(GraphAdjList &graph, Vertex *startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n vector<Vertex *> res;\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n unordered_set<Vertex *> visited = {startVet};\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n queue<Vertex *> que;\n que.push(startVet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (!que.empty()) {\n Vertex *vet = que.front();\n que.pop(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (auto adjVet : graph.adjList[vet]) {\n if (visited.count(adjVet))\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.cs/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = new ArrayList<>();\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n Set<Vertex> visited = new HashSet<>();\n visited.add(startVet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n Queue<Vertex> que = new LinkedList<>();\n que.offer(startVet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (!que.isEmpty()) {\n Vertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex adjVet : graph.adjList.get(vet)) {\n if (visited.contains(adjVet))\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n que.offer(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.go/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> GraphBFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n HashSet<Vertex> visited = [startVet];\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n Queue<Vertex> que = new();\n que.Enqueue(startVet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (que.Count > 0) {\n Vertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.Add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n foreach (Vertex adjVet in graph.adjList[vet]) {\n if (visited.Contains(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.Enqueue(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.Add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.swift/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n res := make([]Vertex, 0)\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n visited := make(map[Vertex]struct{})\n visited[startVet] = struct{}{}\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\n queue := make([]Vertex, 0)\n queue = append(queue, startVet)\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n for len(queue) > 0 {\n // \u961f\u9996\u9876\u70b9\u51fa\u961f\n vet := queue[0]\n queue = queue[1:]\n // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n res = append(res, vet)\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for _, adjVet := range g.adjList[vet] {\n _, isExist := visited[adjVet]\n // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n if !isExist {\n queue = append(queue, adjVet)\n visited[adjVet] = struct{}{}\n }\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res\n}\n
graph_bfs.js/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n var res: [Vertex] = []\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n var visited: Set<Vertex> = [startVet]\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n var que: [Vertex] = [startVet]\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while !que.isEmpty {\n let vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for adjVet in graph.adjList[vet] ?? [] {\n if visited.contains(adjVet) {\n continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res\n}\n
graph_bfs.ts/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n const res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n const visited = new Set();\n visited.add(startVet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n const que = [startVet];\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (que.length) {\n const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (const adjVet of graph.adjList.get(vet) ?? []) {\n if (visited.has(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.dart/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n const res: Vertex[] = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n const visited: Set<Vertex> = new Set();\n visited.add(startVet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n const que = [startVet];\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (que.length) {\n const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (const adjVet of graph.adjList.get(vet) ?? []) {\n if (visited.has(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\n visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.rs/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n // \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n Set<Vertex> visited = {};\n visited.add(startVet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n Queue<Vertex> que = Queue();\n que.add(startVet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (que.isNotEmpty) {\n Vertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex adjVet in graph.adjList[vet]!) {\n if (visited.contains(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res;\n}\n
graph_bfs.c/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> Vec<Vertex> {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n let mut res = vec![];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n let mut visited = HashSet::new();\n visited.insert(start_vet);\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n let mut que = VecDeque::new();\n que.push_back(start_vet);\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while !que.is_empty() {\n let vet = que.pop_front().unwrap(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n if let Some(adj_vets) = graph.adj_list.get(&vet) {\n for &adj_vet in adj_vets {\n if visited.contains(&adj_vet) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n que.push_back(adj_vet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited.insert(adj_vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n }\n }\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n res\n}\n
graph_bfs.zig/* \u8282\u70b9\u961f\u5217\u7ed3\u6784\u4f53 */\ntypedef struct {\n Vertex *vertices[MAX_SIZE];\n int front, rear, size;\n} Queue;\n\n/* \u6784\u9020\u51fd\u6570 */\nQueue *newQueue() {\n Queue *q = (Queue *)malloc(sizeof(Queue));\n q->front = q->rear = q->size = 0;\n return q;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nint isEmpty(Queue *q) {\n return q->size == 0;\n}\n\n/* \u5165\u961f\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\n q->vertices[q->rear] = vet;\n q->rear = (q->rear + 1) % MAX_SIZE;\n q->size++;\n}\n\n/* \u51fa\u961f\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\n Vertex *vet = q->vertices[q->front];\n q->front = (q->front + 1) % MAX_SIZE;\n q->size--;\n return vet;\n}\n\n/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n for (int i = 0; i < size; i++) {\n if (visited[i] == vet)\n return 1;\n }\n return 0;\n}\n\n/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n Queue *queue = newQueue();\n enqueue(queue, startVet);\n visited[(*visitedSize)++] = startVet;\n // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n while (!isEmpty(queue)) {\n Vertex *vet = dequeue(queue); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n res[(*resSize)++] = vet; // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n AdjListNode *node = findNode(graph, vet);\n while (node != NULL) {\n // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n if (!isVisited(visited, *visitedSize, node->vertex)) {\n enqueue(queue, node->vertex); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n visited[(*visitedSize)++] = node->vertex; // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n }\n node = node->next;\n }\n }\n // \u91ca\u653e\u5185\u5b58\n free(queue);\n}\n[class]{}-[func]{graphBFS}\n\u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u56fe 9-10 \u6765\u52a0\u6df1\u7406\u89e3\u3002
<1><2><3><4><5><6><7><8><9><10><11>\u56fe 9-10 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4
\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f
\u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u987a\u5e8f\u904d\u5386\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u3002\u4ee5\u56fe 9-10 \u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\)\u3001\\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\uff0c\u9876\u70b9 \\(2\\)\u3001\\(4\\)\u3001\\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3002
"},{"location":"chapter_graph/graph_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u5e76\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002
\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868
"},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2 \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"res\uff0c\u54c8\u5e0c\u8868visited\uff0c\u961f\u5217que\u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5982\u56fe 9-11 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u5f53\u524d\u9876\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u8fd4\u56de\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5c3d\u5934\u5e76\u8fd4\u56de\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u3002
\u56fe 9-11 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386
"},{"location":"chapter_graph/graph_traversal/#1_1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"\u8fd9\u79cd\u201c\u8d70\u5230\u5c3d\u5934\u518d\u8fd4\u56de\u201d\u7684\u7b97\u6cd5\u8303\u5f0f\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u5728\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u4e2d\uff0c\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868
PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_dfs.pyvisited\u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002
graph_dfs.cppdef dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570\"\"\"\n res.append(vet) # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet) # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for adjVet in graph.adj_list[vet]:\n if adjVet in visited:\n continue # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n # \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]:\n \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS\"\"\"\n # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n res = []\n # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n visited = set[Vertex]()\n dfs(graph, visited, res, start_vet)\n return res\n
graph_dfs.java/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &graph, unordered_set<Vertex *> &visited, vector<Vertex *> &res, Vertex *vet) {\n res.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex *adjVet : graph.adjList[vet]) {\n if (visited.count(adjVet))\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex *> graphDFS(GraphAdjList &graph, Vertex *startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n vector<Vertex *> res;\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n unordered_set<Vertex *> visited;\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.cs/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set<Vertex> visited, List<Vertex> res, Vertex vet) {\n res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex adjVet : graph.adjList.get(vet)) {\n if (visited.contains(adjVet))\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = new ArrayList<>();\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n Set<Vertex> visited = new HashSet<>();\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.go/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid DFS(GraphAdjList graph, HashSet<Vertex> visited, List<Vertex> res, Vertex vet) {\n res.Add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n foreach (Vertex adjVet in graph.adjList[vet]) {\n if (visited.Contains(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9 \n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n DFS(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> GraphDFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n HashSet<Vertex> visited = [];\n DFS(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.swift/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n // append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n *res = append(*res, vet)\n visited[vet] = struct{}{}\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for _, adjVet := range g.adjList[vet] {\n _, isExist := visited[adjVet]\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n if !isExist {\n dfs(g, visited, res, adjVet)\n }\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n res := make([]Vertex, 0)\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n visited := make(map[Vertex]struct{})\n dfs(g, visited, &res, startVet)\n // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n return res\n}\n
graph_dfs.js/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set<Vertex>, res: inout [Vertex], vet: Vertex) {\n res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for adjVet in graph.adjList[vet] ?? [] {\n if visited.contains(adjVet) {\n continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph: graph, visited: &visited, res: &res, vet: adjVet)\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n var res: [Vertex] = []\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n var visited: Set<Vertex> = []\n dfs(graph: graph, visited: &visited, res: &res, vet: startVet)\n return res\n}\n
graph_dfs.ts/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (const adjVet of graph.adjList.get(vet)) {\n if (visited.has(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n const res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n const visited = new Set();\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.dart/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\n graph: GraphAdjList,\n visited: Set<Vertex>,\n res: Vertex[],\n vet: Vertex\n): void {\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (const adjVet of graph.adjList.get(vet)) {\n if (visited.has(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n const res: Vertex[] = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n const visited: Set<Vertex> = new Set();\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.rs/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\n GraphAdjList graph,\n Set<Vertex> visited,\n List<Vertex> res,\n Vertex vet,\n) {\n res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n for (Vertex adjVet in graph.adjList[vet]!) {\n if (visited.contains(adjVet)) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adjVet);\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n List<Vertex> res = [];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n Set<Vertex> visited = {};\n dfs(graph, visited, res, startVet);\n return res;\n}\n
graph_dfs.c/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfn dfs(graph: &GraphAdjList, visited: &mut HashSet<Vertex>, res: &mut Vec<Vertex>, vet: Vertex) {\n res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n visited.insert(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n if let Some(adj_vets) = graph.adj_list.get(&vet) {\n for &adj_vet in adj_vets {\n if visited.contains(&adj_vet) {\n continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n }\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, visited, res, adj_vet);\n }\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> Vec<Vertex> {\n // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n let mut res = vec![];\n // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n let mut visited = HashSet::new();\n dfs(&graph, &mut visited, &mut res, start_vet);\n\n res\n}\n
graph_dfs.zig/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n for (int i = 0; i < size; i++) {\n if (res[i] == vet) {\n return 1;\n }\n }\n return 0;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n res[(*resSize)++] = vet;\n // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n AdjListNode *node = findNode(graph, vet);\n while (node != NULL) {\n // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n if (!isVisited(res, *resSize, node->vertex)) {\n // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n dfs(graph, res, resSize, node->vertex);\n }\n node = node->next;\n }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n dfs(graph, res, resSize, startVet);\n}\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{graphDFS}\n\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 9-12 \u6240\u793a\u3002
- \u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u8868\u793a\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\u3002
- \u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u65b9\u6cd5\u7684\u4f4d\u7f6e\u3002
\u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8bae\u5c06\u56fe 9-12 \u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\u6a21\u62df\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002
<1><2><3><4><5><6><7><8><9><10><11>\u56fe 9-12 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4
\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f
\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u90bb\u63a5\u9876\u70b9\u7684\u987a\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e71\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002
\u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u5b83\u4eec\u5c55\u793a\u4e86\u4e09\u79cd\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u7136\u800c\u8fd9\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002
"},{"location":"chapter_graph/graph_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002
\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868
"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_graph/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"res\uff0c\u54c8\u5e0c\u8868visited\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002- \u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002
- \u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002
- \u6709\u5411\u56fe\u7684\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u5747\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002
- \u90bb\u63a5\u77e9\u9635\u5229\u7528\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u5728\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u95f4\u5360\u7528\u8f83\u591a\u3002
- \u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u4e8e\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002
- \u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002
- \u4ece\u7b97\u6cd5\u601d\u60f3\u7684\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002
- \u56fe\u53ef\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002
- \u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002
- \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002
- \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u65f6\u518d\u56de\u6eaf\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002
\u8def\u5f84\u7684\u5b9a\u4e49\u662f\u9876\u70b9\u5e8f\u5217\u8fd8\u662f\u8fb9\u5e8f\u5217\uff1f
\u7ef4\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8bed\u8a00\u7248\u672c\u7684\u5b9a\u4e49\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. \u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f84\u88ab\u89c6\u4e3a\u4e00\u4e2a\u8fb9\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u3002\u8fd9\u662f\u56e0\u4e3a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u591a\u6761\u8fb9\u8fde\u63a5\uff0c\u6b64\u65f6\u6bcf\u6761\u8fb9\u90fd\u5bf9\u5e94\u4e00\u6761\u8def\u5f84\u3002
\u975e\u8fde\u901a\u56fe\u4e2d\u662f\u5426\u4f1a\u6709\u65e0\u6cd5\u904d\u5386\u5230\u7684\u70b9\uff1f
\u5728\u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002\u904d\u5386\u975e\u8fde\u901a\u56fe\u9700\u8981\u8bbe\u7f6e\u591a\u4e2a\u8d77\u70b9\uff0c\u4ee5\u904d\u5386\u5230\u56fe\u7684\u6240\u6709\u8fde\u901a\u5206\u91cf\u3002
\u5728\u90bb\u63a5\u8868\u4e2d\uff0c\u201c\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u6240\u6709\u9876\u70b9\u201d\u7684\u9876\u70b9\u987a\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f
\u53ef\u4ee5\u662f\u4efb\u610f\u987a\u5e8f\u3002\u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u9700\u8981\u6309\u7167\u6307\u5b9a\u89c4\u5219\u6765\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9876\u70b9\u6dfb\u52a0\u7684\u6b21\u5e8f\uff0c\u6216\u8005\u6309\u7167\u9876\u70b9\u503c\u5927\u5c0f\u7684\u987a\u5e8f\u7b49\uff0c\u8fd9\u6837\u6709\u52a9\u4e8e\u5feb\u901f\u67e5\u627e\u201c\u5e26\u6709\u67d0\u79cd\u6781\u503c\u201d\u7684\u9876\u70b9\u3002
"},{"location":"chapter_greedy/","title":"\u7b2c 15 \u7ae0 \u00a0 \u8d2a\u5fc3","text":"Abstract
\u5411\u65e5\u8475\u671d\u7740\u592a\u9633\u8f6c\u52a8\uff0c\u65f6\u523b\u8ffd\u6c42\u81ea\u8eab\u6210\u957f\u7684\u6700\u5927\u53ef\u80fd\u3002
\u8d2a\u5fc3\u7b56\u7565\u5728\u4e00\u8f6e\u8f6e\u7684\u7b80\u5355\u9009\u62e9\u4e2d\uff0c\u9010\u6b65\u5bfc\u5411\u6700\u4f73\u7b54\u6848\u3002
"},{"location":"chapter_greedy/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5
- 15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898
- 15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898
- 15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898
- 15.5 \u00a0 \u5c0f\u7ed3
Question
\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u4f46\u53ef\u4ee5\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u4ef7\u503c\u6839\u636e\u9009\u62e9\u7684\u91cd\u91cf\u6bd4\u4f8b\u8ba1\u7b97\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 15-3 \u6240\u793a\u3002
\u56fe 15-3 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e
\u5206\u6570\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u6574\u4f53\u4e0a\u975e\u5e38\u76f8\u4f3c\uff0c\u72b6\u6001\u5305\u542b\u5f53\u524d\u7269\u54c1 \\(i\\) \u548c\u5bb9\u91cf \\(c\\) \uff0c\u76ee\u6807\u662f\u6c42\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u6700\u5927\u4ef7\u503c\u3002
\u4e0d\u540c\u70b9\u5728\u4e8e\uff0c\u672c\u9898\u5141\u8bb8\u53ea\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\u3002\u5982\u56fe 15-4 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u7269\u54c1\u4efb\u610f\u5730\u8fdb\u884c\u5207\u5206\uff0c\u5e76\u6309\u7167\u91cd\u91cf\u6bd4\u4f8b\u6765\u8ba1\u7b97\u76f8\u5e94\u4ef7\u503c\u3002
- \u5bf9\u4e8e\u7269\u54c1 \\(i\\) \uff0c\u5b83\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c\u4e3a \\(val[i-1] / wgt[i-1]\\) \uff0c\u7b80\u79f0\u5355\u4f4d\u4ef7\u503c\u3002
- \u5047\u8bbe\u653e\u5165\u4e00\u90e8\u5206\u7269\u54c1 \\(i\\) \uff0c\u91cd\u91cf\u4e3a \\(w\\) \uff0c\u5219\u80cc\u5305\u589e\u52a0\u7684\u4ef7\u503c\u4e3a \\(w \\times val[i-1] / wgt[i-1]\\) \u3002
\u56fe 15-4 \u00a0 \u7269\u54c1\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c
"},{"location":"chapter_greedy/fractional_knapsack_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"\u6700\u5927\u5316\u80cc\u5305\u5185\u7269\u54c1\u603b\u4ef7\u503c\uff0c\u672c\u8d28\u4e0a\u662f\u6700\u5927\u5316\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u7269\u54c1\u4ef7\u503c\u3002\u7531\u6b64\u4fbf\u53ef\u63a8\u7406\u51fa\u56fe 15-5 \u6240\u793a\u7684\u8d2a\u5fc3\u7b56\u7565\u3002
- \u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\u3002
- \u904d\u5386\u6240\u6709\u7269\u54c1\uff0c\u6bcf\u8f6e\u8d2a\u5fc3\u5730\u9009\u62e9\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\u3002
- \u82e5\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u4f7f\u7528\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u586b\u6ee1\u80cc\u5305\u3002
\u56fe 15-5 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565
"},{"location":"chapter_greedy/fractional_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u6211\u4eec\u5efa\u7acb\u4e86\u4e00\u4e2a\u7269\u54c1\u7c7b
PythonC++JavaC#GoSwiftJSTSDartRustCZig fractional_knapsack.pyItem\uff0c\u4ee5\u4fbf\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u8fdb\u884c\u6392\u5e8f\u3002\u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u5f53\u80cc\u5305\u5df2\u6ee1\u65f6\u8df3\u51fa\u5e76\u8fd4\u56de\u89e3\uff1a
fractional_knapsack.cppclass Item:\n \"\"\"\u7269\u54c1\"\"\"\n\n def __init__(self, w: int, v: int):\n self.w = w # \u7269\u54c1\u91cd\u91cf\n self.v = v # \u7269\u54c1\u4ef7\u503c\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -> int:\n \"\"\"\u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3\"\"\"\n # \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n items = [Item(w, v) for w, v in zip(wgt, val)]\n # \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort(key=lambda item: item.v / item.w, reverse=True)\n # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n res = 0\n for item in items:\n if item.w <= cap:\n # \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v\n cap -= item.w\n else:\n # \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (item.v / item.w) * cap\n # \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break\n return res\n
fractional_knapsack.java/* \u7269\u54c1 */\nclass Item {\n public:\n int w; // \u7269\u54c1\u91cd\u91cf\n int v; // \u7269\u54c1\u4ef7\u503c\n\n Item(int w, int v) : w(w), v(v) {\n }\n};\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(vector<int> &wgt, vector<int> &val, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n vector<Item> items;\n for (int i = 0; i < wgt.size(); i++) {\n items.push_back(Item(wgt[i], val[i]));\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n sort(items.begin(), items.end(), [](Item &a, Item &b) { return (double)a.v / a.w > (double)b.v / b.w; });\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n double res = 0;\n for (auto &item : items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (double)item.v / item.w * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.cs/* \u7269\u54c1 */\nclass Item {\n int w; // \u7269\u54c1\u91cd\u91cf\n int v; // \u7269\u54c1\u4ef7\u503c\n\n public Item(int w, int v) {\n this.w = w;\n this.v = v;\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n Item[] items = new Item[wgt.length];\n for (int i = 0; i < wgt.length; i++) {\n items[i] = new Item(wgt[i], val[i]);\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n Arrays.sort(items, Comparator.comparingDouble(item -> -((double) item.v / item.w)));\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n double res = 0;\n for (Item item : items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (double) item.v / item.w * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.go/* \u7269\u54c1 */\nclass Item(int w, int v) {\n public int w = w; // \u7269\u54c1\u91cd\u91cf\n public int v = v; // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n Item[] items = new Item[wgt.Length];\n for (int i = 0; i < wgt.Length; i++) {\n items[i] = new Item(wgt[i], val[i]);\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n Array.Sort(items, (x, y) => (y.v / y.w).CompareTo(x.v / x.w));\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n double res = 0;\n foreach (Item item in items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (double)item.v / item.w * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.swift/* \u7269\u54c1 */\ntype Item struct {\n w int // \u7269\u54c1\u91cd\u91cf\n v int // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n items := make([]Item, len(wgt))\n for i := 0; i < len(wgt); i++ {\n items[i] = Item{wgt[i], val[i]}\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n sort.Slice(items, func(i, j int) bool {\n return float64(items[i].v)/float64(items[i].w) > float64(items[j].v)/float64(items[j].w)\n })\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n res := 0.0\n for _, item := range items {\n if item.w <= cap {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += float64(item.v)\n cap -= item.w\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += float64(item.v) / float64(item.w) * float64(cap)\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break\n }\n }\n return res\n}\n
fractional_knapsack.js/* \u7269\u54c1 */\nclass Item {\n var w: Int // \u7269\u54c1\u91cd\u91cf\n var v: Int // \u7269\u54c1\u4ef7\u503c\n\n init(w: Int, v: Int) {\n self.w = w\n self.v = v\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -> Double {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort(by: { -(Double($0.v) / Double($0.w)) < -(Double($1.v) / Double($1.w)) })\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n var res = 0.0\n var cap = cap\n for item in items {\n if item.w <= cap {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += Double(item.v)\n cap -= item.w\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += Double(item.v) / Double(item.w) * Double(cap)\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break\n }\n }\n return res\n}\n
fractional_knapsack.ts/* \u7269\u54c1 */\nclass Item {\n constructor(w, v) {\n this.w = w; // \u7269\u54c1\u91cd\u91cf\n this.v = v; // \u7269\u54c1\u4ef7\u503c\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt, val, cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n const items = wgt.map((w, i) => new Item(w, val[i]));\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort((a, b) => b.v / b.w - a.v / a.w);\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n let res = 0;\n for (const item of items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (item.v / item.w) * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.dart/* \u7269\u54c1 */\nclass Item {\n w: number; // \u7269\u54c1\u91cd\u91cf\n v: number; // \u7269\u54c1\u4ef7\u503c\n\n constructor(w: number, v: number) {\n this.w = w;\n this.v = v;\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n const items: Item[] = wgt.map((w, i) => new Item(w, val[i]));\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort((a, b) => b.v / b.w - a.v / a.w);\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n let res = 0;\n for (const item of items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (item.v / item.w) * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.rs/* \u7269\u54c1 */\nclass Item {\n int w; // \u7269\u54c1\u91cd\u91cf\n int v; // \u7269\u54c1\u4ef7\u503c\n\n Item(this.w, this.v);\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(List<int> wgt, List<int> val, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n List<Item> items = List.generate(wgt.length, (i) => Item(wgt[i], val[i]));\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort((a, b) => (b.v / b.w).compareTo(a.v / a.w));\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n double res = 0;\n for (Item item in items) {\n if (item.w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += item.v / item.w * cap;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n return res;\n}\n
fractional_knapsack.c/* \u7269\u54c1 */\nstruct Item {\n w: i32, // \u7269\u54c1\u91cd\u91cf\n v: i32, // \u7269\u54c1\u4ef7\u503c\n}\n\nimpl Item {\n fn new(w: i32, v: i32) -> Self {\n Self { w, v }\n }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfn fractional_knapsack(wgt: &[i32], val: &[i32], mut cap: i32) -> f64 {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n let mut items = wgt\n .iter()\n .zip(val.iter())\n .map(|(&w, &v)| Item::new(w, v))\n .collect::<Vec<Item>>();\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n items.sort_by(|a, b| {\n (b.v as f64 / b.w as f64)\n .partial_cmp(&(a.v as f64 / a.w as f64))\n .unwrap()\n });\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n let mut res = 0.0;\n for item in &items {\n if item.w <= cap {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += item.v as f64;\n cap -= item.w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += item.v as f64 / item.w as f64 * cap as f64;\n // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n res\n}\n
fractional_knapsack.zig/* \u7269\u54c1 */\ntypedef struct {\n int w; // \u7269\u54c1\u91cd\u91cf\n int v; // \u7269\u54c1\u4ef7\u503c\n} Item;\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n Item *items = malloc(sizeof(Item) * itemCount);\n for (int i = 0; i < itemCount; i++) {\n items[i] = (Item){.w = wgt[i], .v = val[i]};\n }\n // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n float res = 0.0;\n for (int i = 0; i < itemCount; i++) {\n if (items[i].w <= cap) {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n res += items[i].v;\n cap -= items[i].w;\n } else {\n // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n res += (float)cap / items[i].w * items[i].v;\n cap = 0;\n break;\n }\n }\n free(items);\n return res;\n}\n[class]{Item}-[func]{}\n\n[class]{}-[func]{fractionalKnapsack}\n\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u9700\u8981\u904d\u5386\u6574\u4e2a\u7269\u54c1\u5217\u8868\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7269\u54c1\u6570\u91cf\u3002
\u7531\u4e8e\u521d\u59cb\u5316\u4e86\u4e00\u4e2a
"},{"location":"chapter_greedy/fractional_knapsack_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"Item\u5bf9\u8c61\u5217\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u91c7\u7528\u53cd\u8bc1\u6cd5\u3002\u5047\u8bbe\u7269\u54c1 \\(x\\) \u662f\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\uff0c\u4f7f\u7528\u67d0\u7b97\u6cd5\u6c42\u5f97\u6700\u5927\u4ef7\u503c\u4e3a
res\uff0c\u4f46\u8be5\u89e3\u4e2d\u4e0d\u5305\u542b\u7269\u54c1 \\(x\\) \u3002\u73b0\u5728\u4ece\u80cc\u5305\u4e2d\u62ff\u51fa\u5355\u4f4d\u91cd\u91cf\u7684\u4efb\u610f\u7269\u54c1\uff0c\u5e76\u66ff\u6362\u4e3a\u5355\u4f4d\u91cd\u91cf\u7684\u7269\u54c1 \\(x\\) \u3002\u7531\u4e8e\u7269\u54c1 \\(x\\) \u7684\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\uff0c\u56e0\u6b64\u66ff\u6362\u540e\u7684\u603b\u4ef7\u503c\u4e00\u5b9a\u5927\u4e8e
res\u3002\u8fd9\u4e0eres\u662f\u6700\u4f18\u89e3\u77db\u76fe\uff0c\u8bf4\u660e\u6700\u4f18\u89e3\u4e2d\u5fc5\u987b\u5305\u542b\u7269\u54c1 \\(x\\) \u3002\u5bf9\u4e8e\u8be5\u89e3\u4e2d\u7684\u5176\u4ed6\u7269\u54c1\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u6784\u5efa\u51fa\u4e0a\u8ff0\u77db\u76fe\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u5355\u4f4d\u4ef7\u503c\u66f4\u5927\u7684\u7269\u54c1\u603b\u662f\u66f4\u4f18\u9009\u62e9\uff0c\u8fd9\u8bf4\u660e\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002
\u5982\u56fe 15-6 \u6240\u793a\uff0c\u5982\u679c\u5c06\u7269\u54c1\u91cd\u91cf\u548c\u7269\u54c1\u5355\u4f4d\u4ef7\u503c\u5206\u522b\u770b\u4f5c\u4e00\u5f20\u4e8c\u7ef4\u56fe\u8868\u7684\u6a2a\u8f74\u548c\u7eb5\u8f74\uff0c\u5219\u5206\u6570\u80cc\u5305\u95ee\u9898\u53ef\u8f6c\u5316\u4e3a\u201c\u6c42\u5728\u6709\u9650\u6a2a\u8f74\u533a\u95f4\u4e0b\u7684\u6700\u5927\u56f4\u6210\u9762\u79ef\u201d\u3002\u8fd9\u4e2a\u7c7b\u6bd4\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u4ece\u51e0\u4f55\u89d2\u5ea6\u7406\u89e3\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002
\u56fe 15-6 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u51e0\u4f55\u8868\u793a
"},{"location":"chapter_greedy/greedy_algorithm/","title":"15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5","text":"\u300c\u8d2a\u5fc3\u7b97\u6cd5 greedy algorithm\u300d\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u7684\u7b97\u6cd5\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u95ee\u9898\u7684\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\uff0c\u90fd\u9009\u62e9\u5f53\u524d\u770b\u8d77\u6765\u6700\u4f18\u7684\u9009\u62e9\uff0c\u5373\u8d2a\u5fc3\u5730\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002\u8d2a\u5fc3\u7b97\u6cd5\u7b80\u6d01\u4e14\u9ad8\u6548\uff0c\u5728\u8bb8\u591a\u5b9e\u9645\u95ee\u9898\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002
\u8d2a\u5fc3\u7b97\u6cd5\u548c\u52a8\u6001\u89c4\u5212\u90fd\u5e38\u7528\u4e8e\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u3002\u5b83\u4eec\u4e4b\u95f4\u5b58\u5728\u4e00\u4e9b\u76f8\u4f3c\u4e4b\u5904\uff0c\u6bd4\u5982\u90fd\u4f9d\u8d56\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28\uff0c\u4f46\u5de5\u4f5c\u539f\u7406\u4e0d\u540c\u3002
- \u52a8\u6001\u89c4\u5212\u4f1a\u6839\u636e\u4e4b\u524d\u9636\u6bb5\u7684\u6240\u6709\u51b3\u7b56\u6765\u8003\u8651\u5f53\u524d\u51b3\u7b56\uff0c\u5e76\u4f7f\u7528\u8fc7\u53bb\u5b50\u95ee\u9898\u7684\u89e3\u6765\u6784\u5efa\u5f53\u524d\u5b50\u95ee\u9898\u7684\u89e3\u3002
- \u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4f1a\u8003\u8651\u8fc7\u53bb\u7684\u51b3\u7b56\uff0c\u800c\u662f\u4e00\u8def\u5411\u524d\u5730\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u4e0d\u65ad\u7f29\u5c0f\u95ee\u9898\u8303\u56f4\uff0c\u76f4\u81f3\u95ee\u9898\u88ab\u89e3\u51b3\u3002
\u6211\u4eec\u5148\u901a\u8fc7\u4f8b\u9898\u201c\u96f6\u94b1\u5151\u6362\u201d\u4e86\u89e3\u8d2a\u5fc3\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002\u8fd9\u9053\u9898\u5df2\u7ecf\u5728\u201c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u76f8\u4fe1\u4f60\u5bf9\u5b83\u5e76\u4e0d\u964c\u751f\u3002
Question
\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u5219\u8fd4\u56de \\(-1\\) \u3002
\u672c\u9898\u91c7\u53d6\u7684\u8d2a\u5fc3\u7b56\u7565\u5982\u56fe 15-1 \u6240\u793a\u3002\u7ed9\u5b9a\u76ee\u6807\u91d1\u989d\uff0c\u6211\u4eec\u8d2a\u5fc3\u5730\u9009\u62e9\u4e0d\u5927\u4e8e\u4e14\u6700\u63a5\u8fd1\u5b83\u7684\u786c\u5e01\uff0c\u4e0d\u65ad\u5faa\u73af\u8be5\u6b65\u9aa4\uff0c\u76f4\u81f3\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u4e3a\u6b62\u3002
\u56fe 15-1 \u00a0 \u96f6\u94b1\u5151\u6362\u7684\u8d2a\u5fc3\u7b56\u7565
\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u4f60\u53ef\u80fd\u4f1a\u4e0d\u7531\u5730\u53d1\u51fa\u611f\u53f9\uff1aSo clean \uff01\u8d2a\u5fc3\u7b97\u6cd5\u4ec5\u7528\u7ea6\u5341\u884c\u4ee3\u7801\u5c31\u89e3\u51b3\u4e86\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_greedy.py
coin_change_greedy.cppdef coin_change_greedy(coins: list[int], amt: int) -> int:\n \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3\"\"\"\n # \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n i = len(coins) - 1\n count = 0\n # \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while amt > 0:\n # \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while i > 0 and coins[i] > amt:\n i -= 1\n # \u9009\u62e9 coins[i]\n amt -= coins[i]\n count += 1\n # \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return count if amt == 0 else -1\n
coin_change_greedy.java/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(vector<int> &coins, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = coins.size() - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
coin_change_greedy.cs/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int[] coins, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = coins.length - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
coin_change_greedy.go/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint CoinChangeGreedy(int[] coins, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = coins.Length - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
coin_change_greedy.swift/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins []int, amt int) int {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n i := len(coins) - 1\n count := 0\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n for amt > 0 {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n for i > 0 && coins[i] > amt {\n i--\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i]\n count++\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n if amt != 0 {\n return -1\n }\n return count\n}\n
coin_change_greedy.js/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -> Int {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n var i = coins.count - 1\n var count = 0\n var amt = amt\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while amt > 0 {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while i > 0 && coins[i] > amt {\n i -= 1\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i]\n count += 1\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1\n}\n
coin_change_greedy.ts/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins, amt) {\n // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n let i = coins.length - 1;\n let count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt === 0 ? count : -1;\n}\n
coin_change_greedy.dart/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n let i = coins.length - 1;\n let count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt === 0 ? count : -1;\n}\n
coin_change_greedy.rs/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(List<int> coins, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = coins.length - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
coin_change_greedy.c/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfn coin_change_greedy(coins: &[i32], mut amt: i32) -> i32 {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n let mut i = coins.len() - 1;\n let mut count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while amt > 0 {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while i > 0 && coins[i] > amt {\n i -= 1;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count += 1;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n if amt == 0 {\n count\n } else {\n -1\n }\n}\n
coin_change_greedy.zig/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int *coins, int size, int amt) {\n // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n int i = size - 1;\n int count = 0;\n // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n while (amt > 0) {\n // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n while (i > 0 && coins[i] > amt) {\n i--;\n }\n // \u9009\u62e9 coins[i]\n amt -= coins[i];\n count++;\n }\n // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n return amt == 0 ? count : -1;\n}\n
"},{"location":"chapter_greedy/greedy_algorithm/#1511","title":"15.1.1 \u00a0 \u8d2a\u5fc3\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"[class]{}-[func]{coinChangeGreedy}\n\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u64cd\u4f5c\u76f4\u63a5\u3001\u5b9e\u73b0\u7b80\u5355\uff0c\u800c\u4e14\u901a\u5e38\u6548\u7387\u4e5f\u5f88\u9ad8\u3002\u5728\u4ee5\u4e0a\u4ee3\u7801\u4e2d\uff0c\u8bb0\u786c\u5e01\u6700\u5c0f\u9762\u503c\u4e3a \\(\\min(coins)\\) \uff0c\u5219\u8d2a\u5fc3\u9009\u62e9\u6700\u591a\u5faa\u73af \\(amt / \\min(coins)\\) \u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(amt / \\min(coins))\\) \u3002\u8fd9\u6bd4\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\times amt)\\) \u63d0\u5347\u4e86\u4e00\u4e2a\u6570\u91cf\u7ea7\u3002
\u7136\u800c\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u9762\u503c\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5e76\u4e0d\u80fd\u627e\u5230\u6700\u4f18\u89e3\u3002\u56fe 15-2 \u7ed9\u51fa\u4e86\u4e24\u4e2a\u793a\u4f8b\u3002
- \u6b63\u4f8b \\(coins = [1, 5, 10, 20, 50, 100]\\)\uff1a\u5728\u8be5\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u7ed9\u5b9a\u4efb\u610f \\(amt\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u90fd\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3\u3002
- \u53cd\u4f8b \\(coins = [1, 20, 50]\\)\uff1a\u5047\u8bbe \\(amt = 60\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 10\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(11\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(20 + 20 + 20\\) \uff0c\u4ec5\u9700 \\(3\\) \u679a\u786c\u5e01\u3002
- \u53cd\u4f8b \\(coins = [1, 49, 50]\\)\uff1a\u5047\u8bbe \\(amt = 98\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 48\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(49\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(49 + 49\\) \uff0c\u4ec5\u9700 \\(2\\) \u679a\u786c\u5e01\u3002
\u56fe 15-2 \u00a0 \u8d2a\u5fc3\u65e0\u6cd5\u627e\u51fa\u6700\u4f18\u89e3\u7684\u793a\u4f8b
\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u4e8e\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u65e0\u6cd5\u4fdd\u8bc1\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3\uff0c\u5e76\u4e14\u6709\u53ef\u80fd\u627e\u5230\u975e\u5e38\u5dee\u7684\u89e3\u3002\u5b83\u66f4\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\u3002
\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u9002\u7528\u60c5\u51b5\u5206\u4ee5\u4e0b\u4e24\u79cd\u3002
- \u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u5f80\u5f80\u662f\u6700\u4f18\u9009\u62e9\uff0c\u56e0\u4e3a\u5b83\u5f80\u5f80\u6bd4\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u66f4\u9ad8\u6548\u3002
- \u53ef\u4ee5\u627e\u5230\u8fd1\u4f3c\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u4e5f\u662f\u53ef\u7528\u7684\u3002\u5bf9\u4e8e\u5f88\u591a\u590d\u6742\u95ee\u9898\u6765\u8bf4\uff0c\u5bfb\u627e\u5168\u5c40\u6700\u4f18\u89e3\u975e\u5e38\u56f0\u96be\uff0c\u80fd\u4ee5\u8f83\u9ad8\u6548\u7387\u627e\u5230\u6b21\u4f18\u89e3\u4e5f\u662f\u975e\u5e38\u4e0d\u9519\u7684\u3002
\u90a3\u4e48\u95ee\u9898\u6765\u4e86\uff0c\u4ec0\u4e48\u6837\u7684\u95ee\u9898\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u5462\uff1f\u6216\u8005\u8bf4\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1f
\u76f8\u8f83\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u4f7f\u7528\u6761\u4ef6\u66f4\u52a0\u82db\u523b\uff0c\u5176\u4e3b\u8981\u5173\u6ce8\u95ee\u9898\u7684\u4e24\u4e2a\u6027\u8d28\u3002
- \u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\uff1a\u53ea\u6709\u5f53\u5c40\u90e8\u6700\u4f18\u9009\u62e9\u59cb\u7ec8\u53ef\u4ee5\u5bfc\u81f4\u5168\u5c40\u6700\u4f18\u89e3\u65f6\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u624d\u80fd\u4fdd\u8bc1\u5f97\u5230\u6700\u4f18\u89e3\u3002
- \u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5305\u542b\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002
\u6700\u4f18\u5b50\u7ed3\u6784\u5df2\u7ecf\u5728\u201c\u52a8\u6001\u89c4\u5212\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u8fd9\u91cc\u4e0d\u518d\u8d58\u8ff0\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e9b\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\u5e76\u4e0d\u660e\u663e\uff0c\u4f46\u4ecd\u7136\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u89e3\u51b3\u3002
\u6211\u4eec\u4e3b\u8981\u63a2\u7a76\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u5224\u65ad\u65b9\u6cd5\u3002\u867d\u7136\u5b83\u7684\u63cf\u8ff0\u770b\u4e0a\u53bb\u6bd4\u8f83\u7b80\u5355\uff0c\u4f46\u5b9e\u9645\u4e0a\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\uff0c\u8bc1\u660e\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u5e76\u975e\u6613\u4e8b\u3002
\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u6211\u4eec\u867d\u7136\u80fd\u591f\u5bb9\u6613\u5730\u4e3e\u51fa\u53cd\u4f8b\uff0c\u5bf9\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u8fdb\u884c\u8bc1\u4f2a\uff0c\u4f46\u8bc1\u5b9e\u7684\u96be\u5ea6\u8f83\u5927\u3002\u5982\u679c\u95ee\uff1a\u6ee1\u8db3\u4ec0\u4e48\u6761\u4ef6\u7684\u786c\u5e01\u7ec4\u5408\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\uff1f\u6211\u4eec\u5f80\u5f80\u53ea\u80fd\u51ed\u501f\u76f4\u89c9\u6216\u4e3e\u4f8b\u5b50\u6765\u7ed9\u51fa\u4e00\u4e2a\u6a21\u68f1\u4e24\u53ef\u7684\u7b54\u6848\uff0c\u800c\u96be\u4ee5\u7ed9\u51fa\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\u3002
Quote
\u6709\u4e00\u7bc7\u8bba\u6587\u7ed9\u51fa\u4e86\u4e00\u4e2a \\(O(n^3)\\) \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u7b97\u6cd5\uff0c\u7528\u4e8e\u5224\u65ad\u4e00\u4e2a\u786c\u5e01\u7ec4\u5408\u80fd\u5426\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u627e\u51fa\u4efb\u610f\u91d1\u989d\u7684\u6700\u4f18\u89e3\u3002
Pearson, David. A polynomial-time algorithm for the change-making problem. Operations Research Letters 33.3 (2005): 231-234.
"},{"location":"chapter_greedy/greedy_algorithm/#1513","title":"15.1.3 \u00a0 \u8d2a\u5fc3\u89e3\u9898\u6b65\u9aa4","text":"\u8d2a\u5fc3\u95ee\u9898\u7684\u89e3\u51b3\u6d41\u7a0b\u5927\u4f53\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e09\u6b65\u3002
- \u95ee\u9898\u5206\u6790\uff1a\u68b3\u7406\u4e0e\u7406\u89e3\u95ee\u9898\u7279\u6027\uff0c\u5305\u62ec\u72b6\u6001\u5b9a\u4e49\u3001\u4f18\u5316\u76ee\u6807\u548c\u7ea6\u675f\u6761\u4ef6\u7b49\u3002\u8fd9\u4e00\u6b65\u5728\u56de\u6eaf\u548c\u52a8\u6001\u89c4\u5212\u4e2d\u90fd\u6709\u6d89\u53ca\u3002
- \u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\uff1a\u786e\u5b9a\u5982\u4f55\u5728\u6bcf\u4e00\u6b65\u4e2d\u505a\u51fa\u8d2a\u5fc3\u9009\u62e9\u3002\u8fd9\u4e2a\u7b56\u7565\u80fd\u591f\u5728\u6bcf\u4e00\u6b65\u51cf\u5c0f\u95ee\u9898\u7684\u89c4\u6a21\uff0c\u5e76\u6700\u7ec8\u89e3\u51b3\u6574\u4e2a\u95ee\u9898\u3002
- \u6b63\u786e\u6027\u8bc1\u660e\uff1a\u901a\u5e38\u9700\u8981\u8bc1\u660e\u95ee\u9898\u5177\u6709\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8fd9\u4e2a\u6b65\u9aa4\u53ef\u80fd\u9700\u8981\u7528\u5230\u6570\u5b66\u8bc1\u660e\uff0c\u4f8b\u5982\u5f52\u7eb3\u6cd5\u6216\u53cd\u8bc1\u6cd5\u7b49\u3002
\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6c42\u89e3\u95ee\u9898\u7684\u6838\u5fc3\u6b65\u9aa4\uff0c\u4f46\u5b9e\u65bd\u8d77\u6765\u53ef\u80fd\u5e76\u4e0d\u5bb9\u6613\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002
- \u4e0d\u540c\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\u7684\u5dee\u5f02\u8f83\u5927\u3002\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\u6765\u8bf4\uff0c\u8d2a\u5fc3\u7b56\u7565\u6bd4\u8f83\u6d45\u663e\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e9b\u5927\u6982\u7684\u601d\u8003\u4e0e\u5c1d\u8bd5\u5c31\u80fd\u5f97\u51fa\u3002\u800c\u5bf9\u4e8e\u4e00\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b56\u7565\u53ef\u80fd\u975e\u5e38\u9690\u853d\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u975e\u5e38\u8003\u9a8c\u4e2a\u4eba\u7684\u89e3\u9898\u7ecf\u9a8c\u4e0e\u7b97\u6cd5\u80fd\u529b\u4e86\u3002
- \u67d0\u4e9b\u8d2a\u5fc3\u7b56\u7565\u5177\u6709\u8f83\u5f3a\u7684\u8ff7\u60d1\u6027\u3002\u5f53\u6211\u4eec\u6ee1\u6000\u4fe1\u5fc3\u8bbe\u8ba1\u597d\u8d2a\u5fc3\u7b56\u7565\uff0c\u5199\u51fa\u89e3\u9898\u4ee3\u7801\u5e76\u63d0\u4ea4\u8fd0\u884c\uff0c\u5f88\u53ef\u80fd\u53d1\u73b0\u90e8\u5206\u6d4b\u8bd5\u6837\u4f8b\u65e0\u6cd5\u901a\u8fc7\u3002\u8fd9\u662f\u56e0\u4e3a\u8bbe\u8ba1\u7684\u8d2a\u5fc3\u7b56\u7565\u53ea\u662f\u201c\u90e8\u5206\u6b63\u786e\u201d\u7684\uff0c\u4e0a\u6587\u4ecb\u7ecd\u7684\u96f6\u94b1\u5151\u6362\u5c31\u662f\u4e00\u4e2a\u5178\u578b\u6848\u4f8b\u3002
\u4e3a\u4e86\u4fdd\u8bc1\u6b63\u786e\u6027\uff0c\u6211\u4eec\u5e94\u8be5\u5bf9\u8d2a\u5fc3\u7b56\u7565\u8fdb\u884c\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\uff0c\u901a\u5e38\u9700\u8981\u7528\u5230\u53cd\u8bc1\u6cd5\u6216\u6570\u5b66\u5f52\u7eb3\u6cd5\u3002
\u7136\u800c\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u4e5f\u5f88\u53ef\u80fd\u4e0d\u662f\u4e00\u4ef6\u6613\u4e8b\u3002\u5982\u82e5\u6ca1\u6709\u5934\u7eea\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u9762\u5411\u6d4b\u8bd5\u7528\u4f8b\u8fdb\u884c\u4ee3\u7801\u8c03\u8bd5\uff0c\u4e00\u6b65\u6b65\u4fee\u6539\u4e0e\u9a8c\u8bc1\u8d2a\u5fc3\u7b56\u7565\u3002
"},{"location":"chapter_greedy/greedy_algorithm/#1514","title":"15.1.4 \u00a0 \u8d2a\u5fc3\u5178\u578b\u4f8b\u9898","text":"\u8d2a\u5fc3\u7b97\u6cd5\u5e38\u5e38\u5e94\u7528\u5728\u6ee1\u8db3\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u4ee5\u4e0b\u5217\u4e3e\u4e86\u4e00\u4e9b\u5178\u578b\u7684\u8d2a\u5fc3\u7b97\u6cd5\u95ee\u9898\u3002
- \u786c\u5e01\u627e\u96f6\u95ee\u9898\uff1a\u5728\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u603b\u662f\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002
- \u533a\u95f4\u8c03\u5ea6\u95ee\u9898\uff1a\u5047\u8bbe\u4f60\u6709\u4e00\u4e9b\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u5728\u4e00\u6bb5\u65f6\u95f4\u5185\u8fdb\u884c\uff0c\u4f60\u7684\u76ee\u6807\u662f\u5b8c\u6210\u5c3d\u53ef\u80fd\u591a\u7684\u4efb\u52a1\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u4efb\u52a1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002
- \u5206\u6570\u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u8f7d\u91cd\u91cf\uff0c\u4f60\u7684\u76ee\u6807\u662f\u9009\u62e9\u4e00\u7ec4\u7269\u54c1\uff0c\u4f7f\u5f97\u603b\u91cd\u91cf\u4e0d\u8d85\u8fc7\u8f7d\u91cd\u91cf\uff0c\u4e14\u603b\u4ef7\u503c\u6700\u5927\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u6027\u4ef7\u6bd4\u6700\u9ad8\uff08\u4ef7\u503c / \u91cd\u91cf\uff09\u7684\u7269\u54c1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4e00\u4e9b\u60c5\u51b5\u4e0b\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002
- \u80a1\u7968\u4e70\u5356\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u80a1\u7968\u7684\u5386\u53f2\u4ef7\u683c\uff0c\u4f60\u53ef\u4ee5\u8fdb\u884c\u591a\u6b21\u4e70\u5356\uff0c\u4f46\u5982\u679c\u4f60\u5df2\u7ecf\u6301\u6709\u80a1\u7968\uff0c\u90a3\u4e48\u5728\u5356\u51fa\u4e4b\u524d\u4e0d\u80fd\u518d\u4e70\uff0c\u76ee\u6807\u662f\u83b7\u53d6\u6700\u5927\u5229\u6da6\u3002
- \u970d\u592b\u66fc\u7f16\u7801\uff1a\u970d\u592b\u66fc\u7f16\u7801\u662f\u4e00\u79cd\u7528\u4e8e\u65e0\u635f\u6570\u636e\u538b\u7f29\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002\u901a\u8fc7\u6784\u5efa\u970d\u592b\u66fc\u6811\uff0c\u6bcf\u6b21\u9009\u62e9\u51fa\u73b0\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u8282\u70b9\u5408\u5e76\uff0c\u6700\u540e\u5f97\u5230\u7684\u970d\u592b\u66fc\u6811\u7684\u5e26\u6743\u8def\u5f84\u957f\u5ea6\uff08\u7f16\u7801\u957f\u5ea6\uff09\u6700\u5c0f\u3002
- Dijkstra \u7b97\u6cd5\uff1a\u5b83\u662f\u4e00\u79cd\u89e3\u51b3\u7ed9\u5b9a\u6e90\u9876\u70b9\u5230\u5176\u4f59\u5404\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84\u95ee\u9898\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002
Question
\u8f93\u5165\u4e00\u4e2a\u6570\u7ec4 \\(ht\\) \uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4ee3\u8868\u4e00\u4e2a\u5782\u76f4\u9694\u677f\u7684\u9ad8\u5ea6\u3002\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9694\u677f\uff0c\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u7a7a\u95f4\u53ef\u4ee5\u7ec4\u6210\u4e00\u4e2a\u5bb9\u5668\u3002
\u5bb9\u5668\u7684\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u548c\u5bbd\u5ea6\u7684\u4e58\u79ef\uff08\u9762\u79ef\uff09\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u8f83\u77ed\u7684\u9694\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u4e2a\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002
\u8bf7\u5728\u6570\u7ec4\u4e2d\u9009\u62e9\u4e24\u4e2a\u9694\u677f\uff0c\u4f7f\u5f97\u7ec4\u6210\u7684\u5bb9\u5668\u7684\u5bb9\u91cf\u6700\u5927\uff0c\u8fd4\u56de\u6700\u5927\u5bb9\u91cf\u3002\u793a\u4f8b\u5982\u56fe 15-7 \u6240\u793a\u3002
\u56fe 15-7 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e
\u5bb9\u5668\u7531\u4efb\u610f\u4e24\u4e2a\u9694\u677f\u56f4\u6210\uff0c\u56e0\u6b64\u672c\u9898\u7684\u72b6\u6001\u4e3a\u4e24\u4e2a\u9694\u677f\u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002
\u6839\u636e\u9898\u610f\uff0c\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u4e58\u4ee5\u5bbd\u5ea6\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002\u8bbe\u5bb9\u91cf\u4e3a \\(cap[i, j]\\) \uff0c\u5219\u53ef\u5f97\u8ba1\u7b97\u516c\u5f0f\uff1a
\\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\]\u8bbe\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u4e24\u4e2a\u9694\u677f\u7684\u7ec4\u5408\u6570\u91cf\uff08\u72b6\u6001\u603b\u6570\uff09\u4e3a \\(C_n^2 = \\frac{n(n - 1)}{2}\\) \u4e2a\u3002\u6700\u76f4\u63a5\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u7a77\u4e3e\u6240\u6709\u72b6\u6001\uff0c\u4ece\u800c\u6c42\u5f97\u6700\u5927\u5bb9\u91cf\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002
"},{"location":"chapter_greedy/max_capacity_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"\u8fd9\u9053\u9898\u8fd8\u6709\u66f4\u9ad8\u6548\u7387\u7684\u89e3\u6cd5\u3002\u5982\u56fe 15-8 \u6240\u793a\uff0c\u73b0\u9009\u53d6\u4e00\u4e2a\u72b6\u6001 \\([i, j]\\) \uff0c\u5176\u6ee1\u8db3\u7d22\u5f15 \\(i < j\\) \u4e14\u9ad8\u5ea6 \\(ht[i] < ht[j]\\) \uff0c\u5373 \\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002
\u56fe 15-8 \u00a0 \u521d\u59cb\u72b6\u6001
\u5982\u56fe 15-9 \u6240\u793a\uff0c\u82e5\u6b64\u65f6\u5c06\u957f\u677f \\(j\\) \u5411\u77ed\u677f \\(i\\) \u9760\u8fd1\uff0c\u5219\u5bb9\u91cf\u4e00\u5b9a\u53d8\u5c0f\u3002
\u8fd9\u662f\u56e0\u4e3a\u5728\u79fb\u52a8\u957f\u677f \\(j\\) \u540e\uff0c\u5bbd\u5ea6 \\(j-i\\) \u80af\u5b9a\u53d8\u5c0f\uff1b\u800c\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u56e0\u6b64\u9ad8\u5ea6\u53ea\u53ef\u80fd\u4e0d\u53d8\uff08 \\(i\\) \u4ecd\u4e3a\u77ed\u677f\uff09\u6216\u53d8\u5c0f\uff08\u79fb\u52a8\u540e\u7684 \\(j\\) \u6210\u4e3a\u77ed\u677f\uff09\u3002
\u56fe 15-9 \u00a0 \u5411\u5185\u79fb\u52a8\u957f\u677f\u540e\u7684\u72b6\u6001
\u53cd\u5411\u601d\u8003\uff0c\u6211\u4eec\u53ea\u6709\u5411\u5185\u6536\u7f29\u77ed\u677f \\(i\\) \uff0c\u624d\u6709\u53ef\u80fd\u4f7f\u5bb9\u91cf\u53d8\u5927\u3002\u56e0\u4e3a\u867d\u7136\u5bbd\u5ea6\u4e00\u5b9a\u53d8\u5c0f\uff0c\u4f46\u9ad8\u5ea6\u53ef\u80fd\u4f1a\u53d8\u5927\uff08\u79fb\u52a8\u540e\u7684\u77ed\u677f \\(i\\) \u53ef\u80fd\u4f1a\u53d8\u957f\uff09\u3002\u4f8b\u5982\u5728\u56fe 15-10 \u4e2d\uff0c\u79fb\u52a8\u77ed\u677f\u540e\u9762\u79ef\u53d8\u5927\u3002
\u56fe 15-10 \u00a0 \u5411\u5185\u79fb\u52a8\u77ed\u677f\u540e\u7684\u72b6\u6001
\u7531\u6b64\u4fbf\u53ef\u63a8\u51fa\u672c\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\uff1a\u521d\u59cb\u5316\u4e24\u6307\u9488\u5206\u5217\u5bb9\u5668\u4e24\u7aef\uff0c\u6bcf\u8f6e\u5411\u5185\u6536\u7f29\u77ed\u677f\u5bf9\u5e94\u7684\u6307\u9488\uff0c\u76f4\u81f3\u4e24\u6307\u9488\u76f8\u9047\u3002
\u56fe 15-11 \u5c55\u793a\u4e86\u8d2a\u5fc3\u7b56\u7565\u7684\u6267\u884c\u8fc7\u7a0b\u3002
- \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u5217\u4e0e\u6570\u7ec4\u4e24\u7aef\u3002
- \u8ba1\u7b97\u5f53\u524d\u72b6\u6001\u7684\u5bb9\u91cf \\(cap[i, j]\\) \uff0c\u5e76\u66f4\u65b0\u6700\u5927\u5bb9\u91cf\u3002
- \u6bd4\u8f83\u677f \\(i\\) \u548c \u677f \\(j\\) \u7684\u9ad8\u5ea6\uff0c\u5e76\u5c06\u77ed\u677f\u5411\u5185\u79fb\u52a8\u4e00\u683c\u3002
- \u5faa\u73af\u6267\u884c\u7b2c
2.\u6b65\u548c\u7b2c3.\u6b65\uff0c\u76f4\u81f3 \\(i\\) \u548c \\(j\\) \u76f8\u9047\u65f6\u7ed3\u675f\u3002
\u56fe 15-11 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u8d2a\u5fc3\u8fc7\u7a0b
"},{"location":"chapter_greedy/max_capacity_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u4ee3\u7801\u5faa\u73af\u6700\u591a \\(n\\) \u8f6e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002
\u53d8\u91cf \\(i\\)\u3001\\(j\\)\u3001\\(res\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig max_capacity.py
max_capacity.cppdef max_capacity(ht: list[int]) -> int:\n \"\"\"\u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3\"\"\"\n # \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n i, j = 0, len(ht) - 1\n # \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n res = 0\n # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while i < j:\n # \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n cap = min(ht[i], ht[j]) * (j - i)\n res = max(res, cap)\n # \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if ht[i] < ht[j]:\n i += 1\n else:\n j -= 1\n return res\n
max_capacity.java/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(vector<int> &ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0, j = ht.size() - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int cap = min(ht[i], ht[j]) * (j - i);\n res = max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
max_capacity.cs/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int[] ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0, j = ht.length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int cap = Math.min(ht[i], ht[j]) * (j - i);\n res = Math.max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
max_capacity.go/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint MaxCapacity(int[] ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0, j = ht.Length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int cap = Math.Min(ht[i], ht[j]) * (j - i);\n res = Math.Max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
max_capacity.swift/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht []int) int {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n i, j := 0, len(ht)-1\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n res := 0\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n for i < j {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n res = int(math.Max(float64(res), float64(capacity)))\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if ht[i] < ht[j] {\n i++\n } else {\n j--\n }\n }\n return res\n}\n
max_capacity.js/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht: [Int]) -> Int {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n var i = 0, j = ht.count - 1\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n var res = 0\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while i < j {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n let cap = min(ht[i], ht[j]) * (j - i)\n res = max(res, cap)\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if ht[i] < ht[j] {\n i += 1\n } else {\n j -= 1\n }\n }\n return res\n}\n
max_capacity.ts/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n let i = 0,\n j = ht.length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n let res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n const cap = Math.min(ht[i], ht[j]) * (j - i);\n res = Math.max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i += 1;\n } else {\n j -= 1;\n }\n }\n return res;\n}\n
max_capacity.dart/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht: number[]): number {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n let i = 0,\n j = ht.length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n let res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n res = Math.max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i += 1;\n } else {\n j -= 1;\n }\n }\n return res;\n}\n
max_capacity.rs/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(List<int> ht) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0, j = ht.length - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int cap = min(ht[i], ht[j]) * (j - i);\n res = max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
max_capacity.c/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfn max_capacity(ht: &[i32]) -> i32 {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n let mut i = 0;\n let mut j = ht.len() - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n let mut res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while i < j {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n res = std::cmp::max(res, cap);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if ht[i] < ht[j] {\n i += 1;\n } else {\n j -= 1;\n }\n }\n res\n}\n
max_capacity.zig/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int ht[], int htLength) {\n // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n int i = 0;\n int j = htLength - 1;\n // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n int res = 0;\n // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n while (i < j) {\n // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n int capacity = myMin(ht[i], ht[j]) * (j - i);\n res = myMax(res, capacity);\n // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n if (ht[i] < ht[j]) {\n i++;\n } else {\n j--;\n }\n }\n return res;\n}\n
"},{"location":"chapter_greedy/max_capacity_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"[class]{}-[func]{maxCapacity}\n\u4e4b\u6240\u4ee5\u8d2a\u5fc3\u6bd4\u7a77\u4e3e\u66f4\u5feb\uff0c\u662f\u56e0\u4e3a\u6bcf\u8f6e\u7684\u8d2a\u5fc3\u9009\u62e9\u90fd\u4f1a\u201c\u8df3\u8fc7\u201d\u4e00\u4e9b\u72b6\u6001\u3002
\u6bd4\u5982\u5728\u72b6\u6001 \\(cap[i, j]\\) \u4e0b\uff0c\\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002\u82e5\u8d2a\u5fc3\u5730\u5c06\u77ed\u677f \\(i\\) \u5411\u5185\u79fb\u52a8\u4e00\u683c\uff0c\u4f1a\u5bfc\u81f4\u56fe 15-12 \u6240\u793a\u7684\u72b6\u6001\u88ab\u201c\u8df3\u8fc7\u201d\u3002\u8fd9\u610f\u5473\u7740\u4e4b\u540e\u65e0\u6cd5\u9a8c\u8bc1\u8fd9\u4e9b\u72b6\u6001\u7684\u5bb9\u91cf\u5927\u5c0f\u3002
\\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\]\u56fe 15-12 \u00a0 \u79fb\u52a8\u77ed\u677f\u5bfc\u81f4\u88ab\u8df3\u8fc7\u7684\u72b6\u6001
\u89c2\u5bdf\u53d1\u73b0\uff0c\u8fd9\u4e9b\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u5b9e\u9645\u4e0a\u5c31\u662f\u5c06\u957f\u677f \\(j\\) \u5411\u5185\u79fb\u52a8\u7684\u6240\u6709\u72b6\u6001\u3002\u524d\u9762\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u5185\u79fb\u957f\u677f\u4e00\u5b9a\u4f1a\u5bfc\u81f4\u5bb9\u91cf\u53d8\u5c0f\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u90fd\u4e0d\u53ef\u80fd\u662f\u6700\u4f18\u89e3\uff0c\u8df3\u8fc7\u5b83\u4eec\u4e0d\u4f1a\u5bfc\u81f4\u9519\u8fc7\u6700\u4f18\u89e3\u3002
\u4ee5\u4e0a\u5206\u6790\u8bf4\u660e\uff0c\u79fb\u52a8\u77ed\u677f\u7684\u64cd\u4f5c\u662f\u201c\u5b89\u5168\u201d\u7684\uff0c\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002
"},{"location":"chapter_greedy/max_product_cutting_problem/","title":"15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5c06\u5176\u5207\u5206\u4e3a\u81f3\u5c11\u4e24\u4e2a\u6b63\u6574\u6570\u7684\u548c\uff0c\u6c42\u5207\u5206\u540e\u6240\u6709\u6574\u6570\u7684\u4e58\u79ef\u6700\u5927\u662f\u591a\u5c11\uff0c\u5982\u56fe 15-13 \u6240\u793a\u3002
\u56fe 15-13 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u95ee\u9898\u5b9a\u4e49
\u5047\u8bbe\u6211\u4eec\u5c06 \\(n\\) \u5207\u5206\u4e3a \\(m\\) \u4e2a\u6574\u6570\u56e0\u5b50\uff0c\u5176\u4e2d\u7b2c \\(i\\) \u4e2a\u56e0\u5b50\u8bb0\u4e3a \\(n_i\\) \uff0c\u5373
\\[ n = \\sum_{i=1}^{m}n_i \\]\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u5f97\u6240\u6709\u6574\u6570\u56e0\u5b50\u7684\u6700\u5927\u4e58\u79ef\uff0c\u5373
\\[ \\max(\\prod_{i=1}^{m}n_i) \\]\u6211\u4eec\u9700\u8981\u601d\u8003\u7684\u662f\uff1a\u5207\u5206\u6570\u91cf \\(m\\) \u5e94\u8be5\u591a\u5927\uff0c\u6bcf\u4e2a \\(n_i\\) \u5e94\u8be5\u662f\u591a\u5c11\uff1f
"},{"location":"chapter_greedy/max_product_cutting_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"\u6839\u636e\u7ecf\u9a8c\uff0c\u4e24\u4e2a\u6574\u6570\u7684\u4e58\u79ef\u5f80\u5f80\u6bd4\u5b83\u4eec\u7684\u52a0\u548c\u66f4\u5927\u3002\u5047\u8bbe\u4ece \\(n\\) \u4e2d\u5206\u51fa\u4e00\u4e2a\u56e0\u5b50 \\(2\\) \uff0c\u5219\u5b83\u4eec\u7684\u4e58\u79ef\u4e3a \\(2(n-2)\\) \u3002\u6211\u4eec\u5c06\u8be5\u4e58\u79ef\u4e0e \\(n\\) \u4f5c\u6bd4\u8f83\uff1a
\\[ \\begin{aligned} 2(n-2) & \\geq n \\newline 2n - n - 4 & \\geq 0 \\newline n & \\geq 4 \\end{aligned} \\]\u5982\u56fe 15-14 \u6240\u793a\uff0c\u5f53 \\(n \\geq 4\\) \u65f6\uff0c\u5207\u5206\u51fa\u4e00\u4e2a \\(2\\) \u540e\u4e58\u79ef\u4f1a\u53d8\u5927\uff0c\u8fd9\u8bf4\u660e\u5927\u4e8e\u7b49\u4e8e \\(4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u88ab\u5207\u5206\u3002
\u8d2a\u5fc3\u7b56\u7565\u4e00\uff1a\u5982\u679c\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b \\(\\geq 4\\) \u7684\u56e0\u5b50\uff0c\u90a3\u4e48\u5b83\u5c31\u5e94\u8be5\u88ab\u7ee7\u7eed\u5207\u5206\u3002\u6700\u7ec8\u7684\u5207\u5206\u65b9\u6848\u53ea\u5e94\u51fa\u73b0 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u79cd\u56e0\u5b50\u3002
\u56fe 15-14 \u00a0 \u5207\u5206\u5bfc\u81f4\u4e58\u79ef\u53d8\u5927
\u63a5\u4e0b\u6765\u601d\u8003\u54ea\u4e2a\u56e0\u5b50\u662f\u6700\u4f18\u7684\u3002\u5728 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u4e2a\u56e0\u5b50\u4e2d\uff0c\u663e\u7136 \\(1\\) \u662f\u6700\u5dee\u7684\uff0c\u56e0\u4e3a \\(1 \\times (n-1) < n\\) \u6052\u6210\u7acb\uff0c\u5373\u5207\u5206\u51fa \\(1\\) \u53cd\u800c\u4f1a\u5bfc\u81f4\u4e58\u79ef\u51cf\u5c0f\u3002
\u5982\u56fe 15-15 \u6240\u793a\uff0c\u5f53 \\(n = 6\\) \u65f6\uff0c\u6709 \\(3 \\times 3 > 2 \\times 2 \\times 2\\) \u3002\u8fd9\u610f\u5473\u7740\u5207\u5206\u51fa \\(3\\) \u6bd4\u5207\u5206\u51fa \\(2\\) \u66f4\u4f18\u3002
\u8d2a\u5fc3\u7b56\u7565\u4e8c\uff1a\u5728\u5207\u5206\u65b9\u6848\u4e2d\uff0c\u6700\u591a\u53ea\u5e94\u5b58\u5728\u4e24\u4e2a \\(2\\) \u3002\u56e0\u4e3a\u4e09\u4e2a \\(2\\) \u603b\u662f\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002
\u56fe 15-15 \u00a0 \u6700\u4f18\u5207\u5206\u56e0\u5b50
\u7efc\u4e0a\u6240\u8ff0\uff0c\u53ef\u63a8\u7406\u51fa\u4ee5\u4e0b\u8d2a\u5fc3\u7b56\u7565\u3002
- \u8f93\u5165\u6574\u6570 \\(n\\) \uff0c\u4ece\u5176\u4e0d\u65ad\u5730\u5207\u5206\u51fa\u56e0\u5b50 \\(3\\) \uff0c\u76f4\u81f3\u4f59\u6570\u4e3a \\(0\\)\u3001\\(1\\)\u3001\\(2\\) \u3002
- \u5f53\u4f59\u6570\u4e3a \\(0\\) \u65f6\uff0c\u4ee3\u8868 \\(n\\) \u662f \\(3\\) \u7684\u500d\u6570\uff0c\u56e0\u6b64\u4e0d\u505a\u4efb\u4f55\u5904\u7406\u3002
- \u5f53\u4f59\u6570\u4e3a \\(2\\) \u65f6\uff0c\u4e0d\u7ee7\u7eed\u5212\u5206\uff0c\u4fdd\u7559\u4e4b\u3002
- \u5f53\u4f59\u6570\u4e3a \\(1\\) \u65f6\uff0c\u7531\u4e8e \\(2 \\times 2 > 1 \\times 3\\) \uff0c\u56e0\u6b64\u5e94\u5c06\u6700\u540e\u4e00\u4e2a \\(3\\) \u66ff\u6362\u4e3a \\(2\\) \u3002
\u5982\u56fe 15-16 \u6240\u793a\uff0c\u6211\u4eec\u65e0\u987b\u901a\u8fc7\u5faa\u73af\u6765\u5207\u5206\u6574\u6570\uff0c\u800c\u53ef\u4ee5\u5229\u7528\u5411\u4e0b\u6574\u9664\u8fd0\u7b97\u5f97\u5230 \\(3\\) \u7684\u4e2a\u6570 \\(a\\) \uff0c\u7528\u53d6\u6a21\u8fd0\u7b97\u5f97\u5230\u4f59\u6570 \\(b\\) \uff0c\u6b64\u65f6\u6709\uff1a
\\[ n = 3 a + b \\]\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e \\(n \\leq 3\\) \u7684\u8fb9\u754c\u60c5\u51b5\uff0c\u5fc5\u987b\u62c6\u5206\u51fa\u4e00\u4e2a \\(1\\) \uff0c\u4e58\u79ef\u4e3a \\(1 \\times (n - 1)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig max_product_cutting.py
max_product_cutting.cppdef max_product_cutting(n: int) -> int:\n \"\"\"\u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3\"\"\"\n # \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if n <= 3:\n return 1 * (n - 1)\n # \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n a, b = n // 3, n % 3\n if b == 1:\n # \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return int(math.pow(3, a - 1)) * 2 * 2\n if b == 2:\n # \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return int(math.pow(3, a)) * 2\n # \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return int(math.pow(3, a))\n
max_product_cutting.java/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n / 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return (int)pow(3, a - 1) * 2 * 2;\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int)pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int)pow(3, a);\n}\n
max_product_cutting.cs/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n / 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return (int) Math.pow(3, a - 1) * 2 * 2;\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int) Math.pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int) Math.pow(3, a);\n}\n
max_product_cutting.go/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint MaxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n / 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return (int)Math.Pow(3, a - 1) * 2 * 2;\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int)Math.Pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (int)Math.Pow(3, a);\n}\n
max_product_cutting.swift/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n int) int {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if n <= 3 {\n return 1 * (n - 1)\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n a := n / 3\n b := n % 3\n if b == 1 {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return int(math.Pow(3, float64(a-1))) * 2 * 2\n }\n if b == 2 {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return int(math.Pow(3, float64(a))) * 2\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return int(math.Pow(3, float64(a)))\n}\n
max_product_cutting.js/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n: Int) -> Int {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if n <= 3 {\n return 1 * (n - 1)\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n let a = n / 3\n let b = n % 3\n if b == 1 {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return pow(3, a - 1) * 2 * 2\n }\n if b == 2 {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a) * 2\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a)\n}\n
max_product_cutting.ts/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n let a = Math.floor(n / 3);\n let b = n % 3;\n if (b === 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return Math.pow(3, a - 1) * 2 * 2;\n }\n if (b === 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return Math.pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return Math.pow(3, a);\n}\n
max_product_cutting.dart/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n: number): number {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n let a: number = Math.floor(n / 3);\n let b: number = n % 3;\n if (b === 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return Math.pow(3, a - 1) * 2 * 2;\n }\n if (b === 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return Math.pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return Math.pow(3, a);\n}\n
max_product_cutting.rs/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n ~/ 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return (pow(3, a - 1) * 2 * 2).toInt();\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return (pow(3, a) * 2).toInt();\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a).toInt();\n}\n
max_product_cutting.c/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfn max_product_cutting(n: i32) -> i32 {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if n <= 3 {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n let a = n / 3;\n let b = n % 3;\n if b == 1 {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n 3_i32.pow(a as u32 - 1) * 2 * 2\n } else if b == 2 {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n 3_i32.pow(a as u32) * 2\n } else {\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n 3_i32.pow(a as u32)\n }\n}\n
max_product_cutting.zig/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n // \u5f53 n <= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n if (n <= 3) {\n return 1 * (n - 1);\n }\n // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n int a = n / 3;\n int b = n % 3;\n if (b == 1) {\n // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n return pow(3, a - 1) * 2 * 2;\n }\n if (b == 2) {\n // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a) * 2;\n }\n // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n return pow(3, a);\n}\n[class]{}-[func]{maxProductCutting}\n\u56fe 15-16 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u8ba1\u7b97\u65b9\u6cd5
\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u5e42\u8fd0\u7b97\u7684\u5b9e\u73b0\u65b9\u6cd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5e38\u7528\u7684\u5e42\u8ba1\u7b97\u51fd\u6570\u6709\u4e09\u79cd\u3002
- \u8fd0\u7b97\u7b26
**\u548c\u51fd\u6570pow()\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log\u2061 a)\\) \u3002 - \u51fd\u6570
math.pow()\u5185\u90e8\u8c03\u7528 C \u8bed\u8a00\u5e93\u7684pow()\u51fd\u6570\uff0c\u5176\u6267\u884c\u6d6e\u70b9\u53d6\u5e42\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
\u53d8\u91cf \\(a\\) \u548c \\(b\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
"},{"location":"chapter_greedy/max_product_cutting_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"\u4f7f\u7528\u53cd\u8bc1\u6cd5\uff0c\u53ea\u5206\u6790 \\(n \\geq 3\\) \u7684\u60c5\u51b5\u3002
- \u6240\u6709\u56e0\u5b50 \\(\\leq 3\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728 \\(\\geq 4\\) \u7684\u56e0\u5b50 \\(x\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u5c06\u5176\u7ee7\u7eed\u5212\u5206\u4e3a \\(2(x-2)\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002
- \u5207\u5206\u65b9\u6848\u4e0d\u5305\u542b \\(1\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728\u4e00\u4e2a\u56e0\u5b50 \\(1\\) \uff0c\u90a3\u4e48\u5b83\u4e00\u5b9a\u53ef\u4ee5\u5408\u5e76\u5165\u53e6\u5916\u4e00\u4e2a\u56e0\u5b50\u4e2d\uff0c\u4ee5\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002
- \u5207\u5206\u65b9\u6848\u6700\u591a\u5305\u542b\u4e24\u4e2a \\(2\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b\u4e09\u4e2a \\(2\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4e58\u79ef\u66f4\u5927\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002
- \u8d2a\u5fc3\u7b97\u6cd5\u901a\u5e38\u7528\u4e8e\u89e3\u51b3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5176\u539f\u7406\u662f\u5728\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\u90fd\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002
- \u8d2a\u5fc3\u7b97\u6cd5\u4f1a\u8fed\u4ee3\u5730\u505a\u51fa\u4e00\u4e2a\u53c8\u4e00\u4e2a\u7684\u8d2a\u5fc3\u9009\u62e9\uff0c\u6bcf\u8f6e\u90fd\u5c06\u95ee\u9898\u8f6c\u5316\u6210\u4e00\u4e2a\u89c4\u6a21\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u95ee\u9898\u88ab\u89e3\u51b3\u3002
- \u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u5b9e\u73b0\u7b80\u5355\uff0c\u8fd8\u5177\u6709\u5f88\u9ad8\u7684\u89e3\u9898\u6548\u7387\u3002\u76f8\u6bd4\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u66f4\u4f4e\u3002
- \u5728\u96f6\u94b1\u5151\u6362\u95ee\u9898\u4e2d\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1b\u5bf9\u4e8e\u53e6\u5916\u4e00\u4e9b\u786c\u5e01\u7ec4\u5408\u5219\u4e0d\u7136\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u80fd\u627e\u5230\u5f88\u5dee\u7684\u89e3\u3002
- \u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u7684\u95ee\u9898\u5177\u6709\u4e24\u5927\u6027\u8d28\uff1a\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u4ee3\u8868\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002
- \u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u8bc1\u660e\u5e76\u4e0d\u7b80\u5355\u3002\u76f8\u5bf9\u6765\u8bf4\uff0c\u8bc1\u4f2a\u66f4\u52a0\u5bb9\u6613\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002
- \u6c42\u89e3\u8d2a\u5fc3\u95ee\u9898\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\uff1a\u95ee\u9898\u5206\u6790\u3001\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u3001\u6b63\u786e\u6027\u8bc1\u660e\u3002\u5176\u4e2d\uff0c\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6838\u5fc3\u6b65\u9aa4\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u5f80\u5f80\u662f\u96be\u70b9\u3002
- \u5206\u6570\u80cc\u5305\u95ee\u9898\u5728 0-1 \u80cc\u5305\u7684\u57fa\u7840\u4e0a\uff0c\u5141\u8bb8\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u56e0\u6b64\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u3002\u8d2a\u5fc3\u7b56\u7565\u7684\u6b63\u786e\u6027\u53ef\u4ee5\u4f7f\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\u3002
- \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u53ef\u4f7f\u7528\u7a77\u4e3e\u6cd5\u6c42\u89e3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u901a\u8fc7\u8bbe\u8ba1\u8d2a\u5fc3\u7b56\u7565\uff0c\u6bcf\u8f6e\u5411\u5185\u79fb\u52a8\u77ed\u677f\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \u3002
- \u5728\u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u63a8\u7406\u51fa\u4e24\u4e2a\u8d2a\u5fc3\u7b56\u7565\uff1a\\(\\geq 4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u7ee7\u7eed\u5207\u5206\u3001\u6700\u4f18\u5207\u5206\u56e0\u5b50\u4e3a \\(3\\) \u3002\u4ee3\u7801\u4e2d\u5305\u542b\u5e42\u8fd0\u7b97\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5e42\u8fd0\u7b97\u5b9e\u73b0\u65b9\u6cd5\uff0c\u901a\u5e38\u4e3a \\(O(1)\\) \u6216 \\(O(\\log n)\\) \u3002
Abstract
\u5728\u8ba1\u7b97\u673a\u4e16\u754c\u4e2d\uff0c\u54c8\u5e0c\u8868\u5982\u540c\u4e00\u4f4d\u806a\u6167\u7684\u56fe\u4e66\u7ba1\u7406\u5458\u3002
\u4ed6\u77e5\u9053\u5982\u4f55\u8ba1\u7b97\u7d22\u4e66\u53f7\uff0c\u4ece\u800c\u53ef\u4ee5\u5feb\u901f\u627e\u5230\u76ee\u6807\u56fe\u4e66\u3002
"},{"location":"chapter_hashing/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 6.1 \u00a0 \u54c8\u5e0c\u8868
- 6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81
- 6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5
- 6.4 \u00a0 \u5c0f\u7ed3
\u524d\u4e24\u8282\u4ecb\u7ecd\u4e86\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u539f\u7406\u548c\u54c8\u5e0c\u51b2\u7a81\u7684\u5904\u7406\u65b9\u6cd5\u3002\u7136\u800c\u65e0\u8bba\u662f\u5f00\u653e\u5bfb\u5740\u8fd8\u662f\u94fe\u5f0f\u5730\u5740\uff0c\u5b83\u4eec\u53ea\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u53d1\u751f\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\uff0c\u800c\u65e0\u6cd5\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u3002
\u5982\u679c\u54c8\u5e0c\u51b2\u7a81\u8fc7\u4e8e\u9891\u7e41\uff0c\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u5219\u4f1a\u6025\u5267\u52a3\u5316\u3002\u5982\u56fe 6-8 \u6240\u793a\uff0c\u5bf9\u4e8e\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\uff0c\u7406\u60f3\u60c5\u51b5\u4e0b\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u5404\u4e2a\u6876\u4e2d\uff0c\u8fbe\u5230\u6700\u4f73\u67e5\u8be2\u6548\u7387\uff1b\u6700\u5dee\u60c5\u51b5\u4e0b\u6240\u6709\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002
\u56fe 6-8 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u7684\u6700\u4f73\u60c5\u51b5\u4e0e\u6700\u5dee\u60c5\u51b5
\u952e\u503c\u5bf9\u7684\u5206\u5e03\u60c5\u51b5\u7531\u54c8\u5e0c\u51fd\u6570\u51b3\u5b9a\u3002\u56de\u5fc6\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u6b65\u9aa4\uff0c\u5148\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u518d\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\uff1a
index = hash(key) % capacity\n\u89c2\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf
capacity\u56fa\u5b9a\u65f6\uff0c\u54c8\u5e0c\u7b97\u6cd5hash()\u51b3\u5b9a\u4e86\u8f93\u51fa\u503c\uff0c\u8fdb\u800c\u51b3\u5b9a\u4e86\u952e\u503c\u5bf9\u5728\u54c8\u5e0c\u8868\u4e2d\u7684\u5206\u5e03\u60c5\u51b5\u3002\u8fd9\u610f\u5473\u7740\uff0c\u4e3a\u4e86\u964d\u4f4e\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u6982\u7387\uff0c\u6211\u4eec\u5e94\u5f53\u5c06\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u54c8\u5e0c\u7b97\u6cd5
"},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u76ee\u6807","text":"hash()\u7684\u8bbe\u8ba1\u4e0a\u3002\u4e3a\u4e86\u5b9e\u73b0\u201c\u65e2\u5feb\u53c8\u7a33\u201d\u7684\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u5177\u5907\u4ee5\u4e0b\u7279\u70b9\u3002
- \u786e\u5b9a\u6027\uff1a\u5bf9\u4e8e\u76f8\u540c\u7684\u8f93\u5165\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u59cb\u7ec8\u4ea7\u751f\u76f8\u540c\u7684\u8f93\u51fa\u3002\u8fd9\u6837\u624d\u80fd\u786e\u4fdd\u54c8\u5e0c\u8868\u662f\u53ef\u9760\u7684\u3002
- \u6548\u7387\u9ad8\uff1a\u8ba1\u7b97\u54c8\u5e0c\u503c\u7684\u8fc7\u7a0b\u5e94\u8be5\u8db3\u591f\u5feb\u3002\u8ba1\u7b97\u5f00\u9500\u8d8a\u5c0f\uff0c\u54c8\u5e0c\u8868\u7684\u5b9e\u7528\u6027\u8d8a\u9ad8\u3002
- \u5747\u5300\u5206\u5e03\uff1a\u54c8\u5e0c\u7b97\u6cd5\u5e94\u4f7f\u5f97\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u54c8\u5e0c\u8868\u4e2d\u3002\u5206\u5e03\u8d8a\u5747\u5300\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002
\u5b9e\u9645\u4e0a\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u54c8\u5e0c\u8868\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5176\u4ed6\u9886\u57df\u4e2d\u3002
- \u5bc6\u7801\u5b58\u50a8\uff1a\u4e3a\u4e86\u4fdd\u62a4\u7528\u6237\u5bc6\u7801\u7684\u5b89\u5168\uff0c\u7cfb\u7edf\u901a\u5e38\u4e0d\u4f1a\u76f4\u63a5\u5b58\u50a8\u7528\u6237\u7684\u660e\u6587\u5bc6\u7801\uff0c\u800c\u662f\u5b58\u50a8\u5bc6\u7801\u7684\u54c8\u5e0c\u503c\u3002\u5f53\u7528\u6237\u8f93\u5165\u5bc6\u7801\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5bf9\u8f93\u5165\u7684\u5bc6\u7801\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u7136\u540e\u4e0e\u5b58\u50a8\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u5bc6\u7801\u5c31\u88ab\u89c6\u4e3a\u6b63\u786e\u3002
- \u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5\uff1a\u6570\u636e\u53d1\u9001\u65b9\u53ef\u4ee5\u8ba1\u7b97\u6570\u636e\u7684\u54c8\u5e0c\u503c\u5e76\u5c06\u5176\u4e00\u540c\u53d1\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8ba1\u7b97\u63a5\u6536\u5230\u7684\u6570\u636e\u7684\u54c8\u5e0c\u503c\uff0c\u5e76\u4e0e\u63a5\u6536\u5230\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u6570\u636e\u5c31\u88ab\u89c6\u4e3a\u5b8c\u6574\u3002
\u5bf9\u4e8e\u5bc6\u7801\u5b66\u7684\u76f8\u5173\u5e94\u7528\uff0c\u4e3a\u4e86\u9632\u6b62\u4ece\u54c8\u5e0c\u503c\u63a8\u5bfc\u51fa\u539f\u59cb\u5bc6\u7801\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9700\u8981\u5177\u5907\u66f4\u9ad8\u7b49\u7ea7\u7684\u5b89\u5168\u7279\u6027\u3002
- \u5355\u5411\u6027\uff1a\u65e0\u6cd5\u901a\u8fc7\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u5173\u4e8e\u8f93\u5165\u6570\u636e\u7684\u4efb\u4f55\u4fe1\u606f\u3002
- \u6297\u78b0\u649e\u6027\uff1a\u5e94\u5f53\u6781\u96be\u627e\u5230\u4e24\u4e2a\u4e0d\u540c\u7684\u8f93\u5165\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u54c8\u5e0c\u503c\u76f8\u540c\u3002
- \u96ea\u5d29\u6548\u5e94\uff1a\u8f93\u5165\u7684\u5fae\u5c0f\u53d8\u5316\u5e94\u5f53\u5bfc\u81f4\u8f93\u51fa\u7684\u663e\u8457\u4e14\u4e0d\u53ef\u9884\u6d4b\u7684\u53d8\u5316\u3002
\u8bf7\u6ce8\u610f\uff0c\u201c\u5747\u5300\u5206\u5e03\u201d\u4e0e\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u4e24\u4e2a\u72ec\u7acb\u7684\u6982\u5ff5\uff0c\u6ee1\u8db3\u5747\u5300\u5206\u5e03\u4e0d\u4e00\u5b9a\u6ee1\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u968f\u673a\u8f93\u5165
"},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1","text":"key\u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570key % 100\u53ef\u4ee5\u4ea7\u751f\u5747\u5300\u5206\u5e03\u7684\u8f93\u51fa\u3002\u7136\u800c\u8be5\u54c8\u5e0c\u7b97\u6cd5\u8fc7\u4e8e\u7b80\u5355\uff0c\u6240\u6709\u540e\u4e24\u4f4d\u76f8\u7b49\u7684key\u7684\u8f93\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4ece\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684key\uff0c\u4ece\u800c\u7834\u89e3\u5bc6\u7801\u3002\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u9700\u8981\u8003\u8651\u8bb8\u591a\u56e0\u7d20\u7684\u590d\u6742\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u67d0\u4e9b\u8981\u6c42\u4e0d\u9ad8\u7684\u573a\u666f\uff0c\u6211\u4eec\u4e5f\u80fd\u8bbe\u8ba1\u4e00\u4e9b\u7b80\u5355\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002
- \u52a0\u6cd5\u54c8\u5e0c\uff1a\u5bf9\u8f93\u5165\u7684\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u8fdb\u884c\u76f8\u52a0\uff0c\u5c06\u5f97\u5230\u7684\u603b\u548c\u4f5c\u4e3a\u54c8\u5e0c\u503c\u3002
- \u4e58\u6cd5\u54c8\u5e0c\uff1a\u5229\u7528\u4e58\u6cd5\u7684\u4e0d\u76f8\u5173\u6027\uff0c\u6bcf\u8f6e\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5404\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u54c8\u5e0c\u503c\u4e2d\u3002
- \u5f02\u6216\u54c8\u5e0c\uff1a\u5c06\u8f93\u5165\u6570\u636e\u7684\u6bcf\u4e2a\u5143\u7d20\u901a\u8fc7\u5f02\u6216\u64cd\u4f5c\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\u3002
- \u65cb\u8f6c\u54c8\u5e0c\uff1a\u5c06\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u79ef\u4e4b\u524d\u90fd\u4f1a\u5bf9\u54c8\u5e0c\u503c\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u3002
simple_hash.cppdef add_hash(key: str) -> int:\n \"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\n hash = 0\n modulus = 1000000007\n for c in key:\n hash += ord(c)\n return hash % modulus\n\ndef mul_hash(key: str) -> int:\n \"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\n hash = 0\n modulus = 1000000007\n for c in key:\n hash = 31 * hash + ord(c)\n return hash % modulus\n\ndef xor_hash(key: str) -> int:\n \"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\n hash = 0\n modulus = 1000000007\n for c in key:\n hash ^= ord(c)\n return hash % modulus\n\ndef rot_hash(key: str) -> int:\n \"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\n hash = 0\n modulus = 1000000007\n for c in key:\n hash = (hash << 4) ^ (hash >> 28) ^ ord(c)\n return hash % modulus\n
simple_hash.java/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (unsigned char c : key) {\n hash = (hash + (int)c) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (unsigned char c : key) {\n hash = (31 * hash + (int)c) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\n int hash = 0;\n const int MODULUS = 1000000007;\n for (unsigned char c : key) {\n hash ^= (int)c;\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (unsigned char c : key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ (int)c) % MODULUS;\n }\n return (int)hash;\n}\n
simple_hash.cs/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n long hash = 0;\n final int MODULUS = 1000000007;\n for (char c : key.toCharArray()) {\n hash = (hash + (int) c) % MODULUS;\n }\n return (int) hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n long hash = 0;\n final int MODULUS = 1000000007;\n for (char c : key.toCharArray()) {\n hash = (31 * hash + (int) c) % MODULUS;\n }\n return (int) hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (char c : key.toCharArray()) {\n hash ^= (int) c;\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n long hash = 0;\n final int MODULUS = 1000000007;\n for (char c : key.toCharArray()) {\n hash = ((hash << 4) ^ (hash >> 28) ^ (int) c) % MODULUS;\n }\n return (int) hash;\n}\n
simple_hash.go/* \u52a0\u6cd5\u54c8\u5e0c */\nint AddHash(string key) {\n long hash = 0;\n const int MODULUS = 1000000007;\n foreach (char c in key) {\n hash = (hash + c) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint MulHash(string key) {\n long hash = 0;\n const int MODULUS = 1000000007;\n foreach (char c in key) {\n hash = (31 * hash + c) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint XorHash(string key) {\n int hash = 0;\n const int MODULUS = 1000000007;\n foreach (char c in key) {\n hash ^= c;\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint RotHash(string key) {\n long hash = 0;\n const int MODULUS = 1000000007;\n foreach (char c in key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ c) % MODULUS;\n }\n return (int)hash;\n}\n
simple_hash.swift/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\n var hash int64\n var modulus int64\n\n modulus = 1000000007\n for _, b := range []byte(key) {\n hash = (hash + int64(b)) % modulus\n }\n return int(hash)\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\n var hash int64\n var modulus int64\n\n modulus = 1000000007\n for _, b := range []byte(key) {\n hash = (31*hash + int64(b)) % modulus\n }\n return int(hash)\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\n hash := 0\n modulus := 1000000007\n for _, b := range []byte(key) {\n fmt.Println(int(b))\n hash ^= int(b)\n hash = (31*hash + int(b)) % modulus\n }\n return hash & modulus\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\n var hash int64\n var modulus int64\n\n modulus = 1000000007\n for _, b := range []byte(key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ int64(b)) % modulus\n }\n return int(hash)\n}\n
simple_hash.js/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -> Int {\n var hash = 0\n let MODULUS = 1_000_000_007\n for c in key {\n for scalar in c.unicodeScalars {\n hash = (hash + Int(scalar.value)) % MODULUS\n }\n }\n return hash\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -> Int {\n var hash = 0\n let MODULUS = 1_000_000_007\n for c in key {\n for scalar in c.unicodeScalars {\n hash = (31 * hash + Int(scalar.value)) % MODULUS\n }\n }\n return hash\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -> Int {\n var hash = 0\n let MODULUS = 1_000_000_007\n for c in key {\n for scalar in c.unicodeScalars {\n hash ^= Int(scalar.value)\n }\n }\n return hash & MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -> Int {\n var hash = 0\n let MODULUS = 1_000_000_007\n for c in key {\n for scalar in c.unicodeScalars {\n hash = ((hash << 4) ^ (hash >> 28) ^ Int(scalar.value)) % MODULUS\n }\n }\n return hash\n}\n
simple_hash.ts/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key) {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = (hash + c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key) {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key) {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash ^= c.charCodeAt(0);\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key) {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n
simple_hash.dart/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key: string): number {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = (hash + c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key: string): number {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key: string): number {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash ^= c.charCodeAt(0);\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key: string): number {\n let hash = 0;\n const MODULUS = 1000000007;\n for (const c of key) {\n hash = ((hash << 4) ^ (hash >> 28) ^ c.charCodeAt(0)) % MODULUS;\n }\n return hash;\n}\n
simple_hash.rs/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (int i = 0; i < key.length; i++) {\n hash = (hash + key.codeUnitAt(i)) % MODULUS;\n }\n return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (int i = 0; i < key.length; i++) {\n hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n }\n return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (int i = 0; i < key.length; i++) {\n hash ^= key.codeUnitAt(i);\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n int hash = 0;\n final int MODULUS = 1000000007;\n for (int i = 0; i < key.length; i++) {\n hash = ((hash << 4) ^ (hash >> 28) ^ key.codeUnitAt(i)) % MODULUS;\n }\n return hash;\n}\n
simple_hash.c/* \u52a0\u6cd5\u54c8\u5e0c */\nfn add_hash(key: &str) -> i32 {\n let mut hash = 0_i64;\n const MODULUS: i64 = 1000000007;\n\n for c in key.chars() {\n hash = (hash + c as i64) % MODULUS;\n }\n\n hash as i32\n} \n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfn mul_hash(key: &str) -> i32 {\n let mut hash = 0_i64;\n const MODULUS: i64 = 1000000007;\n\n for c in key.chars() {\n hash = (31 * hash + c as i64) % MODULUS;\n }\n\n hash as i32\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfn xor_hash(key: &str) -> i32 {\n let mut hash = 0_i64;\n const MODULUS: i64 = 1000000007;\n\n for c in key.chars() {\n hash ^= c as i64;\n }\n\n (hash & MODULUS) as i32\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfn rot_hash(key: &str) -> i32 {\n let mut hash = 0_i64;\n const MODULUS: i64 = 1000000007;\n\n for c in key.chars() {\n hash = ((hash << 4) ^ (hash >> 28) ^ c as i64) % MODULUS;\n }\n\n hash as i32\n}\n
simple_hash.zig/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(char *key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (int i = 0; i < strlen(key); i++) {\n hash = (hash + (unsigned char)key[i]) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(char *key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (int i = 0; i < strlen(key); i++) {\n hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n }\n return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(char *key) {\n int hash = 0;\n const int MODULUS = 1000000007;\n\n for (int i = 0; i < strlen(key); i++) {\n hash ^= (unsigned char)key[i];\n }\n return hash & MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(char *key) {\n long long hash = 0;\n const int MODULUS = 1000000007;\n for (int i = 0; i < strlen(key); i++) {\n hash = ((hash << 4) ^ (hash >> 28) ^ (unsigned char)key[i]) % MODULUS;\n }\n\n return (int)hash;\n}\n[class]{}-[func]{addHash}\n\n[class]{}-[func]{mulHash}\n\n[class]{}-[func]{xorHash}\n\n[class]{}-[func]{rotHash}\n\u89c2\u5bdf\u53d1\u73b0\uff0c\u6bcf\u79cd\u54c8\u5e0c\u7b97\u6cd5\u7684\u6700\u540e\u4e00\u6b65\u90fd\u662f\u5bf9\u5927\u8d28\u6570 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u786e\u4fdd\u54c8\u5e0c\u503c\u5728\u5408\u9002\u7684\u8303\u56f4\u5185\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u8981\u5f3a\u8c03\u5bf9\u8d28\u6570\u53d6\u6a21\uff0c\u6216\u8005\u8bf4\u5bf9\u5408\u6570\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u4e48\uff1f\u8fd9\u662f\u4e00\u4e2a\u6709\u8da3\u7684\u95ee\u9898\u3002
\u5148\u629b\u51fa\u7ed3\u8bba\uff1a\u4f7f\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\u3002\u56e0\u4e3a\u8d28\u6570\u4e0d\u4e0e\u5176\u4ed6\u6570\u5b57\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u53ef\u4ee5\u51cf\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u4ea7\u751f\u7684\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u4ece\u800c\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u3002
\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u5408\u6570 \\(9\\) \u4f5c\u4e3a\u6a21\u6570\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\uff0c\u90a3\u4e48\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684
\\[ \\begin{aligned} \\text{modulus} & = 9 \\newline \\text{key} & = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} & = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\]key\u90fd\u4f1a\u88ab\u6620\u5c04\u5230 \\(0\\)\u3001\\(3\\)\u3001\\(6\\) \u8fd9\u4e09\u4e2a\u54c8\u5e0c\u503c\u3002\u5982\u679c\u8f93\u5165
\\[ \\begin{aligned} \\text{modulus} & = 13 \\newline \\text{key} & = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} & = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\]key\u6070\u597d\u6ee1\u8db3\u8fd9\u79cd\u7b49\u5dee\u6570\u5217\u7684\u6570\u636e\u5206\u5e03\uff0c\u90a3\u4e48\u54c8\u5e0c\u503c\u5c31\u4f1a\u51fa\u73b0\u805a\u5806\uff0c\u4ece\u800c\u52a0\u91cd\u54c8\u5e0c\u51b2\u7a81\u3002\u73b0\u5728\uff0c\u5047\u8bbe\u5c06modulus\u66ff\u6362\u4e3a\u8d28\u6570 \\(13\\) \uff0c\u7531\u4e8ekey\u548cmodulus\u4e4b\u95f4\u4e0d\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u56e0\u6b64\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u7684\u5747\u5300\u6027\u4f1a\u660e\u663e\u63d0\u5347\u3002\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5982\u679c\u80fd\u591f\u4fdd\u8bc1
key\u662f\u968f\u673a\u5747\u5300\u5206\u5e03\u7684\uff0c\u90a3\u4e48\u9009\u62e9\u8d28\u6570\u6216\u8005\u5408\u6570\u4f5c\u4e3a\u6a21\u6570\u90fd\u53ef\u4ee5\uff0c\u5b83\u4eec\u90fd\u80fd\u8f93\u51fa\u5747\u5300\u5206\u5e03\u7684\u54c8\u5e0c\u503c\u3002\u800c\u5f53key\u7684\u5206\u5e03\u5b58\u5728\u67d0\u79cd\u5468\u671f\u6027\u65f6\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73b0\u805a\u96c6\u73b0\u8c61\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u901a\u5e38\u9009\u53d6\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u5e76\u4e14\u8fd9\u4e2a\u8d28\u6570\u6700\u597d\u8db3\u591f\u5927\uff0c\u4ee5\u5c3d\u53ef\u80fd\u6d88\u9664\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002
"},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3 \u00a0 \u5e38\u89c1\u54c8\u5e0c\u7b97\u6cd5","text":"\u4e0d\u96be\u53d1\u73b0\uff0c\u4ee5\u4e0a\u4ecb\u7ecd\u7684\u7b80\u5355\u54c8\u5e0c\u7b97\u6cd5\u90fd\u6bd4\u8f83\u201c\u8106\u5f31\u201d\uff0c\u8fdc\u8fdc\u6ca1\u6709\u8fbe\u5230\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u3002\u4f8b\u5982\uff0c\u7531\u4e8e\u52a0\u6cd5\u548c\u5f02\u6216\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u54c8\u5e0c\u548c\u5f02\u6216\u54c8\u5e0c\u65e0\u6cd5\u533a\u5206\u5185\u5bb9\u76f8\u540c\u4f46\u987a\u5e8f\u4e0d\u540c\u7684\u5b57\u7b26\u4e32\uff0c\u8fd9\u53ef\u80fd\u4f1a\u52a0\u5267\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e76\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u95ee\u9898\u3002
\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u7528\u4e00\u4e9b\u6807\u51c6\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5\u3001SHA-1\u3001SHA-2\u3001SHA-3 \u7b49\u3002\u5b83\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u957f\u5ea6\u7684\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u6052\u5b9a\u957f\u5ea6\u7684\u54c8\u5e0c\u503c\u3002
\u8fd1\u4e00\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5904\u5728\u4e0d\u65ad\u5347\u7ea7\u4e0e\u4f18\u5316\u7684\u8fc7\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u52aa\u529b\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u6027\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u548c\u9ed1\u5ba2\u5219\u81f4\u529b\u4e8e\u5bfb\u627e\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u95ee\u9898\u3002\u8868 6-2 \u5c55\u793a\u4e86\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002
- MD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u51fb\uff0c\u56e0\u6b64\u5b83\u4eec\u88ab\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u5f03\u7528\u3002
- SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u54c8\u5e0c\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73b0\u6210\u529f\u7684\u653b\u51fb\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u7528\u5728\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u4e2d\u3002
- SHA-3 \u76f8\u8f83 SHA-2 \u7684\u5b9e\u73b0\u5f00\u9500\u66f4\u4f4e\u3001\u8ba1\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u76d6\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002
\u8868 6-2 \u00a0 \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5
MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u65f6\u95f4 1992 1995 2002 2008 \u8f93\u51fa\u957f\u5ea6 128 bits 160 bits 256/512 bits 224/256/384/512 bits \u54c8\u5e0c\u51b2\u7a81 \u8f83\u591a \u8f83\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7ea7 \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u9ad8 \u9ad8 \u5e94\u7528 \u5df2\u88ab\u5f03\u7528\uff0c\u4ecd\u7528\u4e8e\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5 \u5df2\u88ab\u5f03\u7528 \u52a0\u5bc6\u8d27\u5e01\u4ea4\u6613\u9a8c\u8bc1\u3001\u6570\u5b57\u7b7e\u540d\u7b49 \u53ef\u7528\u4e8e\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u54c8\u5e0c\u503c","text":"\u6211\u4eec\u77e5\u9053\uff0c\u54c8\u5e0c\u8868\u7684
key\u53ef\u4ee5\u662f\u6574\u6570\u3001\u5c0f\u6570\u6216\u5b57\u7b26\u4e32\u7b49\u6570\u636e\u7c7b\u578b\u3002\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u8fd9\u4e9b\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u8c03\u7528hash()\u51fd\u6570\u6765\u8ba1\u7b97\u5404\u79cd\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u3002- \u6574\u6570\u548c\u5e03\u5c14\u91cf\u7684\u54c8\u5e0c\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002
- \u6d6e\u70b9\u6570\u548c\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u8f83\u4e3a\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8bf7\u81ea\u884c\u5b66\u4e60\u3002
- \u5143\u7ec4\u7684\u54c8\u5e0c\u503c\u662f\u5bf9\u5176\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u54c8\u5e0c\uff0c\u7136\u540e\u5c06\u8fd9\u4e9b\u54c8\u5e0c\u503c\u7ec4\u5408\u8d77\u6765\uff0c\u5f97\u5230\u5355\u4e00\u7684\u54c8\u5e0c\u503c\u3002
- \u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u57fa\u4e8e\u5176\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u901a\u8fc7\u91cd\u5199\u5bf9\u8c61\u7684\u54c8\u5e0c\u65b9\u6cd5\uff0c\u53ef\u5b9e\u73b0\u57fa\u4e8e\u5185\u5bb9\u751f\u6210\u54c8\u5e0c\u503c\u3002
Tip
\u8bf7\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u54c8\u5e0c\u503c\u8ba1\u7b97\u51fd\u6570\u7684\u5b9a\u4e49\u548c\u65b9\u6cd5\u4e0d\u540c\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig built_in_hash.py
built_in_hash.cppnum = 3\nhash_num = hash(num)\n# \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u5c14\u91cf True \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 326484311674566659\n\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 4617003410720528961\n\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7ec4 (12836, '\u5c0f\u54c8') \u7684\u54c8\u5e0c\u503c\u4e3a 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u8282\u70b9\u5bf9\u8c61 <ListNode object at 0x1058fd810> \u7684\u54c8\u5e0c\u503c\u4e3a 274267521\n
built_in_hash.javaint num = 3;\nsize_t hashNum = hash<int>()(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbool bol = true;\nsize_t hashBol = hash<bool>()(bol);\n// \u5e03\u5c14\u91cf 1 \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash<double>()(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 4614256650576692846\n\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash<string>()(str);\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 15466937326284535026\n\n// \u5728 C++ \u4e2d\uff0c\u5185\u7f6e std:hash() \u4ec5\u63d0\u4f9b\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\n// \u6570\u7ec4\u3001\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u9700\u8981\u81ea\u884c\u5b9e\u73b0\n
built_in_hash.csint num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -727081396\n\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@7dc5e7b4 \u7684\u54c8\u5e0c\u503c\u4e3a 2110121908\n
built_in_hash.goint num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729;\n\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -586107568;\n\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// \u8282\u70b9\u5bf9\u8c61 0 \u7684\u54c8\u5e0c\u503c\u4e3a 39053774;\n
built_in_hash.swift// Go \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n
built_in_hash.jslet num = 3\nlet hashNum = num.hashValue\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -2465384235396674631\n\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u6570\u7ec4 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u54c8\u5e0c\u503c\u4e3a -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode \u7684\u54c8\u5e0c\u503c\u4e3a -2434780518035996159\n
built_in_hash.ts// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n
built_in_hash.dart// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n
built_in_hash.rsint num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u5c14\u503c true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2570631074981783\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 468167534\n\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u8282\u70b9\u5bf9\u8c61 Instance of 'ListNode' \u7684\u54c8\u5e0c\u503c\u4e3a 1033450432\n
built_in_hash.cuse std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&mut num_hasher);\nlet hash_num = num_hasher.finish();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\nprintln!(\"\u5c0f\u6570 {} \u7684\u54c8\u5e0c\u503c\u4e3a {}\", dec, hash_dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2566941990314602357\n\nlet str = \"Hello \u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&mut str_hasher);\nlet hash_str = str_hasher.finish();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 16092673739211250988\n\nlet arr = (&12836, &\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// \u5143\u7ec4 (12836, \"\u5c0f\u54c8\") \u7684\u54c8\u5e0c\u503c\u4e3a 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&mut hasher);\nlet hash = hasher.finish();\n// \u8282\u70b9\u5bf9\u8c61 RefCell { value: ListNode { val: 42, next: None } } \u7684\u54c8\u5e0c\u503c\u4e3a15387811073369036852\n
built_in_hash.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n\n\u5728\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u624d\u53ef\u4f5c\u4e3a\u54c8\u5e0c\u8868\u7684
key\u3002\u5047\u5982\u6211\u4eec\u5c06\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4f5c\u4e3akey\uff0c\u5f53\u5217\u8868\u7684\u5185\u5bb9\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u5b83\u7684\u54c8\u5e0c\u503c\u4e5f\u968f\u4e4b\u6539\u53d8\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5230\u539f\u5148\u7684value\u4e86\u3002\u867d\u7136\u81ea\u5b9a\u4e49\u5bf9\u8c61\uff08\u6bd4\u5982\u94fe\u8868\u8282\u70b9\uff09\u7684\u6210\u5458\u53d8\u91cf\u662f\u53ef\u53d8\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u54c8\u5e0c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u901a\u5e38\u662f\u57fa\u4e8e\u5185\u5b58\u5730\u5740\u751f\u6210\u7684\uff0c\u5373\u4f7f\u5bf9\u8c61\u7684\u5185\u5bb9\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u4f46\u5b83\u7684\u5185\u5b58\u5730\u5740\u4e0d\u53d8\uff0c\u54c8\u5e0c\u503c\u4ecd\u7136\u662f\u4e0d\u53d8\u7684\u3002
\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u53d1\u73b0\u5728\u4e0d\u540c\u63a7\u5236\u53f0\u4e2d\u8fd0\u884c\u7a0b\u5e8f\u65f6\uff0c\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u662f\u4e0d\u540c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a Python \u89e3\u91ca\u5668\u5728\u6bcf\u6b21\u542f\u52a8\u65f6\uff0c\u90fd\u4f1a\u4e3a\u5b57\u7b26\u4e32\u54c8\u5e0c\u51fd\u6570\u52a0\u5165\u4e00\u4e2a\u968f\u673a\u7684\u76d0\uff08Salt\uff09\u503c\u3002\u8fd9\u79cd\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u51fb\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002
"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"\u4e0a\u4e00\u8282\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u7406\u8bba\u4e0a\u54c8\u5e0c\u51b2\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u6bd4\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u4e3a\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u4e3a\u6570\u7ec4\u5bb9\u91cf\u5927\u5c0f\uff0c\u5219\u5fc5\u7136\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u81f3\u540c\u4e00\u6876\u7d22\u5f15\u3002
\u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002\u4e3a\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u6bcf\u5f53\u9047\u5230\u54c8\u5e0c\u51b2\u7a81\u5c31\u8fdb\u884c\u54c8\u5e0c\u8868\u6269\u5bb9\uff0c\u76f4\u81f3\u51b2\u7a81\u6d88\u5931\u4e3a\u6b62\u3002\u6b64\u65b9\u6cd5\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u6570\u636e\u642c\u8fd0\u4e0e\u54c8\u5e0c\u503c\u8ba1\u7b97\u3002\u4e3a\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u7b56\u7565\u3002
- \u6539\u826f\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u4f7f\u5f97\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\u3002
- \u4ec5\u5728\u5fc5\u8981\u65f6\uff0c\u5373\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u624d\u6267\u884c\u6269\u5bb9\u64cd\u4f5c\u3002
\u54c8\u5e0c\u8868\u7684\u7ed3\u6784\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u201c\u94fe\u5f0f\u5730\u5740\u201d\u548c\u201c\u5f00\u653e\u5bfb\u5740\u201d\u3002
"},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1 \u00a0 \u94fe\u5f0f\u5730\u5740","text":"\u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u4ec5\u80fd\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u300c\u94fe\u5f0f\u5730\u5740 separate chaining\u300d\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u6362\u4e3a\u94fe\u8868\uff0c\u5c06\u952e\u503c\u5bf9\u4f5c\u4e3a\u94fe\u8868\u8282\u70b9\uff0c\u5c06\u6240\u6709\u53d1\u751f\u51b2\u7a81\u7684\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5728\u540c\u4e00\u94fe\u8868\u4e2d\u3002\u56fe 6-5 \u5c55\u793a\u4e86\u4e00\u4e2a\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u4f8b\u5b50\u3002
\u56fe 6-5 \u00a0 \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868
\u57fa\u4e8e\u94fe\u5f0f\u5730\u5740\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u53d1\u751f\u4e86\u4ee5\u4e0b\u53d8\u5316\u3002
- \u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165
key\uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6876\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4key\u4ee5\u67e5\u627e\u76ee\u6807\u952e\u503c\u5bf9\u3002 - \u6dfb\u52a0\u5143\u7d20\uff1a\u9996\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u5c06\u8282\u70b9\uff08\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u4e2d\u3002
- \u5220\u9664\u5143\u7d20\uff1a\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7684\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u63a5\u7740\u904d\u5386\u94fe\u8868\u4ee5\u67e5\u627e\u76ee\u6807\u8282\u70b9\u5e76\u5c06\u5176\u5220\u9664\u3002
\u94fe\u5f0f\u5730\u5740\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002
- \u5360\u7528\u7a7a\u95f4\u589e\u5927\uff0c\u94fe\u8868\u5305\u542b\u8282\u70b9\u6307\u9488\uff0c\u5b83\u76f8\u6bd4\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002
- \u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff0c\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\u3002
\u4ee5\u4e0b\u4ee3\u7801\u7ed9\u51fa\u4e86\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u9700\u8981\u6ce8\u610f\u4e24\u70b9\u3002
- \u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4ee3\u66ff\u94fe\u8868\uff0c\u4ece\u800c\u7b80\u5316\u4ee3\u7801\u3002\u5728\u8fd9\u79cd\u8bbe\u5b9a\u4e0b\uff0c\u54c8\u5e0c\u8868\uff08\u6570\u7ec4\uff09\u5305\u542b\u591a\u4e2a\u6876\uff0c\u6bcf\u4e2a\u6876\u90fd\u662f\u4e00\u4e2a\u5217\u8868\u3002
- \u4ee5\u4e0b\u5b9e\u73b0\u5305\u542b\u54c8\u5e0c\u8868\u6269\u5bb9\u65b9\u6cd5\u3002\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(\\frac{2}{3}\\) \u65f6\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002
hash_map_chaining.cppclass HashMapChaining:\n \"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self.size = 0 # \u952e\u503c\u5bf9\u6570\u91cf\n self.capacity = 4 # \u54c8\u5e0c\u8868\u5bb9\u91cf\n self.load_thres = 2.0 / 3.0 # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n self.extend_ratio = 2 # \u6269\u5bb9\u500d\u6570\n self.buckets = [[] for _ in range(self.capacity)] # \u6876\u6570\u7ec4\n\n def hash_func(self, key: int) -> int:\n \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n return key % self.capacity\n\n def load_factor(self) -> float:\n \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n return self.size / self.capacity\n\n def get(self, key: int) -> str | None:\n \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n index = self.hash_func(key)\n bucket = self.buckets[index]\n # \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for pair in bucket:\n if pair.key == key:\n return pair.val\n # \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\n return None\n\n def put(self, key: int, val: str):\n \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if self.load_factor() > self.load_thres:\n self.extend()\n index = self.hash_func(key)\n bucket = self.buckets[index]\n # \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for pair in bucket:\n if pair.key == key:\n pair.val = val\n return\n # \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n pair = Pair(key, val)\n bucket.append(pair)\n self.size += 1\n\n def remove(self, key: int):\n \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n index = self.hash_func(key)\n bucket = self.buckets[index]\n # \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for pair in bucket:\n if pair.key == key:\n bucket.remove(pair)\n self.size -= 1\n break\n\n def extend(self):\n \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n buckets = self.buckets\n # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n self.capacity *= self.extend_ratio\n self.buckets = [[] for _ in range(self.capacity)]\n self.size = 0\n # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for bucket in buckets:\n for pair in bucket:\n self.put(pair.key, pair.val)\n\n def print(self):\n \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n for bucket in self.buckets:\n res = []\n for pair in bucket:\n res.append(str(pair.key) + \" -> \" + pair.val)\n print(res)\n
hash_map_chaining.java/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n private:\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n vector<vector<Pair *>> buckets; // \u6876\u6570\u7ec4\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n buckets.resize(capacity);\n }\n\n /* \u6790\u6784\u65b9\u6cd5 */\n ~HashMapChaining() {\n for (auto &bucket : buckets) {\n for (Pair *pair : bucket) {\n // \u91ca\u653e\u5185\u5b58\n delete pair;\n }\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return (double)size / (double)capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n string get(int key) {\n int index = hashFunc(key);\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (Pair *pair : buckets[index]) {\n if (pair->key == key) {\n return pair->val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\";\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, string val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n int index = hashFunc(key);\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (Pair *pair : buckets[index]) {\n if (pair->key == key) {\n pair->val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n buckets[index].push_back(new Pair(key, val));\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n int index = hashFunc(key);\n auto &bucket = buckets[index];\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (int i = 0; i < bucket.size(); i++) {\n if (bucket[i]->key == key) {\n Pair *tmp = bucket[i];\n bucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n delete tmp; // \u91ca\u653e\u5185\u5b58\n size--;\n return;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n vector<vector<Pair *>> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets.clear();\n buckets.resize(capacity);\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (auto &bucket : bucketsTmp) {\n for (Pair *pair : bucket) {\n put(pair->key, pair->val);\n // \u91ca\u653e\u5185\u5b58\n delete pair;\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void print() {\n for (auto &bucket : buckets) {\n cout << \"[\";\n for (Pair *pair : bucket) {\n cout << pair->key << \" -> \" << pair->val << \", \";\n }\n cout << \"]\\n\";\n }\n }\n};\n
hash_map_chaining.cs/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n List<List<Pair>> buckets; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public HashMapChaining() {\n size = 0;\n capacity = 4;\n loadThres = 2.0 / 3.0;\n extendRatio = 2;\n buckets = new ArrayList<>(capacity);\n for (int i = 0; i < capacity; i++) {\n buckets.add(new ArrayList<>());\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return (double) size / capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n String get(int key) {\n int index = hashFunc(key);\n List<Pair> bucket = buckets.get(index);\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (Pair pair : bucket) {\n if (pair.key == key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, String val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n int index = hashFunc(key);\n List<Pair> bucket = buckets.get(index);\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (Pair pair : bucket) {\n if (pair.key == key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n Pair pair = new Pair(key, val);\n bucket.add(pair);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n int index = hashFunc(key);\n List<Pair> bucket = buckets.get(index);\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (Pair pair : bucket) {\n if (pair.key == key) {\n bucket.remove(pair);\n size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n List<List<Pair>> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = new ArrayList<>(capacity);\n for (int i = 0; i < capacity; i++) {\n buckets.add(new ArrayList<>());\n }\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (List<Pair> bucket : bucketsTmp) {\n for (Pair pair : bucket) {\n put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void print() {\n for (List<Pair> bucket : buckets) {\n List<String> res = new ArrayList<>();\n for (Pair pair : bucket) {\n res.add(pair.key + \" -> \" + pair.val);\n }\n System.out.println(res);\n }\n }\n}\n
hash_map_chaining.go/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n List<List<Pair>> buckets; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public HashMapChaining() {\n size = 0;\n capacity = 4;\n loadThres = 2.0 / 3.0;\n extendRatio = 2;\n buckets = new List<List<Pair>>(capacity);\n for (int i = 0; i < capacity; i++) {\n buckets.Add([]);\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int HashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double LoadFactor() {\n return (double)size / capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public string? Get(int key) {\n int index = HashFunc(key);\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n foreach (Pair pair in buckets[index]) {\n if (pair.key == key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void Put(int key, string val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (LoadFactor() > loadThres) {\n Extend();\n }\n int index = HashFunc(key);\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n foreach (Pair pair in buckets[index]) {\n if (pair.key == key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n buckets[index].Add(new Pair(key, val));\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void Remove(int key) {\n int index = HashFunc(key);\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n foreach (Pair pair in buckets[index].ToList()) {\n if (pair.key == key) {\n buckets[index].Remove(pair);\n size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void Extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n List<List<Pair>> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = new List<List<Pair>>(capacity);\n for (int i = 0; i < capacity; i++) {\n buckets.Add([]);\n }\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n foreach (List<Pair> bucket in bucketsTmp) {\n foreach (Pair pair in bucket) {\n Put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void Print() {\n foreach (List<Pair> bucket in buckets) {\n List<string> res = [];\n foreach (Pair pair in bucket) {\n res.Add(pair.key + \" -> \" + pair.val);\n }\n foreach (string kv in res) {\n Console.WriteLine(kv);\n }\n }\n }\n}\n
hash_map_chaining.swift/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\n size int // \u952e\u503c\u5bf9\u6570\u91cf\n capacity int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n loadThres float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n extendRatio int // \u6269\u5bb9\u500d\u6570\n buckets [][]pair // \u6876\u6570\u7ec4\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\n buckets := make([][]pair, 4)\n for i := 0; i < 4; i++ {\n buckets[i] = make([]pair, 0)\n }\n return &hashMapChaining{\n size: 0,\n capacity: 4,\n loadThres: 2.0 / 3.0,\n extendRatio: 2,\n buckets: buckets,\n }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\n idx := m.hashFunc(key)\n bucket := m.buckets[idx]\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for _, p := range bucket {\n if p.key == key {\n return p.val\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if m.loadFactor() > m.loadThres {\n m.extend()\n }\n idx := m.hashFunc(key)\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for i := range m.buckets[idx] {\n if m.buckets[idx][i].key == key {\n m.buckets[idx][i].val = val\n return\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n p := pair{\n key: key,\n val: val,\n }\n m.buckets[idx] = append(m.buckets[idx], p)\n m.size += 1\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\n idx := m.hashFunc(key)\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for i, p := range m.buckets[idx] {\n if p.key == key {\n // \u5207\u7247\u5220\u9664\n m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n m.size -= 1\n break\n }\n }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n tmpBuckets := make([][]pair, len(m.buckets))\n for i := 0; i < len(m.buckets); i++ {\n tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n copy(tmpBuckets[i], m.buckets[i])\n }\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n m.capacity *= m.extendRatio\n m.buckets = make([][]pair, m.capacity)\n for i := 0; i < m.capacity; i++ {\n m.buckets[i] = make([]pair, 0)\n }\n m.size = 0\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for _, bucket := range tmpBuckets {\n for _, p := range bucket {\n m.put(p.key, p.val)\n }\n }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\n var builder strings.Builder\n\n for _, bucket := range m.buckets {\n builder.WriteString(\"[\")\n for _, p := range bucket {\n builder.WriteString(strconv.Itoa(p.key) + \" -> \" + p.val + \" \")\n }\n builder.WriteString(\"]\")\n fmt.Println(builder.String())\n builder.Reset()\n }\n}\n
hash_map_chaining.js/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n var buckets: [[Pair]] // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init() {\n size = 0\n capacity = 4\n loadThres = 2.0 / 3.0\n extendRatio = 2\n buckets = Array(repeating: [], count: capacity)\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n func hashFunc(key: Int) -> Int {\n key % capacity\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n func loadFactor() -> Double {\n Double(size / capacity)\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n func get(key: Int) -> String? {\n let index = hashFunc(key: key)\n let bucket = buckets[index]\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for pair in bucket {\n if pair.key == key {\n return pair.val\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de nil\n return nil\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n func put(key: Int, val: String) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if loadFactor() > loadThres {\n extend()\n }\n let index = hashFunc(key: key)\n let bucket = buckets[index]\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for pair in bucket {\n if pair.key == key {\n pair.val = val\n return\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n let pair = Pair(key: key, val: val)\n buckets[index].append(pair)\n size += 1\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n func remove(key: Int) {\n let index = hashFunc(key: key)\n let bucket = buckets[index]\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (pairIndex, pair) in bucket.enumerated() {\n if pair.key == key {\n buckets[index].remove(at: pairIndex)\n }\n }\n size -= 1\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n func extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n let bucketsTmp = buckets\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio\n buckets = Array(repeating: [], count: capacity)\n size = 0\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for bucket in bucketsTmp {\n for pair in bucket {\n put(key: pair.key, val: pair.val)\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n func print() {\n for bucket in buckets {\n let res = bucket.map { \"\\($0.key) -> \\($0.val)\" }\n Swift.print(res)\n }\n }\n}\n
hash_map_chaining.ts/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n #size; // \u952e\u503c\u5bf9\u6570\u91cf\n #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n #extendRatio; // \u6269\u5bb9\u500d\u6570\n #buckets; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.#size = 0;\n this.#capacity = 4;\n this.#loadThres = 2.0 / 3.0;\n this.#extendRatio = 2;\n this.#buckets = new Array(this.#capacity).fill(null).map((x) => []);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n #hashFunc(key) {\n return key % this.#capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n #loadFactor() {\n return this.#size / this.#capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key) {\n const index = this.#hashFunc(key);\n const bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (const pair of bucket) {\n if (pair.key === key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n put(key, val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (this.#loadFactor() > this.#loadThres) {\n this.#extend();\n }\n const index = this.#hashFunc(key);\n const bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (const pair of bucket) {\n if (pair.key === key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n const pair = new Pair(key, val);\n bucket.push(pair);\n this.#size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n remove(key) {\n const index = this.#hashFunc(key);\n let bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (let i = 0; i < bucket.length; i++) {\n if (bucket[i].key === key) {\n bucket.splice(i, 1);\n this.#size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n #extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n const bucketsTmp = this.#buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n this.#capacity *= this.#extendRatio;\n this.#buckets = new Array(this.#capacity).fill(null).map((x) => []);\n this.#size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (const bucket of bucketsTmp) {\n for (const pair of bucket) {\n this.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print() {\n for (const bucket of this.#buckets) {\n let res = [];\n for (const pair of bucket) {\n res.push(pair.key + ' -> ' + pair.val);\n }\n console.log(res);\n }\n }\n}\n
hash_map_chaining.dart/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n #size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n #capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n #loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n #extendRatio: number; // \u6269\u5bb9\u500d\u6570\n #buckets: Pair[][]; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.#size = 0;\n this.#capacity = 4;\n this.#loadThres = 2.0 / 3.0;\n this.#extendRatio = 2;\n this.#buckets = new Array(this.#capacity).fill(null).map((x) => []);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n #hashFunc(key: number): number {\n return key % this.#capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n #loadFactor(): number {\n return this.#size / this.#capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key: number): string | null {\n const index = this.#hashFunc(key);\n const bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (const pair of bucket) {\n if (pair.key === key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n put(key: number, val: string): void {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (this.#loadFactor() > this.#loadThres) {\n this.#extend();\n }\n const index = this.#hashFunc(key);\n const bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (const pair of bucket) {\n if (pair.key === key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n const pair = new Pair(key, val);\n bucket.push(pair);\n this.#size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n remove(key: number): void {\n const index = this.#hashFunc(key);\n let bucket = this.#buckets[index];\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (let i = 0; i < bucket.length; i++) {\n if (bucket[i].key === key) {\n bucket.splice(i, 1);\n this.#size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n #extend(): void {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n const bucketsTmp = this.#buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n this.#capacity *= this.#extendRatio;\n this.#buckets = new Array(this.#capacity).fill(null).map((x) => []);\n this.#size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (const bucket of bucketsTmp) {\n for (const pair of bucket) {\n this.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print(): void {\n for (const bucket of this.#buckets) {\n let res = [];\n for (const pair of bucket) {\n res.push(pair.key + ' -> ' + pair.val);\n }\n console.log(res);\n }\n }\n}\n
hash_map_chaining.rs/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n late int size; // \u952e\u503c\u5bf9\u6570\u91cf\n late int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n late double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n late int extendRatio; // \u6269\u5bb9\u500d\u6570\n late List<List<Pair>> buckets; // \u6876\u6570\u7ec4\n\n /* \u6784\u9020\u65b9\u6cd5 */\n HashMapChaining() {\n size = 0;\n capacity = 4;\n loadThres = 2.0 / 3.0;\n extendRatio = 2;\n buckets = List.generate(capacity, (_) => []);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return size / capacity;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n String? get(int key) {\n int index = hashFunc(key);\n List<Pair> bucket = buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for (Pair pair in bucket) {\n if (pair.key == key) {\n return pair.val;\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, String val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n int index = hashFunc(key);\n List<Pair> bucket = buckets[index];\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for (Pair pair in bucket) {\n if (pair.key == key) {\n pair.val = val;\n return;\n }\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n Pair pair = Pair(key, val);\n bucket.add(pair);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n int index = hashFunc(key);\n List<Pair> bucket = buckets[index];\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for (Pair pair in bucket) {\n if (pair.key == key) {\n bucket.remove(pair);\n size--;\n break;\n }\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n List<List<Pair>> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = List.generate(capacity, (_) => []);\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (List<Pair> bucket in bucketsTmp) {\n for (Pair pair in bucket) {\n put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void printHashMap() {\n for (List<Pair> bucket in buckets) {\n List<String> res = [];\n for (Pair pair in bucket) {\n res.add(\"${pair.key} -> ${pair.val}\");\n }\n print(res);\n }\n }\n}\n
hash_map_chaining.c/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapChaining {\n size: i32,\n capacity: i32,\n load_thres: f32,\n extend_ratio: i32,\n buckets: Vec<Vec<Pair>>,\n}\n\nimpl HashMapChaining {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new() -> Self {\n Self {\n size: 0,\n capacity: 4,\n load_thres: 2.0 / 3.0,\n extend_ratio: 2,\n buckets: vec![vec![]; 4],\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n fn hash_func(&self, key: i32) -> usize {\n key as usize % self.capacity as usize\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n fn load_factor(&self) -> f32 {\n self.size as f32 / self.capacity as f32\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n fn remove(&mut self, key: i32) -> Option<String> {\n let index = self.hash_func(key);\n let bucket = &mut self.buckets[index];\n\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n for i in 0..bucket.len() {\n if bucket[i].key == key {\n let pair = bucket.remove(i);\n self.size -= 1;\n return Some(pair.val);\n }\n }\n\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\n None\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n fn extend(&mut self) {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n let buckets_tmp = std::mem::replace(&mut self.buckets, vec![]);\n\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n self.capacity *= self.extend_ratio;\n self.buckets = vec![Vec::new(); self.capacity as usize];\n self.size = 0;\n\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for bucket in buckets_tmp {\n for pair in bucket {\n self.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n fn print(&self) {\n for bucket in &self.buckets {\n let mut res = Vec::new();\n for pair in bucket {\n res.push(format!(\"{} -> {}\", pair.key, pair.val));\n }\n println!(\"{:?}\", res);\n }\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n fn put(&mut self, key: i32, val: String) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if self.load_factor() > self.load_thres {\n self.extend();\n }\n\n let index = self.hash_func(key);\n let bucket = &mut self.buckets[index];\n\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n for pair in bucket {\n if pair.key == key {\n pair.val = val.clone();\n return;\n }\n }\n let bucket = &mut self.buckets[index];\n\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n let pair = Pair {\n key,\n val: val.clone(),\n };\n bucket.push(pair);\n self.size += 1;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n fn get(&self, key: i32) -> Option<&str> {\n let index = self.hash_func(key);\n let bucket = &self.buckets[index];\n\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n for pair in bucket {\n if pair.key == key {\n return Some(&pair.val);\n }\n }\n\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\n None\n }\n}\n
hash_map_chaining.zig/* \u94fe\u8868\u8282\u70b9 */\ntypedef struct Node {\n Pair *pair;\n struct Node *next;\n} Node;\n\n/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n Node **buckets; // \u6876\u6570\u7ec4\n} HashMapChaining;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapChaining *newHashMapChaining() {\n HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n hashMap->size = 0;\n hashMap->capacity = 4;\n hashMap->loadThres = 2.0 / 3.0;\n hashMap->extendRatio = 2;\n hashMap->buckets = (Node **)malloc(hashMap->capacity * sizeof(Node *));\n for (int i = 0; i < hashMap->capacity; i++) {\n hashMap->buckets[i] = NULL;\n }\n return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n for (int i = 0; i < hashMap->capacity; i++) {\n Node *cur = hashMap->buckets[i];\n while (cur) {\n Node *tmp = cur;\n cur = cur->next;\n free(tmp->pair);\n free(tmp);\n }\n }\n free(hashMap->buckets);\n free(hashMap);\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapChaining *hashMap, int key) {\n return key % hashMap->capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\n return (double)hashMap->size / (double)hashMap->capacity;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\n int index = hashFunc(hashMap, key);\n // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n Node *cur = hashMap->buckets[index];\n while (cur) {\n if (cur->pair->key == key) {\n return cur->pair->val;\n }\n cur = cur->next;\n }\n return \"\"; // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor(hashMap) > hashMap->loadThres) {\n extend(hashMap);\n }\n int index = hashFunc(hashMap, key);\n // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n Node *cur = hashMap->buckets[index];\n while (cur) {\n if (cur->pair->key == key) {\n strcpy(cur->pair->val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n return;\n }\n cur = cur->next;\n }\n // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n Pair *newPair = (Pair *)malloc(sizeof(Pair));\n newPair->key = key;\n strcpy(newPair->val, val);\n Node *newNode = (Node *)malloc(sizeof(Node));\n newNode->pair = newPair;\n newNode->next = hashMap->buckets[index];\n hashMap->buckets[index] = newNode;\n hashMap->size++;\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n int oldCapacity = hashMap->capacity;\n Node **oldBuckets = hashMap->buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n hashMap->capacity *= hashMap->extendRatio;\n hashMap->buckets = (Node **)malloc(hashMap->capacity * sizeof(Node *));\n for (int i = 0; i < hashMap->capacity; i++) {\n hashMap->buckets[i] = NULL;\n }\n hashMap->size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (int i = 0; i < oldCapacity; i++) {\n Node *cur = oldBuckets[i];\n while (cur) {\n put(hashMap, cur->pair->key, cur->pair->val);\n Node *temp = cur;\n cur = cur->next;\n // \u91ca\u653e\u5185\u5b58\n free(temp->pair);\n free(temp);\n }\n }\n\n free(oldBuckets);\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n int index = hashFunc(hashMap, key);\n Node *cur = hashMap->buckets[index];\n Node *pre = NULL;\n while (cur) {\n if (cur->pair->key == key) {\n // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n if (pre) {\n pre->next = cur->next;\n } else {\n hashMap->buckets[index] = cur->next;\n }\n // \u91ca\u653e\u5185\u5b58\n free(cur->pair);\n free(cur);\n hashMap->size--;\n return;\n }\n pre = cur;\n cur = cur->next;\n }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapChaining *hashMap) {\n for (int i = 0; i < hashMap->capacity; i++) {\n Node *cur = hashMap->buckets[i];\n printf(\"[\");\n while (cur) {\n printf(\"%d -> %s, \", cur->pair->key, cur->pair->val);\n cur = cur->next;\n }\n printf(\"]\\n\");\n }\n}\n[class]{HashMapChaining}-[func]{}\n\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5f53\u94fe\u8868\u5f88\u957f\u65f6\uff0c\u67e5\u8be2\u6548\u7387 \\(O(n)\\) \u5f88\u5dee\u3002\u6b64\u65f6\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u201cAVL \u6811\u201d\u6216\u201c\u7ea2\u9ed1\u6811\u201d\uff0c\u4ece\u800c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002
"},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2 \u00a0 \u5f00\u653e\u5bfb\u5740","text":"\u300c\u5f00\u653e\u5bfb\u5740 open addressing\u300d\u4e0d\u5f15\u5165\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\uff0c\u63a2\u6d4b\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u3001\u591a\u6b21\u54c8\u5e0c\u7b49\u3002
\u4e0b\u9762\u4ee5\u7ebf\u6027\u63a2\u6d4b\u4e3a\u4f8b\uff0c\u4ecb\u7ecd\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u673a\u5236\u3002
"},{"location":"chapter_hashing/hash_collision/#1","title":"1. \u00a0 \u7ebf\u6027\u63a2\u6d4b","text":"\u7ebf\u6027\u63a2\u6d4b\u91c7\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u641c\u7d22\u6765\u8fdb\u884c\u63a2\u6d4b\uff0c\u5176\u64cd\u4f5c\u65b9\u6cd5\u4e0e\u666e\u901a\u54c8\u5e0c\u8868\u6709\u6240\u4e0d\u540c\u3002
- \u63d2\u5165\u5143\u7d20\uff1a\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u82e5\u53d1\u73b0\u6876\u5185\u5df2\u6709\u5143\u7d20\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u901a\u5e38\u4e3a \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u6876\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002
- \u67e5\u627e\u5143\u7d20\uff1a\u82e5\u53d1\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u5411\u540e\u7ebf\u6027\u904d\u5386\uff0c\u76f4\u5230\u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de
value\u5373\u53ef\uff1b\u5982\u679c\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u76ee\u6807\u5143\u7d20\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8fd4\u56de \\(\\text{None}\\) \u3002
\u56fe 6-6 \u5c55\u793a\u4e86\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u952e\u503c\u5bf9\u5206\u5e03\u3002\u6839\u636e\u6b64\u54c8\u5e0c\u51fd\u6570\uff0c\u6700\u540e\u4e24\u4f4d\u76f8\u540c\u7684
key\u90fd\u4f1a\u88ab\u6620\u5c04\u5230\u76f8\u540c\u7684\u6876\u3002\u800c\u901a\u8fc7\u7ebf\u6027\u63a2\u6d4b\uff0c\u5b83\u4eec\u88ab\u4f9d\u6b21\u5b58\u50a8\u5728\u8be5\u6876\u4ee5\u53ca\u4e4b\u4e0b\u7684\u6876\u4e2d\u3002\u56fe 6-6 \u00a0 \u5f00\u653e\u5bfb\u5740\u548c\u7ebf\u6027\u63a2\u6d4b
\u7136\u800c\uff0c\u7ebf\u6027\u63a2\u6d4b\u5bb9\u6613\u4ea7\u751f\u201c\u805a\u96c6\u73b0\u8c61\u201d\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6570\u7ec4\u4e2d\u8fde\u7eed\u88ab\u5360\u7528\u7684\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u4ece\u800c\u8fdb\u4e00\u6b65\u4fc3\u4f7f\u8be5\u4f4d\u7f6e\u7684\u805a\u5806\u751f\u957f\uff0c\u5f62\u6210\u6076\u6027\u5faa\u73af\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u52a3\u5316\u3002
\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6211\u4eec\u4e0d\u80fd\u5728\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u4e2d\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u8fd9\u662f\u56e0\u4e3a\u5220\u9664\u5143\u7d20\u4f1a\u5728\u6570\u7ec4\u5185\u4ea7\u751f\u4e00\u4e2a\u7a7a\u6876 \\(\\text{None}\\) \uff0c\u800c\u5f53\u67e5\u8be2\u5143\u7d20\u65f6\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230\u8be5\u7a7a\u6876\u5c31\u4f1a\u8fd4\u56de\uff0c\u56e0\u6b64\u5728\u8be5\u7a7a\u6876\u4e4b\u4e0b\u7684\u5143\u7d20\u90fd\u65e0\u6cd5\u518d\u88ab\u8bbf\u95ee\u5230\uff0c\u7a0b\u5e8f\u53ef\u80fd\u8bef\u5224\u8fd9\u4e9b\u5143\u7d20\u4e0d\u5b58\u5728\u3002
\u56fe 6-7 \u00a0 \u5728\u5f00\u653e\u5bfb\u5740\u4e2d\u5220\u9664\u5143\u7d20\u5bfc\u81f4\u7684\u67e5\u8be2\u95ee\u9898
\u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u300c\u61d2\u5220\u9664 lazy deletion\u300d\u673a\u5236\uff1a\u5b83\u4e0d\u76f4\u63a5\u4ece\u54c8\u5e0c\u8868\u4e2d\u79fb\u9664\u5143\u7d20\uff0c\u800c\u662f\u5229\u7528\u4e00\u4e2a\u5e38\u91cf
TOMBSTONE\u6765\u6807\u8bb0\u8fd9\u4e2a\u6876\u3002\u5728\u8be5\u673a\u5236\u4e0b\uff0c\\(\\text{None}\\) \u548cTOMBSTONE\u90fd\u4ee3\u8868\u7a7a\u6876\uff0c\u90fd\u53ef\u4ee5\u653e\u7f6e\u952e\u503c\u5bf9\u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230TOMBSTONE\u65f6\u5e94\u8be5\u7ee7\u7eed\u904d\u5386\uff0c\u56e0\u4e3a\u5176\u4e4b\u4e0b\u53ef\u80fd\u8fd8\u5b58\u5728\u952e\u503c\u5bf9\u3002\u7136\u800c\uff0c\u61d2\u5220\u9664\u53ef\u80fd\u4f1a\u52a0\u901f\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u9000\u5316\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6b21\u5220\u9664\u64cd\u4f5c\u90fd\u4f1a\u4ea7\u751f\u4e00\u4e2a\u5220\u9664\u6807\u8bb0\uff0c\u968f\u7740
TOMBSTONE\u7684\u589e\u52a0\uff0c\u641c\u7d22\u65f6\u95f4\u4e5f\u4f1a\u589e\u52a0\uff0c\u56e0\u4e3a\u7ebf\u6027\u63a2\u6d4b\u53ef\u80fd\u9700\u8981\u8df3\u8fc7\u591a\u4e2aTOMBSTONE\u624d\u80fd\u627e\u5230\u76ee\u6807\u5143\u7d20\u3002\u4e3a\u6b64\uff0c\u8003\u8651\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a
TOMBSTONE\u7684\u7d22\u5f15\uff0c\u5e76\u5c06\u641c\u7d22\u5230\u7684\u76ee\u6807\u5143\u7d20\u4e0e\u8be5TOMBSTONE\u4ea4\u6362\u4f4d\u7f6e\u3002\u8fd9\u6837\u505a\u7684\u597d\u5904\u662f\u5f53\u6bcf\u6b21\u67e5\u8be2\u6216\u6dfb\u52a0\u5143\u7d20\u65f6\uff0c\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u81f3\u8ddd\u79bb\u7406\u60f3\u4f4d\u7f6e\uff08\u63a2\u6d4b\u8d77\u59cb\u70b9\uff09\u66f4\u8fd1\u7684\u6876\uff0c\u4ece\u800c\u4f18\u5316\u67e5\u8be2\u6548\u7387\u3002\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u5305\u542b\u61d2\u5220\u9664\u7684\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u3002\u4e3a\u4e86\u66f4\u52a0\u5145\u5206\u5730\u4f7f\u7528\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u770b\u4f5c\u4e00\u4e2a\u201c\u73af\u5f62\u6570\u7ec4\u201d\uff0c\u5f53\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map_open_addressing.py
hash_map_open_addressing.cppclass HashMapOpenAddressing:\n \"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self.size = 0 # \u952e\u503c\u5bf9\u6570\u91cf\n self.capacity = 4 # \u54c8\u5e0c\u8868\u5bb9\u91cf\n self.load_thres = 2.0 / 3.0 # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n self.extend_ratio = 2 # \u6269\u5bb9\u500d\u6570\n self.buckets: list[Pair | None] = [None] * self.capacity # \u6876\u6570\u7ec4\n self.TOMBSTONE = Pair(-1, \"-1\") # \u5220\u9664\u6807\u8bb0\n\n def hash_func(self, key: int) -> int:\n \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n return key % self.capacity\n\n def load_factor(self) -> float:\n \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n return self.size / self.capacity\n\n def find_bucket(self, key: int) -> int:\n \"\"\"\u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\"\"\"\n index = self.hash_func(key)\n first_tombstone = -1\n # \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while self.buckets[index] is not None:\n # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if self.buckets[index].key == key:\n # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if first_tombstone != -1:\n self.buckets[first_tombstone] = self.buckets[index]\n self.buckets[index] = self.TOMBSTONE\n return first_tombstone # \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n return index # \u8fd4\u56de\u6876\u7d22\u5f15\n # \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n first_tombstone = index\n # \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % self.capacity\n # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return index if first_tombstone == -1 else first_tombstone\n\n def get(self, key: int) -> str:\n \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n index = self.find_bucket(key)\n # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if self.buckets[index] not in [None, self.TOMBSTONE]:\n return self.buckets[index].val\n # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de None\n return None\n\n def put(self, key: int, val: str):\n \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if self.load_factor() > self.load_thres:\n self.extend()\n # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n index = self.find_bucket(key)\n # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if self.buckets[index] not in [None, self.TOMBSTONE]:\n self.buckets[index].val = val\n return\n # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n self.buckets[index] = Pair(key, val)\n self.size += 1\n\n def remove(self, key: int):\n \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n index = self.find_bucket(key)\n # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if self.buckets[index] not in [None, self.TOMBSTONE]:\n self.buckets[index] = self.TOMBSTONE\n self.size -= 1\n\n def extend(self):\n \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n buckets_tmp = self.buckets\n # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n self.capacity *= self.extend_ratio\n self.buckets = [None] * self.capacity\n self.size = 0\n # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for pair in buckets_tmp:\n if pair not in [None, self.TOMBSTONE]:\n self.put(pair.key, pair.val)\n\n def print(self):\n \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n for pair in self.buckets:\n if pair is None:\n print(\"None\")\n elif pair is self.TOMBSTONE:\n print(\"TOMBSTONE\")\n else:\n print(pair.key, \"->\", pair.val)\n
hash_map_open_addressing.java/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n private:\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n const double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n const int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n vector<Pair *> buckets; // \u6876\u6570\u7ec4\n Pair *TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n }\n\n /* \u6790\u6784\u65b9\u6cd5 */\n ~HashMapOpenAddressing() {\n for (Pair *pair : buckets) {\n if (pair != nullptr && pair != TOMBSTONE) {\n delete pair;\n }\n }\n delete TOMBSTONE;\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return (double)size / capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n int findBucket(int key) {\n int index = hashFunc(key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (buckets[index] != nullptr) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (buckets[index]->key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n buckets[firstTombstone] = buckets[index];\n buckets[index] = TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && buckets[index] == TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n string get(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) {\n return buckets[index]->val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\";\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, string val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) {\n buckets[index]->val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n buckets[index] = new Pair(key, val);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) {\n delete buckets[index];\n buckets[index] = TOMBSTONE;\n size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n vector<Pair *> bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = vector<Pair *>(capacity, nullptr);\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (Pair *pair : bucketsTmp) {\n if (pair != nullptr && pair != TOMBSTONE) {\n put(pair->key, pair->val);\n delete pair;\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void print() {\n for (Pair *pair : buckets) {\n if (pair == nullptr) {\n cout << \"nullptr\" << endl;\n } else if (pair == TOMBSTONE) {\n cout << \"TOMBSTONE\" << endl;\n } else {\n cout << pair->key << \" -> \" << pair->val << endl;\n }\n }\n }\n};\n
hash_map_open_addressing.cs/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n private int size; // \u952e\u503c\u5bf9\u6570\u91cf\n private int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n private final double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n private final int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n private Pair[] buckets; // \u6876\u6570\u7ec4\n private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public HashMapOpenAddressing() {\n size = 0;\n buckets = new Pair[capacity];\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private int hashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n private double loadFactor() {\n return (double) size / capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n private int findBucket(int key) {\n int index = hashFunc(key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (buckets[index] != null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (buckets[index].key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n buckets[firstTombstone] = buckets[index];\n buckets[index] = TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && buckets[index] == TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public String get(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n return buckets[index].val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void put(int key, String val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > loadThres) {\n extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n buckets[index].val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n buckets[index] = new Pair(key, val);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void remove(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n buckets[index] = TOMBSTONE;\n size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n private void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n Pair[] bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = new Pair[capacity];\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (Pair pair : bucketsTmp) {\n if (pair != null && pair != TOMBSTONE) {\n put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void print() {\n for (Pair pair : buckets) {\n if (pair == null) {\n System.out.println(\"null\");\n } else if (pair == TOMBSTONE) {\n System.out.println(\"TOMBSTONE\");\n } else {\n System.out.println(pair.key + \" -> \" + pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.go/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n Pair[] buckets; // \u6876\u6570\u7ec4\n Pair TOMBSTONE = new(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public HashMapOpenAddressing() {\n size = 0;\n buckets = new Pair[capacity];\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int HashFunc(int key) {\n return key % capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double LoadFactor() {\n return (double)size / capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n int FindBucket(int key) {\n int index = HashFunc(key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (buckets[index] != null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (buckets[index].key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n buckets[firstTombstone] = buckets[index];\n buckets[index] = TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && buckets[index] == TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public string? Get(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = FindBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n return buckets[index].val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void Put(int key, string val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (LoadFactor() > loadThres) {\n Extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = FindBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n buckets[index].val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n buckets[index] = new Pair(key, val);\n size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void Remove(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = FindBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (buckets[index] != null && buckets[index] != TOMBSTONE) {\n buckets[index] = TOMBSTONE;\n size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void Extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n Pair[] bucketsTmp = buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio;\n buckets = new Pair[capacity];\n size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n foreach (Pair pair in bucketsTmp) {\n if (pair != null && pair != TOMBSTONE) {\n Put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void Print() {\n foreach (Pair pair in buckets) {\n if (pair == null) {\n Console.WriteLine(\"null\");\n } else if (pair == TOMBSTONE) {\n Console.WriteLine(\"TOMBSTONE\");\n } else {\n Console.WriteLine(pair.key + \" -> \" + pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.swift/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\n size int // \u952e\u503c\u5bf9\u6570\u91cf\n capacity int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n loadThres float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n extendRatio int // \u6269\u5bb9\u500d\u6570\n buckets []pair // \u6876\u6570\u7ec4\n removed pair // \u5220\u9664\u6807\u8bb0\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n buckets := make([]pair, 4)\n return &hashMapOpenAddressing{\n size: 0,\n capacity: 4,\n loadThres: 2.0 / 3.0,\n extendRatio: 2,\n buckets: buckets,\n removed: pair{\n key: -1,\n val: \"-1\",\n },\n }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapOpenAddressing) hashFunc(key int) int {\n return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapOpenAddressing) loadFactor() float64 {\n return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) get(key int) string {\n idx := m.hashFunc(key)\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\n for i := 0; i < m.capacity; i++ {\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n j := (idx + i) % m.capacity\n // \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\n if m.buckets[j] == (pair{}) {\n return \"\"\n }\n // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if m.buckets[j].key == key && m.buckets[j] != m.removed {\n return m.buckets[j].val\n }\n }\n // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) put(key int, val string) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if m.loadFactor() > m.loadThres {\n m.extend()\n }\n idx := m.hashFunc(key)\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\n for i := 0; i < m.capacity; i++ {\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n j := (idx + i) % m.capacity\n // \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\n if m.buckets[j] == (pair{}) || m.buckets[j] == m.removed {\n m.buckets[j] = pair{\n key: key,\n val: val,\n }\n m.size += 1\n return\n }\n // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\n if m.buckets[j].key == key {\n m.buckets[j].val = val\n return\n }\n }\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) remove(key int) {\n idx := m.hashFunc(key)\n // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\n for i := 0; i < m.capacity; i++ {\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n j := (idx + i) % m.capacity\n // \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if m.buckets[j] == (pair{}) {\n return\n }\n // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\n if m.buckets[j].key == key {\n m.buckets[j] = m.removed\n m.size -= 1\n }\n }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n tmpBuckets := make([]pair, len(m.buckets))\n copy(tmpBuckets, m.buckets)\n\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n m.capacity *= m.extendRatio\n m.buckets = make([]pair, m.capacity)\n m.size = 0\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for _, p := range tmpBuckets {\n if p != (pair{}) && p != m.removed {\n m.put(p.key, p.val)\n }\n }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) print() {\n for _, p := range m.buckets {\n if p != (pair{}) {\n fmt.Println(strconv.Itoa(p.key) + \" -> \" + p.val)\n } else {\n fmt.Println(\"nil\")\n }\n }\n}\n
hash_map_open_addressing.js/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n var buckets: [Pair?] // \u6876\u6570\u7ec4\n var TOMBSTONE: Pair // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init() {\n size = 0\n capacity = 4\n loadThres = 2.0 / 3.0\n extendRatio = 2\n buckets = Array(repeating: nil, count: capacity)\n TOMBSTONE = Pair(key: -1, val: \"-1\")\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n func hashFunc(key: Int) -> Int {\n key % capacity\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n func loadFactor() -> Double {\n Double(size / capacity)\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n func findBucket(key: Int) -> Int {\n var index = hashFunc(key: key)\n var firstTombstone = -1\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while buckets[index] != nil {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if buckets[index]!.key == key {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if firstTombstone != -1 {\n buckets[firstTombstone] = buckets[index]\n buckets[index] = TOMBSTONE\n return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if firstTombstone == -1 && buckets[index] == TOMBSTONE {\n firstTombstone = index\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % capacity\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n func get(key: Int) -> String? {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = findBucket(key: key)\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if buckets[index] != nil, buckets[index] != TOMBSTONE {\n return buckets[index]!.val\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return nil\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n func put(key: Int, val: String) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if loadFactor() > loadThres {\n extend()\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = findBucket(key: key)\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if buckets[index] != nil, buckets[index] != TOMBSTONE {\n buckets[index]!.val = val\n return\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n buckets[index] = Pair(key: key, val: val)\n size += 1\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n func remove(key: Int) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = findBucket(key: key)\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if buckets[index] != nil, buckets[index] != TOMBSTONE {\n buckets[index] = TOMBSTONE\n size -= 1\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n func extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n let bucketsTmp = buckets\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n capacity *= extendRatio\n buckets = Array(repeating: nil, count: capacity)\n size = 0\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for pair in bucketsTmp {\n if let pair, pair != TOMBSTONE {\n put(key: pair.key, val: pair.val)\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n func print() {\n for pair in buckets {\n if pair == nil {\n Swift.print(\"null\")\n } else if pair == TOMBSTONE {\n Swift.print(\"TOMBSTONE\")\n } else {\n Swift.print(\"\\(pair!.key) -> \\(pair!.val)\")\n }\n }\n }\n}\n
hash_map_open_addressing.ts/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n #size; // \u952e\u503c\u5bf9\u6570\u91cf\n #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n #extendRatio; // \u6269\u5bb9\u500d\u6570\n #buckets; // \u6876\u6570\u7ec4\n #TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.#size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n this.#capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n this.#loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n this.#extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n this.#buckets = Array(this.#capacity).fill(null); // \u6876\u6570\u7ec4\n this.#TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n #hashFunc(key) {\n return key % this.#capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n #loadFactor() {\n return this.#size / this.#capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n #findBucket(key) {\n let index = this.#hashFunc(key);\n let firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (this.#buckets[index] !== null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (this.#buckets[index].key === key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone !== -1) {\n this.#buckets[firstTombstone] = this.#buckets[index];\n this.#buckets[index] = this.#TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (\n firstTombstone === -1 &&\n this.#buckets[index] === this.#TOMBSTONE\n ) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % this.#capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone === -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.#findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (\n this.#buckets[index] !== null &&\n this.#buckets[index] !== this.#TOMBSTONE\n ) {\n return this.#buckets[index].val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n put(key, val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (this.#loadFactor() > this.#loadThres) {\n this.#extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.#findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (\n this.#buckets[index] !== null &&\n this.#buckets[index] !== this.#TOMBSTONE\n ) {\n this.#buckets[index].val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n this.#buckets[index] = new Pair(key, val);\n this.#size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n remove(key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.#findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (\n this.#buckets[index] !== null &&\n this.#buckets[index] !== this.#TOMBSTONE\n ) {\n this.#buckets[index] = this.#TOMBSTONE;\n this.#size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n #extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n const bucketsTmp = this.#buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n this.#capacity *= this.#extendRatio;\n this.#buckets = Array(this.#capacity).fill(null);\n this.#size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (const pair of bucketsTmp) {\n if (pair !== null && pair !== this.#TOMBSTONE) {\n this.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print() {\n for (const pair of this.#buckets) {\n if (pair === null) {\n console.log('null');\n } else if (pair === this.#TOMBSTONE) {\n console.log('TOMBSTONE');\n } else {\n console.log(pair.key + ' -> ' + pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.dart/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n private size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n private capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n private loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n private extendRatio: number; // \u6269\u5bb9\u500d\u6570\n private buckets: Array<Pair | null>; // \u6876\u6570\u7ec4\n private TOMBSTONE: Pair; // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor() {\n this.size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n this.capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n this.loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n this.extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n this.buckets = Array(this.capacity).fill(null); // \u6876\u6570\u7ec4\n this.TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private hashFunc(key: number): number {\n return key % this.capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n private loadFactor(): number {\n return this.size / this.capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n private findBucket(key: number): number {\n let index = this.hashFunc(key);\n let firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (this.buckets[index] !== null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (this.buckets[index]!.key === key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone !== -1) {\n this.buckets[firstTombstone] = this.buckets[index];\n this.buckets[index] = this.TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (\n firstTombstone === -1 &&\n this.buckets[index] === this.TOMBSTONE\n ) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % this.capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone === -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key: number): string | null {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (\n this.buckets[index] !== null &&\n this.buckets[index] !== this.TOMBSTONE\n ) {\n return this.buckets[index]!.val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n put(key: number, val: string): void {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (this.loadFactor() > this.loadThres) {\n this.extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (\n this.buckets[index] !== null &&\n this.buckets[index] !== this.TOMBSTONE\n ) {\n this.buckets[index]!.val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n this.buckets[index] = new Pair(key, val);\n this.size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n remove(key: number): void {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n const index = this.findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (\n this.buckets[index] !== null &&\n this.buckets[index] !== this.TOMBSTONE\n ) {\n this.buckets[index] = this.TOMBSTONE;\n this.size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n private extend(): void {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n const bucketsTmp = this.buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n this.capacity *= this.extendRatio;\n this.buckets = Array(this.capacity).fill(null);\n this.size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (const pair of bucketsTmp) {\n if (pair !== null && pair !== this.TOMBSTONE) {\n this.put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print(): void {\n for (const pair of this.buckets) {\n if (pair === null) {\n console.log('null');\n } else if (pair === this.TOMBSTONE) {\n console.log('TOMBSTONE');\n } else {\n console.log(pair.key + ' -> ' + pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.rs/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n late int _size; // \u952e\u503c\u5bf9\u6570\u91cf\n int _capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double _loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int _extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n late List<Pair?> _buckets; // \u6876\u6570\u7ec4\n Pair _TOMBSTONE = Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n /* \u6784\u9020\u65b9\u6cd5 */\n HashMapOpenAddressing() {\n _size = 0;\n _buckets = List.generate(_capacity, (index) => null);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n return key % _capacity;\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n double loadFactor() {\n return _size / _capacity;\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n int findBucket(int key) {\n int index = hashFunc(key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (_buckets[index] != null) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (_buckets[index]!.key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n _buckets[firstTombstone] = _buckets[index];\n _buckets[index] = _TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && _buckets[index] == _TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % _capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n String? get(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (_buckets[index] != null && _buckets[index] != _TOMBSTONE) {\n return _buckets[index]!.val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n return null;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, String val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor() > _loadThres) {\n extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (_buckets[index] != null && _buckets[index] != _TOMBSTONE) {\n _buckets[index]!.val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n _buckets[index] = new Pair(key, val);\n _size++;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (_buckets[index] != null && _buckets[index] != _TOMBSTONE) {\n _buckets[index] = _TOMBSTONE;\n _size--;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n void extend() {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n List<Pair?> bucketsTmp = _buckets;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n _capacity *= _extendRatio;\n _buckets = List.generate(_capacity, (index) => null);\n _size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (Pair? pair in bucketsTmp) {\n if (pair != null && pair != _TOMBSTONE) {\n put(pair.key, pair.val);\n }\n }\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void printHashMap() {\n for (Pair? pair in _buckets) {\n if (pair == null) {\n print(\"null\");\n } else if (pair == _TOMBSTONE) {\n print(\"TOMBSTONE\");\n } else {\n print(\"${pair.key} -> ${pair.val}\");\n }\n }\n }\n}\n
hash_map_open_addressing.c/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapOpenAddressing {\n size: usize, // \u952e\u503c\u5bf9\u6570\u91cf\n capacity: usize, // \u54c8\u5e0c\u8868\u5bb9\u91cf\n load_thres: f64, // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n extend_ratio: usize, // \u6269\u5bb9\u500d\u6570\n buckets: Vec<Option<Pair>>, // \u6876\u6570\u7ec4\n TOMBSTONE: Option<Pair>, // \u5220\u9664\u6807\u8bb0\n}\n\n\nimpl HashMapOpenAddressing {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new() -> Self {\n Self {\n size: 0,\n capacity: 4,\n load_thres: 2.0 / 3.0,\n extend_ratio: 2,\n buckets: vec![None; 4],\n TOMBSTONE: Some(Pair {key: -1, val: \"-1\".to_string()}),\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n fn hash_func(&self, key: i32) -> usize {\n (key % self.capacity as i32) as usize\n }\n\n /* \u8d1f\u8f7d\u56e0\u5b50 */\n fn load_factor(&self) -> f64 {\n self.size as f64 / self.capacity as f64\n }\n\n /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n fn find_bucket(&mut self, key: i32) -> usize {\n let mut index = self.hash_func(key);\n let mut first_tombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while self.buckets[index].is_some() {\n // \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n if self.buckets[index].as_ref().unwrap().key == key {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u5efa\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if first_tombstone != -1 {\n self.buckets[first_tombstone as usize] = self.buckets[index].take();\n self.buckets[index] = self.TOMBSTONE.clone();\n return first_tombstone as usize; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if first_tombstone == -1 && self.buckets[index] == self.TOMBSTONE {\n first_tombstone = index as i32;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % self.capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n if first_tombstone == -1 { index } else { first_tombstone as usize }\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n fn get(&mut self, key: i32) -> Option<&str> {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = self.find_bucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if self.buckets[index].is_some() && self.buckets[index] != self.TOMBSTONE {\n return self.buckets[index].as_ref().map(|pair| &pair.val as &str);\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n None\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n fn put(&mut self, key: i32, val: String) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if self.load_factor() > self.load_thres {\n self.extend();\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = self.find_bucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if self.buckets[index].is_some() && self.buckets[index] != self.TOMBSTONE {\n self.buckets[index].as_mut().unwrap().val = val;\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n self.buckets[index] = Some(Pair { key, val });\n self.size += 1;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n fn remove(&mut self, key: i32) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n let index = self.find_bucket(key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if self.buckets[index].is_some() && self.buckets[index] != self.TOMBSTONE {\n self.buckets[index] = self.TOMBSTONE.clone();\n self.size -= 1;\n }\n }\n\n /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n fn extend(&mut self) {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n let buckets_tmp = self.buckets.clone();\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n self.capacity *= self.extend_ratio;\n self.buckets = vec![None; self.capacity];\n self.size = 0;\n\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for pair in buckets_tmp {\n if pair.is_none() || pair == self.TOMBSTONE {\n continue;\n }\n let pair = pair.unwrap();\n\n self.put(pair.key, pair.val);\n }\n }\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n fn print(&self) {\n for pair in &self.buckets {\n if pair.is_none() {\n println!(\"null\");\n } else if pair == &self.TOMBSTONE {\n println!(\"TOMBSTONE\");\n } else {\n let pair = pair.as_ref().unwrap();\n println!(\"{} -> {}\", pair.key, pair.val);\n }\n }\n }\n}\n
hash_map_open_addressing.zig/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n int size; // \u952e\u503c\u5bf9\u6570\u91cf\n int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n int extendRatio; // \u6269\u5bb9\u500d\u6570\n Pair **buckets; // \u6876\u6570\u7ec4\n Pair *TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n} HashMapOpenAddressing;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n hashMap->size = 0;\n hashMap->capacity = 4;\n hashMap->loadThres = 2.0 / 3.0;\n hashMap->extendRatio = 2;\n hashMap->buckets = (Pair **)malloc(sizeof(Pair *) * hashMap->capacity);\n hashMap->TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n hashMap->TOMBSTONE->key = -1;\n hashMap->TOMBSTONE->val = \"-1\";\n\n return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n for (int i = 0; i < hashMap->capacity; i++) {\n Pair *pair = hashMap->buckets[i];\n if (pair != NULL && pair != hashMap->TOMBSTONE) {\n free(pair->val);\n free(pair);\n }\n }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n return key % hashMap->capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n return (double)hashMap->size / (double)hashMap->capacity;\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n int index = hashFunc(hashMap, key);\n int firstTombstone = -1;\n // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n while (hashMap->buckets[index] != NULL) {\n // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n if (hashMap->buckets[index]->key == key) {\n // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n if (firstTombstone != -1) {\n hashMap->buckets[firstTombstone] = hashMap->buckets[index];\n hashMap->buckets[index] = hashMap->TOMBSTONE;\n return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n }\n return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n }\n // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n if (firstTombstone == -1 && hashMap->buckets[index] == hashMap->TOMBSTONE) {\n firstTombstone = index;\n }\n // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n index = (index + 1) % hashMap->capacity;\n }\n // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(hashMap, key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n if (hashMap->buckets[index] != NULL && hashMap->buckets[index] != hashMap->TOMBSTONE) {\n return hashMap->buckets[index]->val;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n return \"\";\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n if (loadFactor(hashMap) > hashMap->loadThres) {\n extend(hashMap);\n }\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(hashMap, key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n if (hashMap->buckets[index] != NULL && hashMap->buckets[index] != hashMap->TOMBSTONE) {\n free(hashMap->buckets[index]->val);\n hashMap->buckets[index]->val = (char *)malloc(sizeof(strlen(val + 1)));\n strcpy(hashMap->buckets[index]->val, val);\n hashMap->buckets[index]->val[strlen(val)] = '\\0';\n return;\n }\n // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n Pair *pair = (Pair *)malloc(sizeof(Pair));\n pair->key = key;\n pair->val = (char *)malloc(sizeof(strlen(val + 1)));\n strcpy(pair->val, val);\n pair->val[strlen(val)] = '\\0';\n\n hashMap->buckets[index] = pair;\n hashMap->size++;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n int index = findBucket(hashMap, key);\n // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n if (hashMap->buckets[index] != NULL && hashMap->buckets[index] != hashMap->TOMBSTONE) {\n Pair *pair = hashMap->buckets[index];\n free(pair->val);\n free(pair);\n hashMap->buckets[index] = hashMap->TOMBSTONE;\n hashMap->size--;\n }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n Pair **bucketsTmp = hashMap->buckets;\n int oldCapacity = hashMap->capacity;\n // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n hashMap->capacity *= hashMap->extendRatio;\n hashMap->buckets = (Pair **)malloc(sizeof(Pair *) * hashMap->capacity);\n hashMap->size = 0;\n // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n for (int i = 0; i < oldCapacity; i++) {\n Pair *pair = bucketsTmp[i];\n if (pair != NULL && pair != hashMap->TOMBSTONE) {\n put(hashMap, pair->key, pair->val);\n free(pair->val);\n free(pair);\n }\n }\n free(bucketsTmp);\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\n for (int i = 0; i < hashMap->capacity; i++) {\n Pair *pair = hashMap->buckets[i];\n if (pair == NULL) {\n printf(\"NULL\\n\");\n } else if (pair == hashMap->TOMBSTONE) {\n printf(\"TOMBSTONE\\n\");\n } else {\n printf(\"%d -> %s\\n\", pair->key, pair->val);\n }\n }\n}\n
"},{"location":"chapter_hashing/hash_collision/#2","title":"2. \u00a0 \u5e73\u65b9\u63a2\u6d4b","text":"[class]{HashMapOpenAddressing}-[func]{}\n\u5e73\u65b9\u63a2\u6d4b\u4e0e\u7ebf\u6027\u63a2\u6d4b\u7c7b\u4f3c\uff0c\u90fd\u662f\u5f00\u653e\u5bfb\u5740\u7684\u5e38\u89c1\u7b56\u7565\u4e4b\u4e00\u3002\u5f53\u53d1\u751f\u51b2\u7a81\u65f6\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e0d\u662f\u7b80\u5355\u5730\u8df3\u8fc7\u4e00\u4e2a\u56fa\u5b9a\u7684\u6b65\u6570\uff0c\u800c\u662f\u8df3\u8fc7\u201c\u63a2\u6d4b\u6b21\u6570\u7684\u5e73\u65b9\u201d\u7684\u6b65\u6570\uff0c\u5373 \\(1, 4, 9, \\dots\\) \u6b65\u3002
\u5e73\u65b9\u63a2\u6d4b\u4e3b\u8981\u5177\u6709\u4ee5\u4e0b\u4f18\u52bf\u3002
- \u5e73\u65b9\u63a2\u6d4b\u901a\u8fc7\u8df3\u8fc7\u63a2\u6d4b\u6b21\u6570\u5e73\u65b9\u7684\u8ddd\u79bb\uff0c\u8bd5\u56fe\u7f13\u89e3\u7ebf\u6027\u63a2\u6d4b\u7684\u805a\u96c6\u6548\u5e94\u3002
- \u5e73\u65b9\u63a2\u6d4b\u4f1a\u8df3\u8fc7\u66f4\u5927\u7684\u8ddd\u79bb\u6765\u5bfb\u627e\u7a7a\u4f4d\u7f6e\uff0c\u6709\u52a9\u4e8e\u6570\u636e\u5206\u5e03\u5f97\u66f4\u52a0\u5747\u5300\u3002
\u7136\u800c\uff0c\u5e73\u65b9\u63a2\u6d4b\u5e76\u4e0d\u662f\u5b8c\u7f8e\u7684\u3002
- \u4ecd\u7136\u5b58\u5728\u805a\u96c6\u73b0\u8c61\uff0c\u5373\u67d0\u4e9b\u4f4d\u7f6e\u6bd4\u5176\u4ed6\u4f4d\u7f6e\u66f4\u5bb9\u6613\u88ab\u5360\u7528\u3002
- \u7531\u4e8e\u5e73\u65b9\u7684\u589e\u957f\uff0c\u5e73\u65b9\u63a2\u6d4b\u53ef\u80fd\u4e0d\u4f1a\u63a2\u6d4b\u6574\u4e2a\u54c8\u5e0c\u8868\uff0c\u8fd9\u610f\u5473\u7740\u5373\u4f7f\u54c8\u5e0c\u8868\u4e2d\u6709\u7a7a\u6876\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e5f\u53ef\u80fd\u65e0\u6cd5\u8bbf\u95ee\u5230\u5b83\u3002
\u987e\u540d\u601d\u4e49\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\)\u3001\\(f_2(x)\\)\u3001\\(f_3(x)\\)\u3001\\(\\dots\\) \u8fdb\u884c\u63a2\u6d4b\u3002
- \u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u6876\u540e\u63d2\u5165\u5143\u7d20\u3002
- \u67e5\u627e\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u4e0b\u8fdb\u884c\u67e5\u627e\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u65f6\u8fd4\u56de\uff1b\u82e5\u9047\u5230\u7a7a\u6876\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u4e0d\u5b58\u5728\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(\\text{None}\\) \u3002
\u4e0e\u7ebf\u6027\u63a2\u6d4b\u76f8\u6bd4\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u4f1a\u5e26\u6765\u989d\u5916\u7684\u8ba1\u7b97\u91cf\u3002
Tip
\u8bf7\u6ce8\u610f\uff0c\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u548c\u591a\u6b21\u54c8\u5e0c\uff09\u54c8\u5e0c\u8868\u90fd\u5b58\u5728\u201c\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u201d\u7684\u95ee\u9898\u3002
"},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u9009\u62e9","text":"\u5404\u79cd\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u7b56\u7565\uff0c\u4e0b\u9762\u4e3e\u51e0\u4e2a\u4f8b\u5b50\u3002
- Python \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002\u5b57\u5178 dict \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002
- Java \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002\u81ea JDK 1.8 \u4ee5\u6765\uff0c\u5f53 HashMap \u5185\u6570\u7ec4\u957f\u5ea6\u8fbe\u5230 64 \u4e14\u94fe\u8868\u957f\u5ea6\u8fbe\u5230 8 \u65f6\uff0c\u94fe\u8868\u4f1a\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002
- Go \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\u3002\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u786e\u4fdd\u6027\u80fd\u3002
\u300c\u54c8\u5e0c\u8868 hash table\u300d\uff0c\u53c8\u79f0\u300c\u6563\u5217\u8868\u300d\uff0c\u5176\u901a\u8fc7\u5efa\u7acb\u952e
key\u4e0e\u503cvalue\u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8be2\u3002\u5177\u4f53\u800c\u8a00\uff0c\u6211\u4eec\u5411\u54c8\u5e0c\u8868\u8f93\u5165\u4e00\u4e2a\u952ekey\uff0c\u5219\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5bf9\u5e94\u7684\u503cvalue\u3002\u5982\u56fe 6-1 \u6240\u793a\uff0c\u7ed9\u5b9a \\(n\\) \u4e2a\u5b66\u751f\uff0c\u6bcf\u4e2a\u5b66\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b66\u53f7\u201d\u4e24\u9879\u6570\u636e\u3002\u5047\u5982\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u201c\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\u201d\u7684\u67e5\u8be2\u529f\u80fd\uff0c\u5219\u53ef\u4ee5\u91c7\u7528\u56fe 6-1 \u6240\u793a\u7684\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002
\u56fe 6-1 \u00a0 \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a
\u9664\u54c8\u5e0c\u8868\u5916\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u4e5f\u53ef\u4ee5\u5b9e\u73b0\u67e5\u8be2\u529f\u80fd\uff0c\u5b83\u4eec\u7684\u6548\u7387\u5bf9\u6bd4\u5982\u8868 6-1 \u6240\u793a\u3002
- \u6dfb\u52a0\u5143\u7d20\uff1a\u4ec5\u9700\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u6570\u7ec4\uff08\u94fe\u8868\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
- \u67e5\u8be2\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\uff08\u94fe\u8868\uff09\u662f\u4e71\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u904d\u5386\u5176\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
- \u5220\u9664\u5143\u7d20\uff1a\u9700\u8981\u5148\u67e5\u8be2\u5230\u5143\u7d20\uff0c\u518d\u4ece\u6570\u7ec4\uff08\u94fe\u8868\uff09\u4e2d\u5220\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
\u8868 6-1 \u00a0 \u5143\u7d20\u67e5\u8be2\u6548\u7387\u5bf9\u6bd4
\u6570\u7ec4 \u94fe\u8868 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\)\u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u8fdb\u884c\u589e\u5220\u67e5\u6539\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
"},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1 \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"\u54c8\u5e0c\u8868\u7684\u5e38\u89c1\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u548c\u5220\u9664\u952e\u503c\u5bf9\u7b49\uff0c\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map.py
hash_map.cpp# \u521d\u59cb\u5316\u54c8\u5e0c\u8868\nhmap: dict = {}\n\n# \u6dfb\u52a0\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n# \u67e5\u8be2\u64cd\u4f5c\n# \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = hmap[15937]\n\n# \u5220\u9664\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nhmap.pop(10583)\n
hash_map.java/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map<int, string> map;\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n
hash_map.cs/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<Integer, String> map = new HashMap<>();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\"); \nmap.put(15937, \"\u5c0f\u5570\"); \nmap.put(16750, \"\u5c0f\u7b97\"); \nmap.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n
hash_map_test.go/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary<int, string> map = new() {\n /* \u6dfb\u52a0\u64cd\u4f5c */\n // \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\n { 12836, \"\u5c0f\u54c8\" },\n { 15937, \"\u5c0f\u5570\" },\n { 16750, \"\u5c0f\u7b97\" },\n { 13276, \"\u5c0f\u6cd5\" },\n { 10583, \"\u5c0f\u9e2d\" }\n};\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n
hash_map.swift/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nhmap := make(map[int]string)\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := hmap[15937]\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(hmap, 10583)\n
hash_map.js/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n
hash_map.ts/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n
hash_map.dart/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map<number, string>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n
hash_map.rs/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<int, String> map = {};\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n
hash_map.cuse std::collections::HashMap;\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nlet mut map: HashMap<i32, String> = HashMap::new();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.insert(12836, \"\u5c0f\u54c8\".to_string());\nmap.insert(15937, \"\u5c0f\u5570\".to_string());\nmap.insert(16750, \"\u5c0f\u7b97\".to_string());\nmap.insert(13279, \"\u5c0f\u6cd5\".to_string());\nmap.insert(10583, \"\u5c0f\u9e2d\".to_string());\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet _name: Option<&String> = map.get(&15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nlet _removed_value: Option<String> = map.remove(&10583);\n
hash_map.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n\n\u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u5f0f\uff1a\u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u548c\u904d\u5386\u503c\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map.py
hash_map.cpp# \u904d\u5386\u54c8\u5e0c\u8868\n# \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value in hmap.items():\n print(key, \"->\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in hmap.keys():\n print(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in hmap.values():\n print(value)\n
hash_map.java/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (auto kv: map) {\n cout << kv.first << \" -> \" << kv.second << endl;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u5668\u904d\u5386 key->value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n cout << iter->first << \"->\" << iter->second << endl;\n}\n
hash_map.cs/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (Map.Entry <Integer, String> kv: map.entrySet()) {\n System.out.println(kv.getKey() + \" -> \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\n System.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\n System.out.println(val);\n}\n
hash_map_test.go/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nforeach (var kv in map) {\n Console.WriteLine(kv.Key + \" -> \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys) {\n Console.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (string val in map.Values) {\n Console.WriteLine(val);\n}\n
hash_map.swift/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value := range hmap {\n fmt.Println(key, \"->\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range hmap {\n fmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range hmap {\n fmt.Println(value)\n}\n
hash_map.js/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nfor (key, value) in map {\n print(\"\\(key) -> \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\n print(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\n print(value)\n}\n
hash_map.ts/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\n console.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n console.info(v);\n}\n
hash_map.dart/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\n console.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n console.info(v);\n}\n
hash_map.rs/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nmap.forEach((key, value) {\n print('$key -> $value');\n});\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nmap.keys.forEach((key) {\n print(key);\n});\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nmap.values.forEach((value) {\n print(value);\n});\n
hash_map.c/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nfor (key, value) in &map {\n println!(\"{key} -> {value}\");\n}\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys() {\n println!(\"{key}\"); \n}\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values() {\n println!(\"{value}\");\n}\n
hash_map.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n
"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2 \u00a0 \u54c8\u5e0c\u8868\u7b80\u5355\u5b9e\u73b0","text":"\n\u6211\u4eec\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u300c\u6876 bucket\u300d\uff0c\u6bcf\u4e2a\u6876\u53ef\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u56e0\u6b64\uff0c\u67e5\u8be2\u64cd\u4f5c\u5c31\u662f\u627e\u5230
key\u5bf9\u5e94\u7684\u6876\uff0c\u5e76\u5728\u6876\u4e2d\u83b7\u53d6value\u3002\u90a3\u4e48\uff0c\u5982\u4f55\u57fa\u4e8e
key\u5b9a\u4f4d\u5bf9\u5e94\u7684\u6876\u5462\uff1f\u8fd9\u662f\u901a\u8fc7\u300c\u54c8\u5e0c\u51fd\u6570 hash function\u300d\u5b9e\u73b0\u7684\u3002\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u8f93\u51fa\u7a7a\u95f4\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u6240\u6709key\uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u6240\u6709\u6876\uff08\u6570\u7ec4\u7d22\u5f15\uff09\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8f93\u5165\u4e00\u4e2akey\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u8be5key\u5bf9\u5e94\u7684\u952e\u503c\u5bf9\u5728\u6570\u7ec4\u4e2d\u7684\u5b58\u50a8\u4f4d\u7f6e\u3002\u8f93\u5165\u4e00\u4e2a
key\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u8fc7\u7a0b\u5206\u4e3a\u4ee5\u4e0b\u4e24\u6b65\u3002- \u901a\u8fc7\u67d0\u79cd\u54c8\u5e0c\u7b97\u6cd5
hash()\u8ba1\u7b97\u5f97\u5230\u54c8\u5e0c\u503c\u3002 - \u5c06\u54c8\u5e0c\u503c\u5bf9\u6876\u6570\u91cf\uff08\u6570\u7ec4\u957f\u5ea6\uff09
capacity\u53d6\u6a21\uff0c\u4ece\u800c\u83b7\u53d6\u8be5key\u5bf9\u5e94\u7684\u6570\u7ec4\u7d22\u5f15index\u3002
index = hash(key) % capacity\n\u968f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528
index\u5728\u54c8\u5e0c\u8868\u4e2d\u8bbf\u95ee\u5bf9\u5e94\u7684\u6876\uff0c\u4ece\u800c\u83b7\u53d6value\u3002\u8bbe\u6570\u7ec4\u957f\u5ea6
capacity = 100\u3001\u54c8\u5e0c\u7b97\u6cd5hash(key) = key\uff0c\u6613\u5f97\u54c8\u5e0c\u51fd\u6570\u4e3akey % 100\u3002\u56fe 6-2 \u4ee5key\u5b66\u53f7\u548cvalue\u59d3\u540d\u4e3a\u4f8b\uff0c\u5c55\u793a\u4e86\u54c8\u5e0c\u51fd\u6570\u7684\u5de5\u4f5c\u539f\u7406\u3002\u56fe 6-2 \u00a0 \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406
\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u7b80\u5355\u54c8\u5e0c\u8868\u3002\u5176\u4e2d\uff0c\u6211\u4eec\u5c06
PythonC++JavaC#GoSwiftJSTSDartRustCZig array_hash_map.pykey\u548cvalue\u5c01\u88c5\u6210\u4e00\u4e2a\u7c7bPair\uff0c\u4ee5\u8868\u793a\u952e\u503c\u5bf9\u3002
array_hash_map.cppclass Pair:\n \"\"\"\u952e\u503c\u5bf9\"\"\"\n\n def __init__(self, key: int, val: str):\n self.key = key\n self.val = val\n\nclass ArrayHashMap:\n \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n # \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n self.buckets: list[Pair | None] = [None] * 100\n\n def hash_func(self, key: int) -> int:\n \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n index = key % 100\n return index\n\n def get(self, key: int) -> str:\n \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n index: int = self.hash_func(key)\n pair: Pair = self.buckets[index]\n if pair is None:\n return None\n return pair.val\n\n def put(self, key: int, val: str):\n \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n pair = Pair(key, val)\n index: int = self.hash_func(key)\n self.buckets[index] = pair\n\n def remove(self, key: int):\n \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n index: int = self.hash_func(key)\n # \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n self.buckets[index] = None\n\n def entry_set(self) -> list[Pair]:\n \"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\n result: list[Pair] = []\n for pair in self.buckets:\n if pair is not None:\n result.append(pair)\n return result\n\n def key_set(self) -> list[int]:\n \"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\n result = []\n for pair in self.buckets:\n if pair is not None:\n result.append(pair.key)\n return result\n\n def value_set(self) -> list[str]:\n \"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\n result = []\n for pair in self.buckets:\n if pair is not None:\n result.append(pair.val)\n return result\n\n def print(self):\n \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n for pair in self.buckets:\n if pair is not None:\n print(pair.key, \"->\", pair.val)\n
array_hash_map.java/* \u952e\u503c\u5bf9 */\nstruct Pair {\n public:\n int key;\n string val;\n Pair(int key, string val) {\n this->key = key;\n this->val = val;\n }\n};\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n private:\n vector<Pair *> buckets;\n\n public:\n ArrayHashMap() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n buckets = vector<Pair *>(100);\n }\n\n ~ArrayHashMap() {\n // \u91ca\u653e\u5185\u5b58\n for (const auto &bucket : buckets) {\n delete bucket;\n }\n buckets.clear();\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int hashFunc(int key) {\n int index = key % 100;\n return index;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n string get(int key) {\n int index = hashFunc(key);\n Pair *pair = buckets[index];\n if (pair == nullptr)\n return \"\";\n return pair->val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, string val) {\n Pair *pair = new Pair(key, val);\n int index = hashFunc(key);\n buckets[index] = pair;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n int index = hashFunc(key);\n // \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\n delete buckets[index];\n buckets[index] = nullptr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n vector<Pair *> pairSet() {\n vector<Pair *> pairSet;\n for (Pair *pair : buckets) {\n if (pair != nullptr) {\n pairSet.push_back(pair);\n }\n }\n return pairSet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n vector<int> keySet() {\n vector<int> keySet;\n for (Pair *pair : buckets) {\n if (pair != nullptr) {\n keySet.push_back(pair->key);\n }\n }\n return keySet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n vector<string> valueSet() {\n vector<string> valueSet;\n for (Pair *pair : buckets) {\n if (pair != nullptr) {\n valueSet.push_back(pair->val);\n }\n }\n return valueSet;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void print() {\n for (Pair *kv : pairSet()) {\n cout << kv->key << \" -> \" << kv->val << endl;\n }\n }\n};\n
array_hash_map.cs/* \u952e\u503c\u5bf9 */\nclass Pair {\n public int key;\n public String val;\n\n public Pair(int key, String val) {\n this.key = key;\n this.val = val;\n }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n private List<Pair> buckets;\n\n public ArrayHashMap() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n buckets = new ArrayList<>();\n for (int i = 0; i < 100; i++) {\n buckets.add(null);\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private int hashFunc(int key) {\n int index = key % 100;\n return index;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public String get(int key) {\n int index = hashFunc(key);\n Pair pair = buckets.get(index);\n if (pair == null)\n return null;\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void put(int key, String val) {\n Pair pair = new Pair(key, val);\n int index = hashFunc(key);\n buckets.set(index, pair);\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void remove(int key) {\n int index = hashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n buckets.set(index, null);\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n public List<Pair> pairSet() {\n List<Pair> pairSet = new ArrayList<>();\n for (Pair pair : buckets) {\n if (pair != null)\n pairSet.add(pair);\n }\n return pairSet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n public List<Integer> keySet() {\n List<Integer> keySet = new ArrayList<>();\n for (Pair pair : buckets) {\n if (pair != null)\n keySet.add(pair.key);\n }\n return keySet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n public List<String> valueSet() {\n List<String> valueSet = new ArrayList<>();\n for (Pair pair : buckets) {\n if (pair != null)\n valueSet.add(pair.val);\n }\n return valueSet;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void print() {\n for (Pair kv : pairSet()) {\n System.out.println(kv.key + \" -> \" + kv.val);\n }\n }\n}\n
array_hash_map.go/* \u952e\u503c\u5bf9 int->string */\nclass Pair(int key, string val) {\n public int key = key;\n public string val = val;\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n List<Pair?> buckets;\n public ArrayHashMap() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n buckets = [];\n for (int i = 0; i < 100; i++) {\n buckets.Add(null);\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int HashFunc(int key) {\n int index = key % 100;\n return index;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public string? Get(int key) {\n int index = HashFunc(key);\n Pair? pair = buckets[index];\n if (pair == null) return null;\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public void Put(int key, string val) {\n Pair pair = new(key, val);\n int index = HashFunc(key);\n buckets[index] = pair;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public void Remove(int key) {\n int index = HashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n buckets[index] = null;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n public List<Pair> PairSet() {\n List<Pair> pairSet = [];\n foreach (Pair? pair in buckets) {\n if (pair != null)\n pairSet.Add(pair);\n }\n return pairSet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n public List<int> KeySet() {\n List<int> keySet = [];\n foreach (Pair? pair in buckets) {\n if (pair != null)\n keySet.Add(pair.key);\n }\n return keySet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n public List<string> ValueSet() {\n List<string> valueSet = [];\n foreach (Pair? pair in buckets) {\n if (pair != null)\n valueSet.Add(pair.val);\n }\n return valueSet;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public void Print() {\n foreach (Pair kv in PairSet()) {\n Console.WriteLine(kv.key + \" -> \" + kv.val);\n }\n }\n}\n
array_hash_map.swift/* \u952e\u503c\u5bf9 */\ntype pair struct {\n key int\n val string\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\n buckets []*pair\n}\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n buckets := make([]*pair, 100)\n return &arrayHashMap{buckets: buckets}\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n index := key % 100\n return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\n index := a.hashFunc(key)\n pair := a.buckets[index]\n if pair == nil {\n return \"Not Found\"\n }\n return pair.val\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\n pair := &pair{key: key, val: val}\n index := a.hashFunc(key)\n a.buckets[index] = pair\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\n index := a.hashFunc(key)\n // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n a.buckets[index] = nil\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\n var pairs []*pair\n for _, pair := range a.buckets {\n if pair != nil {\n pairs = append(pairs, pair)\n }\n }\n return pairs\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\n var keys []int\n for _, pair := range a.buckets {\n if pair != nil {\n keys = append(keys, pair.key)\n }\n }\n return keys\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\n var values []string\n for _, pair := range a.buckets {\n if pair != nil {\n values = append(values, pair.val)\n }\n }\n return values\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\n for _, pair := range a.buckets {\n if pair != nil {\n fmt.Println(pair.key, \"->\", pair.val)\n }\n }\n}\n
array_hash_map.js/* \u952e\u503c\u5bf9 */\nclass Pair: Equatable {\n public var key: Int\n public var val: String\n\n public init(key: Int, val: String) {\n self.key = key\n self.val = val\n }\n\n public static func == (lhs: Pair, rhs: Pair) -> Bool {\n lhs.key == rhs.key && lhs.val == rhs.val\n }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n private var buckets: [Pair?] = []\n\n init() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n for _ in 0 ..< 100 {\n buckets.append(nil)\n }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private func hashFunc(key: Int) -> Int {\n let index = key % 100\n return index\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n func get(key: Int) -> String? {\n let index = hashFunc(key: key)\n let pair = buckets[index]\n return pair?.val\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n func put(key: Int, val: String) {\n let pair = Pair(key: key, val: val)\n let index = hashFunc(key: key)\n buckets[index] = pair\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n func remove(key: Int) {\n let index = hashFunc(key: key)\n // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n buckets[index] = nil\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n func pairSet() -> [Pair] {\n var pairSet: [Pair] = []\n for pair in buckets {\n if let pair = pair {\n pairSet.append(pair)\n }\n }\n return pairSet\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n func keySet() -> [Int] {\n var keySet: [Int] = []\n for pair in buckets {\n if let pair = pair {\n keySet.append(pair.key)\n }\n }\n return keySet\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n func valueSet() -> [String] {\n var valueSet: [String] = []\n for pair in buckets {\n if let pair = pair {\n valueSet.append(pair.val)\n }\n }\n return valueSet\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n func print() {\n for pair in pairSet() {\n Swift.print(\"\\(pair.key) -> \\(pair.val)\")\n }\n }\n}\n
array_hash_map.ts/* \u952e\u503c\u5bf9 Number -> String */\nclass Pair {\n constructor(key, val) {\n this.key = key;\n this.val = val;\n }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n #buckets;\n constructor() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n this.#buckets = new Array(100).fill(null);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n #hashFunc(key) {\n return key % 100;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n get(key) {\n let index = this.#hashFunc(key);\n let pair = this.#buckets[index];\n if (pair === null) return null;\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n set(key, val) {\n let index = this.#hashFunc(key);\n this.#buckets[index] = new Pair(key, val);\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n delete(key) {\n let index = this.#hashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n this.#buckets[index] = null;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n entries() {\n let arr = [];\n for (let i = 0; i < this.#buckets.length; i++) {\n if (this.#buckets[i]) {\n arr.push(this.#buckets[i]);\n }\n }\n return arr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n keys() {\n let arr = [];\n for (let i = 0; i < this.#buckets.length; i++) {\n if (this.#buckets[i]) {\n arr.push(this.#buckets[i].key);\n }\n }\n return arr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n values() {\n let arr = [];\n for (let i = 0; i < this.#buckets.length; i++) {\n if (this.#buckets[i]) {\n arr.push(this.#buckets[i].val);\n }\n }\n return arr;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n print() {\n let pairSet = this.entries();\n for (const pair of pairSet) {\n console.info(`${pair.key} -> ${pair.val}`);\n }\n }\n}\n
array_hash_map.dart/* \u952e\u503c\u5bf9 Number -> String */\nclass Pair {\n public key: number;\n public val: string;\n\n constructor(key: number, val: string) {\n this.key = key;\n this.val = val;\n }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n private readonly buckets: (Pair | null)[];\n\n constructor() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n this.buckets = new Array(100).fill(null);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n private hashFunc(key: number): number {\n return key % 100;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n public get(key: number): string | null {\n let index = this.hashFunc(key);\n let pair = this.buckets[index];\n if (pair === null) return null;\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n public set(key: number, val: string) {\n let index = this.hashFunc(key);\n this.buckets[index] = new Pair(key, val);\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n public delete(key: number) {\n let index = this.hashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n this.buckets[index] = null;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n public entries(): (Pair | null)[] {\n let arr: (Pair | null)[] = [];\n for (let i = 0; i < this.buckets.length; i++) {\n if (this.buckets[i]) {\n arr.push(this.buckets[i]);\n }\n }\n return arr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n public keys(): (number | undefined)[] {\n let arr: (number | undefined)[] = [];\n for (let i = 0; i < this.buckets.length; i++) {\n if (this.buckets[i]) {\n arr.push(this.buckets[i].key);\n }\n }\n return arr;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n public values(): (string | undefined)[] {\n let arr: (string | undefined)[] = [];\n for (let i = 0; i < this.buckets.length; i++) {\n if (this.buckets[i]) {\n arr.push(this.buckets[i].val);\n }\n }\n return arr;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n public print() {\n let pairSet = this.entries();\n for (const pair of pairSet) {\n console.info(`${pair.key} -> ${pair.val}`);\n }\n }\n}\n
array_hash_map.rs/* \u952e\u503c\u5bf9 */\nclass Pair {\n int key;\n String val;\n Pair(this.key, this.val);\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n late List<Pair?> _buckets;\n\n ArrayHashMap() {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n _buckets = List.filled(100, null);\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n int _hashFunc(int key) {\n final int index = key % 100;\n return index;\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n String? get(int key) {\n final int index = _hashFunc(key);\n final Pair? pair = _buckets[index];\n if (pair == null) {\n return null;\n }\n return pair.val;\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n void put(int key, String val) {\n final Pair pair = Pair(key, val);\n final int index = _hashFunc(key);\n _buckets[index] = pair;\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n void remove(int key) {\n final int index = _hashFunc(key);\n _buckets[index] = null;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n List<Pair> pairSet() {\n List<Pair> pairSet = [];\n for (final Pair? pair in _buckets) {\n if (pair != null) {\n pairSet.add(pair);\n }\n }\n return pairSet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n List<int> keySet() {\n List<int> keySet = [];\n for (final Pair? pair in _buckets) {\n if (pair != null) {\n keySet.add(pair.key);\n }\n }\n return keySet;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n List<String> values() {\n List<String> valueSet = [];\n for (final Pair? pair in _buckets) {\n if (pair != null) {\n valueSet.add(pair.val);\n }\n }\n return valueSet;\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n void printHashMap() {\n for (final Pair kv in pairSet()) {\n print(\"${kv.key} -> ${kv.val}\");\n }\n }\n}\n
array_hash_map.c/* \u952e\u503c\u5bf9 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n pub key: i32,\n pub val: String,\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\npub struct ArrayHashMap {\n buckets: Vec<Option<Pair>>\n}\n\nimpl ArrayHashMap {\n pub fn new() -> ArrayHashMap {\n // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n Self { buckets: vec![None; 100] }\n }\n\n /* \u54c8\u5e0c\u51fd\u6570 */\n fn hash_func(&self, key: i32) -> usize {\n key as usize % 100\n }\n\n /* \u67e5\u8be2\u64cd\u4f5c */\n pub fn get(&self, key: i32) -> Option<&String> {\n let index = self.hash_func(key);\n self.buckets[index].as_ref().map(|pair| &pair.val)\n }\n\n /* \u6dfb\u52a0\u64cd\u4f5c */\n pub fn put(&mut self, key: i32, val: &str) {\n let index = self.hash_func(key);\n self.buckets[index] = Some(Pair {\n key,\n val: val.to_string(),\n });\n }\n\n /* \u5220\u9664\u64cd\u4f5c */\n pub fn remove(&mut self, key: i32) {\n let index = self.hash_func(key);\n // \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n self.buckets[index] = None;\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n pub fn entry_set(&self) -> Vec<&Pair> {\n self.buckets.iter().filter_map(|pair| pair.as_ref()).collect()\n }\n\n /* \u83b7\u53d6\u6240\u6709\u952e */\n pub fn key_set(&self) -> Vec<&i32> {\n self.buckets.iter().filter_map(|pair| pair.as_ref().map(|pair| &pair.key)).collect()\n }\n\n /* \u83b7\u53d6\u6240\u6709\u503c */\n pub fn value_set(&self) -> Vec<&String> {\n self.buckets.iter().filter_map(|pair| pair.as_ref().map(|pair| &pair.val)).collect()\n }\n\n /* \u6253\u5370\u54c8\u5e0c\u8868 */\n pub fn print(&self) {\n for pair in self.entry_set() {\n println!(\"{} -> {}\", pair.key, pair.val);\n }\n }\n}\n
array_hash_map.zig/* \u952e\u503c\u5bf9 int->string */\ntypedef struct {\n int key;\n char *val;\n} Pair;\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntypedef struct {\n Pair *buckets[HASHTABLE_CAPACITY];\n} ArrayHashMap;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayHashMap *newArrayHashMap() {\n ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n return hmap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n for (int i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n free(hmap->buckets[i]->val);\n free(hmap->buckets[i]);\n }\n }\n free(hmap);\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n Pair *Pair = malloc(sizeof(Pair));\n Pair->key = key;\n Pair->val = malloc(strlen(val) + 1);\n strcpy(Pair->val, val);\n\n int index = hashFunc(key);\n hmap->buckets[index] = Pair;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n int index = hashFunc(key);\n free(hmap->buckets[index]->val);\n free(hmap->buckets[index]);\n hmap->buckets[index] = NULL;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n Pair *entries;\n int i = 0, index = 0;\n int total = 0;\n /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n total++;\n }\n }\n entries = malloc(sizeof(Pair) * total);\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n entries[index].key = hmap->buckets[i]->key;\n entries[index].val = malloc(strlen(hmap->buckets[i]->val) + 1);\n strcpy(entries[index].val, hmap->buckets[i]->val);\n index++;\n }\n }\n set->set = entries;\n set->len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n int *keys;\n int i = 0, index = 0;\n int total = 0;\n /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n total++;\n }\n }\n keys = malloc(total * sizeof(int));\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n keys[index] = hmap->buckets[i]->key;\n index++;\n }\n }\n set->set = keys;\n set->len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n char **vals;\n int i = 0, index = 0;\n int total = 0;\n /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n total++;\n }\n }\n vals = malloc(total * sizeof(char *));\n for (i = 0; i < HASHTABLE_CAPACITY; i++) {\n if (hmap->buckets[i] != NULL) {\n vals[index] = hmap->buckets[i]->val;\n index++;\n }\n }\n set->set = vals;\n set->len = total;\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(ArrayHashMap *hmap) {\n int i;\n MapSet set;\n pairSet(hmap, &set);\n Pair *entries = (Pair *)set.set;\n for (i = 0; i < set.len; i++) {\n printf(\"%d -> %s\\n\", entries[i].key, entries[i].val);\n }\n free(set.set);\n}\n
"},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u4e0e\u6269\u5bb9","text":"// \u952e\u503c\u5bf9\nconst Pair = struct {\n key: usize = undefined,\n val: []const u8 = undefined,\n\n pub fn init(key: usize, val: []const u8) Pair {\n return Pair {\n .key = key,\n .val = val,\n };\n }\n};\n\n// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\n return struct {\n bucket: ?std.ArrayList(?T) = null,\n mem_allocator: std.mem.Allocator = undefined,\n\n const Self = @This();\n\n // \u6784\u9020\u51fd\u6570\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n self.mem_allocator = allocator;\n // \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\n self.bucket = std.ArrayList(?T).init(self.mem_allocator);\n var i: i32 = 0;\n while (i < 100) : (i += 1) {\n try self.bucket.?.append(null);\n }\n }\n\n // \u6790\u6784\u51fd\u6570\n pub fn deinit(self: *Self) void {\n if (self.bucket != null) self.bucket.?.deinit();\n }\n\n // \u54c8\u5e0c\u51fd\u6570\n fn hashFunc(key: usize) usize {\n var index = key % 100;\n return index;\n }\n\n // \u67e5\u8be2\u64cd\u4f5c\n pub fn get(self: *Self, key: usize) []const u8 {\n var index = hashFunc(key);\n var pair = self.bucket.?.items[index];\n return pair.?.val;\n }\n\n // \u6dfb\u52a0\u64cd\u4f5c\n pub fn put(self: *Self, key: usize, val: []const u8) !void {\n var pair = Pair.init(key, val);\n var index = hashFunc(key);\n self.bucket.?.items[index] = pair;\n }\n\n // \u5220\u9664\u64cd\u4f5c\n pub fn remove(self: *Self, key: usize) !void {\n var index = hashFunc(key);\n // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n self.bucket.?.items[index] = null;\n } \n\n // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\n pub fn pairSet(self: *Self) !std.ArrayList(T) {\n var entry_set = std.ArrayList(T).init(self.mem_allocator);\n for (self.bucket.?.items) |item| {\n if (item == null) continue;\n try entry_set.append(item.?);\n }\n return entry_set;\n } \n\n // \u83b7\u53d6\u6240\u6709\u952e\n pub fn keySet(self: *Self) !std.ArrayList(usize) {\n var key_set = std.ArrayList(usize).init(self.mem_allocator);\n for (self.bucket.?.items) |item| {\n if (item == null) continue;\n try key_set.append(item.?.key);\n }\n return key_set;\n } \n\n // \u83b7\u53d6\u6240\u6709\u503c\n pub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\n var value_set = std.ArrayList([]const u8).init(self.mem_allocator);\n for (self.bucket.?.items) |item| {\n if (item == null) continue;\n try value_set.append(item.?.val);\n }\n return value_set;\n }\n\n // \u6253\u5370\u54c8\u5e0c\u8868\n pub fn print(self: *Self) !void {\n var entry_set = try self.pairSet();\n defer entry_set.deinit();\n for (entry_set.items) |item| {\n std.debug.print(\"{} -> {s}\\n\", .{item.key, item.val});\n }\n }\n };\n}\n\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u6240\u6709
key\u6784\u6210\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u6570\u7ec4\u6240\u6709\u7d22\u5f15\u6784\u6210\u7684\u8f93\u51fa\u7a7a\u95f4\uff0c\u800c\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7406\u8bba\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u76f8\u540c\u8f93\u51fa\u201d\u7684\u60c5\u51b5\u3002\u5bf9\u4e8e\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u54c8\u5e0c\u51fd\u6570\uff0c\u5f53\u8f93\u5165\u7684
key\u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u51fa\u7ed3\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8be2\u5b66\u53f7\u4e3a 12836 \u548c 20336 \u7684\u4e24\u4e2a\u5b66\u751f\u65f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a12836 % 100 = 36\n20336 % 100 = 36\n\u5982\u56fe 6-3 \u6240\u793a\uff0c\u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002\u6211\u4eec\u5c06\u8fd9\u79cd\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u540c\u4e00\u8f93\u51fa\u7684\u60c5\u51b5\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 hash collision\u300d\u3002
\u56fe 6-3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b
\u5bb9\u6613\u60f3\u5230\uff0c\u54c8\u5e0c\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u4e2a
key\u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002\u5982\u56fe 6-4 \u6240\u793a\uff0c\u6269\u5bb9\u524d\u952e\u503c\u5bf9
(136, A)\u548c(236, D)\u53d1\u751f\u51b2\u7a81\uff0c\u6269\u5bb9\u540e\u51b2\u7a81\u6d88\u5931\u3002\u56fe 6-4 \u00a0 \u54c8\u5e0c\u8868\u6269\u5bb9
\u7c7b\u4f3c\u4e8e\u6570\u7ec4\u6269\u5bb9\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u8fc1\u79fb\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u975e\u5e38\u8017\u65f6\uff1b\u5e76\u4e14\u7531\u4e8e\u54c8\u5e0c\u8868\u5bb9\u91cf
capacity\u6539\u53d8\uff0c\u6211\u4eec\u9700\u8981\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u6765\u91cd\u65b0\u8ba1\u7b97\u6240\u6709\u952e\u503c\u5bf9\u7684\u5b58\u50a8\u4f4d\u7f6e\uff0c\u8fd9\u8fdb\u4e00\u6b65\u63d0\u9ad8\u4e86\u6269\u5bb9\u8fc7\u7a0b\u7684\u8ba1\u7b97\u5f00\u9500\u3002\u4e3a\u6b64\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u9884\u7559\u8db3\u591f\u5927\u7684\u54c8\u5e0c\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u9891\u7e41\u6269\u5bb9\u3002\u300c\u8d1f\u8f7d\u56e0\u5b50 load factor\u300d\u662f\u54c8\u5e0c\u8868\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u7684\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u7528\u4e8e\u8861\u91cf\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u4f5c\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002
"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_hashing/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"- \u8f93\u5165
key\uff0c\u54c8\u5e0c\u8868\u80fd\u591f\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u67e5\u8be2\u5230value\uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002 - \u5e38\u89c1\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u3001\u5220\u9664\u952e\u503c\u5bf9\u548c\u904d\u5386\u54c8\u5e0c\u8868\u7b49\u3002
- \u54c8\u5e0c\u51fd\u6570\u5c06
key\u6620\u5c04\u4e3a\u6570\u7ec4\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5bf9\u5e94\u6876\u5e76\u83b7\u53d6value\u3002 - \u4e24\u4e2a\u4e0d\u540c\u7684
key\u53ef\u80fd\u5728\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u540e\u5f97\u5230\u76f8\u540c\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u51fa\u9519\uff0c\u8fd9\u79cd\u73b0\u8c61\u88ab\u79f0\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002 - \u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u7684\u5f00\u9500\u5f88\u5927\u3002
- \u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u53cd\u6620\u4e86\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89e6\u53d1\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u6761\u4ef6\u3002
- \u94fe\u5f0f\u5730\u5740\u901a\u8fc7\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u4e3a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u5b58\u50a8\u5728\u540c\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002\u7136\u800c\uff0c\u94fe\u8868\u8fc7\u957f\u4f1a\u964d\u4f4e\u67e5\u8be2\u6548\u7387\uff0c\u53ef\u4ee5\u901a\u8fc7\u8fdb\u4e00\u6b65\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6765\u63d0\u9ad8\u6548\u7387\u3002
- \u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u8f83\u7ebf\u6027\u63a2\u6d4b\u66f4\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002
- \u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u4f8b\u5982\uff0cJava \u7684
HashMap\u4f7f\u7528\u94fe\u5f0f\u5730\u5740\uff0c\u800c Python \u7684Dict\u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002 - \u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u54c8\u5e0c\u7b97\u6cd5\u5177\u6709\u786e\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u5300\u5206\u5e03\u7684\u7279\u70b9\u3002\u5728\u5bc6\u7801\u5b66\u4e2d\uff0c\u54c8\u5e0c\u7b97\u6cd5\u8fd8\u5e94\u8be5\u5177\u5907\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u5e94\u3002
- \u54c8\u5e0c\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u5747\u5300\u5206\u5e03\uff0c\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002
- \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u5305\u62ec MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002MD5 \u5e38\u7528\u4e8e\u6821\u9a8c\u6587\u4ef6\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u4e8e\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u3002
- \u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u662f\u53ef\u54c8\u5e0c\u7684\u3002
\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f \\(O(n)\\) \uff1f
\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u5f53\u54c8\u5e0c\u51fd\u6570\u8bbe\u8ba1\u5f97\u6bd4\u8f83\u597d\u3001\u5bb9\u91cf\u8bbe\u7f6e\u6bd4\u8f83\u5408\u7406\u3001\u51b2\u7a81\u6bd4\u8f83\u5e73\u5747\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u4eec\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u54c8\u5e0c\u8868\u65f6\uff0c\u901a\u5e38\u8ba4\u4e3a\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002
\u4e3a\u4ec0\u4e48\u4e0d\u4f7f\u7528\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x\\) \u5462\uff1f\u8fd9\u6837\u5c31\u4e0d\u4f1a\u6709\u51b2\u7a81\u4e86
\u5728 \\(f(x) = x\\) \u54c8\u5e0c\u51fd\u6570\u4e0b\uff0c\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u8fd9\u4e0e\u6570\u7ec4\u7b49\u4ef7\u3002\u7136\u800c\uff0c\u8f93\u5165\u7a7a\u95f4\u901a\u5e38\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u957f\u5ea6\uff09\uff0c\u56e0\u6b64\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\u7684\u76ee\u6807\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u72b6\u6001\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u7a7a\u95f4\uff0c\u5e76\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8be2\u6548\u7387\u3002
\u54c8\u5e0c\u8868\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff0c\u4f46\u4e3a\u4ec0\u4e48\u6548\u7387\u53ef\u4ee5\u6bd4\u5b83\u4eec\u66f4\u9ad8\u5462\uff1f
\u9996\u5148\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\uff0c\u4f46\u7a7a\u95f4\u6548\u7387\u53d8\u4f4e\u4e86\u3002\u54c8\u5e0c\u8868\u6709\u76f8\u5f53\u4e00\u90e8\u5206\u5185\u5b58\u672a\u4f7f\u7528\u3002
\u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u573a\u666f\u4e0b\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u4e2a\u529f\u80fd\u80fd\u591f\u5728\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\uff0c\u90a3\u4e48\u901a\u5e38\u6bd4\u54c8\u5e0c\u8868\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u9700\u8981\u5f00\u9500\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u6570\u9879\u66f4\u5927\u3002
\u6700\u540e\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u80fd\u53d1\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u94fe\u5f0f\u5730\u5740\u4e2d\uff0c\u6211\u4eec\u91c7\u53d6\u5728\u94fe\u8868\u6216\u7ea2\u9ed1\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u65f6\u95f4\u7684\u98ce\u9669\u3002
\u591a\u6b21\u54c8\u5e0c\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\u5417\uff1f\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u8fd8\u80fd\u518d\u6b21\u4f7f\u7528\u5417\uff1f
\u591a\u6b21\u54c8\u5e0c\u662f\u5f00\u653e\u5bfb\u5740\u7684\u4e00\u79cd\uff0c\u5f00\u653e\u5bfb\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u901a\u8fc7\u6807\u8bb0\u5220\u9664\u3002\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u53ef\u4ee5\u518d\u6b21\u4f7f\u7528\u3002\u5f53\u5c06\u65b0\u5143\u7d20\u63d2\u5165\u54c8\u5e0c\u8868\uff0c\u5e76\u4e14\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u4f4d\u7f6e\u65f6\uff0c\u8be5\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u5143\u7d20\u4f7f\u7528\u3002\u8fd9\u6837\u505a\u65e2\u80fd\u4fdd\u6301\u54c8\u5e0c\u8868\u7684\u63a2\u6d4b\u5e8f\u5217\u4e0d\u53d8\uff0c\u53c8\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\u4f7f\u7528\u7387\u3002
\u4e3a\u4ec0\u4e48\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\uff0c\u67e5\u627e\u5143\u7d20\u7684\u65f6\u5019\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f
\u67e5\u627e\u7684\u65f6\u5019\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u5bf9\u5e94\u7684\u6876\u548c\u952e\u503c\u5bf9\uff0c\u53d1\u73b0
key\u4e0d\u5339\u914d\uff0c\u8fd9\u5c31\u4ee3\u8868\u6709\u54c8\u5e0c\u51b2\u7a81\u3002\u56e0\u6b64\uff0c\u7ebf\u6027\u63a2\u6d4b\u6cd5\u4f1a\u6839\u636e\u9884\u5148\u8bbe\u5b9a\u7684\u6b65\u957f\u4f9d\u6b21\u5411\u4e0b\u67e5\u627e\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u786e\u7684\u952e\u503c\u5bf9\u6216\u65e0\u6cd5\u627e\u5230\u8df3\u51fa\u4e3a\u6b62\u3002\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u6269\u5bb9\u80fd\u591f\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff1f
\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d6\u4f59\uff0c\u8ba9\u8f93\u51fa\u503c\u843d\u5728\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\u5185\uff1b\u5728\u6269\u5bb9\u540e\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d1\u751f\u53d8\u5316\uff0c\u800c
"},{"location":"chapter_heap/","title":"\u7b2c 8 \u7ae0 \u00a0 \u5806","text":"key\u5bf9\u5e94\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u4e2a\u6876\u7684\u591a\u4e2akey\uff0c\u5728\u6269\u5bb9\u540e\u53ef\u80fd\u4f1a\u88ab\u5206\u914d\u5230\u591a\u4e2a\u6876\u4e2d\uff0c\u4ece\u800c\u5b9e\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u7f13\u89e3\u3002Abstract
\u5806\u5c31\u50cf\u662f\u5c71\u5cb3\u5cf0\u5ce6\uff0c\u5c42\u53e0\u8d77\u4f0f\u3001\u5f62\u6001\u5404\u5f02\u3002
\u5ea7\u5ea7\u5c71\u5cf0\u9ad8\u4f4e\u9519\u843d\uff0c\u800c\u6700\u9ad8\u7684\u5c71\u5cf0\u603b\u662f\u6700\u5148\u6620\u5165\u773c\u5e18\u3002
"},{"location":"chapter_heap/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 8.1 \u00a0 \u5806
- 8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c
- 8.3 \u00a0 Top-K \u95ee\u9898
- 8.4 \u00a0 \u5c0f\u7ed3
\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u6765\u6784\u5efa\u4e00\u4e2a\u5806\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u3002
"},{"location":"chapter_heap/build_heap/#821","title":"8.2.1 \u00a0 \u501f\u52a9\u5165\u5806\u64cd\u4f5c\u5b9e\u73b0","text":"\u6211\u4eec\u9996\u5148\u521b\u5efa\u4e00\u4e2a\u7a7a\u5806\uff0c\u7136\u540e\u904d\u5386\u5217\u8868\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u5143\u7d20\u6267\u884c\u201c\u5165\u5806\u64cd\u4f5c\u201d\uff0c\u5373\u5148\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u5806\u7684\u5c3e\u90e8\uff0c\u518d\u5bf9\u8be5\u5143\u7d20\u6267\u884c\u201c\u4ece\u5e95\u81f3\u9876\u201d\u5806\u5316\u3002
\u6bcf\u5f53\u4e00\u4e2a\u5143\u7d20\u5165\u5806\uff0c\u5806\u7684\u957f\u5ea6\u5c31\u52a0\u4e00\u3002\u7531\u4e8e\u8282\u70b9\u662f\u4ece\u9876\u5230\u5e95\u4f9d\u6b21\u88ab\u6dfb\u52a0\u8fdb\u4e8c\u53c9\u6811\u7684\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u6784\u5efa\u7684\u3002
\u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6bcf\u4e2a\u5143\u7d20\u7684\u5165\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(\\log{n})\\) \u65f6\u95f4\uff0c\u56e0\u6b64\u8be5\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
"},{"location":"chapter_heap/build_heap/#822","title":"8.2.2 \u00a0 \u901a\u8fc7\u904d\u5386\u5806\u5316\u5b9e\u73b0","text":"\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9e\u73b0\u4e00\u79cd\u66f4\u4e3a\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u5171\u5206\u4e3a\u4e24\u6b65\u3002
- \u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u5730\u6dfb\u52a0\u5230\u5806\u4e2d\uff0c\u6b64\u65f6\u5806\u7684\u6027\u8d28\u5c1a\u672a\u5f97\u5230\u6ee1\u8db3\u3002
- \u5012\u5e8f\u904d\u5386\u5806\uff08\u5c42\u5e8f\u904d\u5386\u7684\u5012\u5e8f\uff09\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u975e\u53f6\u8282\u70b9\u6267\u884c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u3002
\u6bcf\u5f53\u5806\u5316\u4e00\u4e2a\u8282\u70b9\u540e\uff0c\u4ee5\u8be5\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\u5c31\u5f62\u6210\u4e00\u4e2a\u5408\u6cd5\u7684\u5b50\u5806\u3002\u800c\u7531\u4e8e\u662f\u5012\u5e8f\u904d\u5386\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u6784\u5efa\u7684\u3002
\u4e4b\u6240\u4ee5\u9009\u62e9\u5012\u5e8f\u904d\u5386\uff0c\u662f\u56e0\u4e3a\u8fd9\u6837\u80fd\u591f\u4fdd\u8bc1\u5f53\u524d\u8282\u70b9\u4e4b\u4e0b\u7684\u5b50\u6811\u5df2\u7ecf\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u8fd9\u6837\u5806\u5316\u5f53\u524d\u8282\u70b9\u624d\u662f\u6709\u6548\u7684\u3002
\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u7531\u4e8e\u53f6\u8282\u70b9\u6ca1\u6709\u5b50\u8282\u70b9\uff0c\u56e0\u6b64\u5b83\u4eec\u5929\u7136\u5c31\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u65e0\u987b\u5806\u5316\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u6700\u540e\u4e00\u4e2a\u975e\u53f6\u8282\u70b9\u662f\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u6211\u4eec\u4ece\u5b83\u5f00\u59cb\u5012\u5e8f\u904d\u5386\u5e76\u6267\u884c\u5806\u5316\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef __init__(self, nums: list[int]):\n \"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n # \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n self.max_heap = nums\n # \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in range(self.parent(self.size() - 1), -1, -1):\n self.sift_down(i)\n
my_heap.java/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector<int> nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n maxHeap = nums;\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = parent(size() - 1); i >= 0; i--) {\n siftDown(i);\n }\n}\n
my_heap.cs/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<Integer> nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n maxHeap = new ArrayList<>(nums);\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = parent(size() - 1); i >= 0; i--) {\n siftDown(i);\n }\n}\n
my_heap.go/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable<int> nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n maxHeap = new List<int>(nums);\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n var size = Parent(this.Size() - 1);\n for (int i = size; i >= 0; i--) {\n SiftDown(i);\n }\n}\n
my_heap.swift/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n h := &maxHeap{data: nums}\n for i := h.parent(len(h.data) - 1); i >= 0; i-- {\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n h.siftDown(i)\n }\n return h\n}\n
my_heap.js/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n maxHeap = nums\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in stride(from: parent(i: size() - 1), through: 0, by: -1) {\n siftDown(i: i)\n }\n}\n
my_heap.ts/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n this.#maxHeap = nums === undefined ? [] : [...nums];\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (let i = this.#parent(this.size() - 1); i >= 0; i--) {\n this.#siftDown(i);\n }\n}\n
my_heap.dart/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n this.maxHeap = nums === undefined ? [] : [...nums];\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (let i = this.parent(this.size() - 1); i >= 0; i--) {\n this.siftDown(i);\n }\n}\n
my_heap.rs/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<int> nums) {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n _maxHeap = nums;\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = _parent(size() - 1); i >= 0; i--) {\n siftDown(i);\n }\n}\n
my_heap.c/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nfn new(nums: Vec<i32>) -> Self {\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n let mut heap = MaxHeap { max_heap: nums };\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in (0..=Self::parent(heap.size() - 1)).rev() {\n heap.sift_down(i);\n }\n heap\n}\n
my_heap.zig/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n // \u6240\u6709\u5143\u7d20\u5165\u5806\n MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n maxHeap->size = size;\n memcpy(maxHeap->data, nums, size * sizeof(int));\n for (int i = parent(maxHeap, size - 1); i >= 0; i--) {\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n siftDown(maxHeap, i);\n }\n return maxHeap;\n}\n
"},{"location":"chapter_heap/build_heap/#823","title":"8.2.3 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"// \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\n if (self.max_heap != null) return;\n self.max_heap = std.ArrayList(T).init(allocator);\n // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n try self.max_heap.?.appendSlice(nums);\n // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n var i: usize = parent(self.size() - 1) + 1;\n while (i > 0) : (i -= 1) {\n try self.siftDown(i - 1);\n }\n}\n\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u63a8\u7b97\u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002
- \u5047\u8bbe\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u53f6\u8282\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u9700\u8981\u5806\u5316\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\((n - 1) / 2\\) \u3002
- \u5728\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u6700\u591a\u5806\u5316\u5230\u53f6\u8282\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(\\log n\\) \u3002
\u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u4f46\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u5e76\u4e0d\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230\u4e8c\u53c9\u6811\u5e95\u5c42\u8282\u70b9\u6570\u91cf\u8fdc\u591a\u4e8e\u9876\u5c42\u8282\u70b9\u7684\u6027\u8d28\u3002
\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8fdb\u884c\u66f4\u4e3a\u51c6\u786e\u7684\u8ba1\u7b97\u3002\u4e3a\u4e86\u964d\u4f4e\u8ba1\u7b97\u96be\u5ea6\uff0c\u5047\u8bbe\u7ed9\u5b9a\u4e00\u4e2a\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \u3001\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002
\u56fe 8-5 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u8282\u70b9\u6570\u91cf
\u5982\u56fe 8-5 \u6240\u793a\uff0c\u8282\u70b9\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8be5\u8ddd\u79bb\u6b63\u662f\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u5404\u5c42\u7684\u201c\u8282\u70b9\u6570\u91cf \\(\\times\\) \u8282\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5f97\u5230\u6240\u6709\u8282\u70b9\u7684\u5806\u5316\u8fed\u4ee3\u6b21\u6570\u7684\u603b\u548c\u3002
\\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\]\u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5bf9 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230\uff1a
\\[ \\begin{aligned} T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\]\u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u7528\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97\uff1a
\\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\]\u89c2\u5bdf\u4e0a\u5f0f\uff0c\u53d1\u73b0 \\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a
\\[ \\begin{aligned} T(h) & = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline & = 2^{h+1} - h - 2 \\newline & = O(2^h) \\end{aligned} \\]\u8fdb\u4e00\u6b65\u5730\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 \u5806","text":"\u300c\u5806 heap\u300d\u662f\u4e00\u79cd\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u4e3b\u8981\u53ef\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff0c\u5982\u56fe 8-1 \u6240\u793a\u3002
- \u300c\u5927\u9876\u5806 max heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002
- \u300c\u5c0f\u9876\u5806 min heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002
\u56fe 8-1 \u00a0 \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806
\u5806\u4f5c\u4e3a\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002
- \u6700\u5e95\u5c42\u8282\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u3002
- \u6211\u4eec\u5c06\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u9876\u201d\uff0c\u5c06\u5e95\u5c42\u6700\u9760\u53f3\u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u5e95\u201d\u3002
- \u5bf9\u4e8e\u5927\u9876\u5806\uff08\u5c0f\u9876\u5806\uff09\uff0c\u5806\u9876\u5143\u7d20\uff08\u6839\u8282\u70b9\uff09\u7684\u503c\u5206\u522b\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002
\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u300c\u4f18\u5148\u961f\u5217 priority queue\u300d\uff0c\u8fd9\u662f\u4e00\u79cd\u62bd\u8c61\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u4f18\u5148\u7ea7\u6392\u5e8f\u7684\u961f\u5217\u3002
\u5b9e\u9645\u4e0a\uff0c\u5806\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u4f18\u5148\u961f\u5217\uff0c\u5927\u9876\u5806\u76f8\u5f53\u4e8e\u5143\u7d20\u6309\u4ece\u5927\u5230\u5c0f\u7684\u987a\u5e8f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u6765\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u201c\u4f18\u5148\u961f\u5217\u201d\u548c\u201c\u5806\u201d\u770b\u4f5c\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u4e66\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u79f0\u4f5c\u201c\u5806\u201d\u3002
\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u8868 8-1 \uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002
\u8868 8-1 \u00a0 \u5806\u7684\u64cd\u4f5c\u6548\u7387
\u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) pop() \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) peek() \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) size() \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) isEmpty() \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\)\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002
\u7c7b\u4f3c\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e2a
PythonC++JavaC#GoSwiftJSTSDartRustCZig heap.pyflag\u6216\u4fee\u6539Comparator\u5b9e\u73b0\u201c\u5c0f\u9876\u5806\u201d\u4e0e\u201c\u5927\u9876\u5806\u201d\u4e4b\u95f4\u7684\u8f6c\u6362\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
heap.cpp# \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n
heap.java/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue<int, vector<int>, greater<int>> minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue<int, vector<int>, less<int>> maxHeap;\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector<int> input{1, 3, 2, 5, 4};\npriority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());\n
heap.cs/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue<Integer> minHeap = new PriorityQueue<>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4));\n
heap.go/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue<int, int> minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue<int, int> maxHeap = new(Comparer<int>.Create((x, y) => y - x));\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue(); // 5\npeek = maxHeap.Dequeue(); // 4\npeek = maxHeap.Dequeue(); // 3\npeek = maxHeap.Dequeue(); // 2\npeek = maxHeap.Dequeue(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<int, int>([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n
heap.swift// Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n // Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n // \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n *h = append(*h, x.(int))\n}\n\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n // \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\n last := (*h)[len(*h)-1]\n *h = (*h)[:len(*h)-1]\n return last\n}\n\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\n return len(*h)\n}\n\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n // \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\n return (*h)[i].(int) > (*h)[j].(int)\n}\n\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\n return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n /* \u521d\u59cb\u5316\u5806 */\n // \u521d\u59cb\u5316\u5927\u9876\u5806\n maxHeap := &intHeap{}\n heap.Init(maxHeap)\n /* \u5143\u7d20\u5165\u5806 */\n // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\n heap.Push(maxHeap, 1)\n heap.Push(maxHeap, 3)\n heap.Push(maxHeap, 2)\n heap.Push(maxHeap, 4)\n heap.Push(maxHeap, 5)\n\n /* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\n top := maxHeap.Top()\n fmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n\n /* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\n heap.Pop(maxHeap) // 5\n heap.Pop(maxHeap) // 4\n heap.Pop(maxHeap) // 3\n heap.Pop(maxHeap) // 2\n heap.Pop(maxHeap) // 1\n\n /* \u83b7\u53d6\u5806\u5927\u5c0f */\n size := len(*maxHeap)\n fmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n\n /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n isEmpty := len(*maxHeap) == 0\n fmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n
heap.js// Swift \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
heap.ts// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
heap.dart// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
heap.rs// Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
heap.cuse std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nlet mut min_heap = BinaryHeap::<Reverse<i32>>::new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nlet mut max_heap = BinaryHeap::new();\n\n/* \u5143\u7d20\u5165\u5806 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap(); // 5\nlet peek = max_heap.pop().unwrap(); // 4\nlet peek = max_heap.pop().unwrap(); // 3\nlet peek = max_heap.pop().unwrap(); // 2\nlet peek = max_heap.pop().unwrap(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = max_heap.len();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = max_heap.is_empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n
heap.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
"},{"location":"chapter_heap/heap/#812","title":"8.1.2 \u00a0 \u5806\u7684\u5b9e\u73b0","text":"\n\u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u5927\u9876\u5806\u3002\u82e5\u8981\u5c06\u5176\u8f6c\u6362\u4e3a\u5c0f\u9876\u5806\uff0c\u53ea\u9700\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\uff0c\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002
"},{"location":"chapter_heap/heap/#1","title":"1. \u00a0 \u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u8bb2\u8fc7\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u7531\u4e8e\u5806\u6b63\u662f\u4e00\u79cd\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u6b64\u6211\u4eec\u5c06\u91c7\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002
\u5f53\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u8282\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u8282\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\u3002\u8282\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002
\u5982\u56fe 8-2 \u6240\u793a\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \uff0c\u7236\u8282\u70b9\u7d22\u5f15\u4e3a \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u8868\u793a\u7a7a\u8282\u70b9\u6216\u8282\u70b9\u4e0d\u5b58\u5728\u3002
\u56fe 8-2 \u00a0 \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8
\u6211\u4eec\u53ef\u4ee5\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef left(self, i: int) -> int:\n \"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\n return 2 * i + 1\n\ndef right(self, i: int) -> int:\n \"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\n return 2 * i + 2\n\ndef parent(self, i: int) -> int:\n \"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\"\"\"\n return (i - 1) // 2 # \u5411\u4e0b\u6574\u9664\n
my_heap.java/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\n return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.cs/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\n return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.go/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint Left(int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint Right(int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint Parent(int i) {\n return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.swift/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\n return 2*i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\n return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n // \u5411\u4e0b\u6574\u9664\n return (i - 1) / 2\n}\n
my_heap.js/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc left(i: Int) -> Int {\n 2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc right(i: Int) -> Int {\n 2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc parent(i: Int) -> Int {\n (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.ts/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\n#left(i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\n#right(i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\n#parent(i) {\n return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.dart/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nleft(i: number): number {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nright(i: number): number {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nparent(i: number): number {\n return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.rs/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _left(int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _right(int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint _parent(int i) {\n return (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.c/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfn left(i: usize) -> usize {\n 2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfn right(i: usize) -> usize {\n 2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfn parent(i: usize) -> usize {\n (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n
my_heap.zig/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\n return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\n return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\n return (i - 1) / 2;\n}\n
"},{"location":"chapter_heap/heap/#2","title":"2. \u00a0 \u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"// \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\nfn left(i: usize) usize {\n return 2 * i + 1;\n}\n\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\nfn right(i: usize) usize {\n return 2 * i + 2;\n}\n\n// \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\nfn parent(i: usize) usize {\n // return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n return @divFloor(i - 1, 2);\n}\n\u5806\u9876\u5143\u7d20\u5373\u4e3a\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u4e5f\u5c31\u662f\u5217\u8868\u7684\u9996\u4e2a\u5143\u7d20\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\n return self.max_heap[0]\n
my_heap.java/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n return maxHeap[0];\n}\n
my_heap.cs/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n return maxHeap.get(0);\n}\n
my_heap.go/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint Peek() {\n return maxHeap[0];\n}\n
my_heap.swift/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\n return h.data[0]\n}\n
my_heap.js/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -> Int {\n maxHeap[0]\n}\n
my_heap.ts/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\n return this.#maxHeap[0];\n}\n
my_heap.dart/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\n return this.maxHeap[0];\n}\n
my_heap.rs/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n return _maxHeap[0];\n}\n
my_heap.c/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfn peek(&self) -> Option<i32> {\n self.max_heap.first().copied()\n}\n
my_heap.zig/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\n return maxHeap->data[0];\n}\n
"},{"location":"chapter_heap/heap/#3","title":"3. \u00a0 \u5143\u7d20\u5165\u5806","text":"// \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\n return self.max_heap.?.items[0];\n} \n\u7ed9\u5b9a\u5143\u7d20
val\uff0c\u6211\u4eec\u9996\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u4e4b\u540e\uff0c\u7531\u4e8e val \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u574f\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5806\u5316 heapify\u300d\u3002\u8003\u8651\u4ece\u5165\u5806\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5982\u56fe 8-3 \u6240\u793a\uff0c\u6211\u4eec\u6bd4\u8f83\u63d2\u5165\u8282\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u8282\u70b9\u66f4\u5927\uff0c\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\u3002\u7136\u540e\u7ee7\u7eed\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u76f4\u81f3\u8d8a\u8fc7\u6839\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002
<1><2><3><4><5><6><7><8><9>\u56fe 8-3 \u00a0 \u5143\u7d20\u5165\u5806\u6b65\u9aa4
\u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef push(self, val: int):\n \"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n # \u6dfb\u52a0\u8282\u70b9\n self.max_heap.append(val)\n # \u4ece\u5e95\u81f3\u9876\u5806\u5316\n self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\n while True:\n # \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n p = self.parent(i)\n # \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if p < 0 or self.max_heap[i] <= self.max_heap[p]:\n break\n # \u4ea4\u6362\u4e24\u8282\u70b9\n self.swap(i, p)\n # \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p\n
my_heap.java/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap.push_back(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || maxHeap[i] <= maxHeap[p])\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(maxHeap[i], maxHeap[p]);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.cs/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap.add(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || maxHeap.get(i) <= maxHeap.get(p))\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.go/* \u5143\u7d20\u5165\u5806 */\nvoid Push(int val) {\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap.Add(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n SiftUp(Size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid SiftUp(int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = Parent(i);\n // \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n if (p < 0 || maxHeap[i] <= maxHeap[p])\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n Swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.swift/* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n // \u6dfb\u52a0\u8282\u70b9\n h.data = append(h.data, val)\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n h.siftUp(len(h.data) - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\n for true {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n p := h.parent(i)\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if p < 0 || h.data[i].(int) <= h.data[p].(int) {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n h.swap(i, p)\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p\n }\n}\n
my_heap.js/* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap.append(val)\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(i: size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\n var i = i\n while true {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n let p = parent(i: i)\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if p < 0 || maxHeap[i] <= maxHeap[p] {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(i: i, j: p)\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p\n }\n}\n
my_heap.ts/* \u5143\u7d20\u5165\u5806 */\npush(val) {\n // \u6dfb\u52a0\u8282\u70b9\n this.#maxHeap.push(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n this.#siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n const p = this.#parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || this.#maxHeap[i] <= this.#maxHeap[p]) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n this.#swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.dart/* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n // \u6dfb\u52a0\u8282\u70b9\n this.maxHeap.push(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n this.siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n const p = this.parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || this.maxHeap[i] <= this.maxHeap[p]) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n this.swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.rs/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n // \u6dfb\u52a0\u8282\u70b9\n _maxHeap.add(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = _parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || _maxHeap[i] <= _maxHeap[p]) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n _swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.c/* \u5143\u7d20\u5165\u5806 */\nfn push(&mut self, val: i32) {\n // \u6dfb\u52a0\u8282\u70b9\n self.max_heap.push(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n self.sift_up(self.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfn sift_up(&mut self, mut i: usize) {\n loop {\n // \u8282\u70b9 i \u5df2\u7ecf\u662f\u5806\u9876\u8282\u70b9\u4e86\uff0c\u7ed3\u675f\u5806\u5316\n if i == 0 {\n break;\n }\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n let p = Self::parent(i);\n // \u5f53\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if self.max_heap[i] <= self.max_heap[p] {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n self.swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
my_heap.zig/* \u5143\u7d20\u5165\u5806 */\nvoid push(MaxHeap *maxHeap, int val) {\n // \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\n if (maxHeap->size == MAX_SIZE) {\n printf(\"heap is full!\");\n return;\n }\n // \u6dfb\u52a0\u8282\u70b9\n maxHeap->data[maxHeap->size] = val;\n maxHeap->size++;\n\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n siftUp(maxHeap, maxHeap->size - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n int p = parent(maxHeap, i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 || maxHeap->data[i] <= maxHeap->data[p]) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(maxHeap, i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n
"},{"location":"chapter_heap/heap/#4","title":"4. \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806","text":"// \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n // \u6dfb\u52a0\u8282\u70b9\n try self.max_heap.?.append(val);\n // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n try self.siftUp(self.size() - 1);\n} \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\n var i = i_;\n while (true) {\n // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n var p = parent(i);\n // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n if (p < 0 or self.max_heap.?.items[i] <= self.max_heap.?.items[p]) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n try self.swap(i, p);\n // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n i = p;\n }\n}\n\u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u76f4\u63a5\u4ece\u5217\u8868\u4e2d\u5220\u9664\u9996\u5143\u7d20\uff0c\u90a3\u4e48\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u7d22\u5f15\u90fd\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u8fd9\u5c06\u4f7f\u5f97\u540e\u7eed\u4f7f\u7528\u5806\u5316\u8fdb\u884c\u4fee\u590d\u53d8\u5f97\u56f0\u96be\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u53d8\u52a8\uff0c\u6211\u4eec\u91c7\u7528\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\u3002
- \u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff09\u3002
- \u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u7531\u4e8e\u5df2\u7ecf\u4ea4\u6362\uff0c\u56e0\u6b64\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\u3002
- \u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\u3002
\u5982\u56fe 8-4 \u6240\u793a\uff0c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u201c\u4ece\u5e95\u81f3\u9876\u5806\u5316\u201d\u76f8\u53cd\uff0c\u6211\u4eec\u5c06\u6839\u8282\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u503c\u8fdb\u884c\u6bd4\u8f83\uff0c\u5c06\u6700\u5927\u7684\u5b50\u8282\u70b9\u4e0e\u6839\u8282\u70b9\u4ea4\u6362\u3002\u7136\u540e\u5faa\u73af\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002
<1><2><3><4><5><6><7><8><9><10>\u56fe 8-4 \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806\u6b65\u9aa4
\u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py
my_heap.cppdef pop(self) -> int:\n \"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n # \u5224\u7a7a\u5904\u7406\n if self.is_empty():\n raise IndexError(\"\u5806\u4e3a\u7a7a\")\n # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n self.swap(0, self.size() - 1)\n # \u5220\u9664\u8282\u70b9\n val = self.max_heap.pop()\n # \u4ece\u9876\u81f3\u5e95\u5806\u5316\n self.sift_down(0)\n # \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val\n\ndef sift_down(self, i: int):\n \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n while True:\n # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n l, r, ma = self.left(i), self.right(i), i\n if l < self.size() and self.max_heap[l] > self.max_heap[ma]:\n ma = l\n if r < self.size() and self.max_heap[r] > self.max_heap[ma]:\n ma = r\n # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i:\n break\n # \u4ea4\u6362\u4e24\u8282\u70b9\n self.swap(i, ma)\n # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n
my_heap.java/* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n // \u5224\u7a7a\u5904\u7406\n if (isEmpty()) {\n throw out_of_range(\"\u5806\u4e3a\u7a7a\");\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(maxHeap[0], maxHeap[size() - 1]);\n // \u5220\u9664\u8282\u70b9\n maxHeap.pop_back();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(0);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = left(i), r = right(i), ma = i;\n if (l < size() && maxHeap[l] > maxHeap[ma])\n ma = l;\n if (r < size() && maxHeap[r] > maxHeap[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i)\n break;\n swap(maxHeap[i], maxHeap[ma]);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.cs/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n // \u5224\u7a7a\u5904\u7406\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(0, size() - 1);\n // \u5220\u9664\u8282\u70b9\n int val = maxHeap.remove(size() - 1);\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = left(i), r = right(i), ma = i;\n if (l < size() && maxHeap.get(l) > maxHeap.get(ma))\n ma = l;\n if (r < size() && maxHeap.get(r) > maxHeap.get(ma))\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i)\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.go/* \u5143\u7d20\u51fa\u5806 */\nint Pop() {\n // \u5224\u7a7a\u5904\u7406\n if (IsEmpty())\n throw new IndexOutOfRangeException();\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n Swap(0, Size() - 1);\n // \u5220\u9664\u8282\u70b9\n int val = maxHeap.Last();\n maxHeap.RemoveAt(Size() - 1);\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n SiftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = Left(i), r = Right(i), ma = i;\n if (l < Size() && maxHeap[l] > maxHeap[ma])\n ma = l;\n if (r < Size() && maxHeap[r] > maxHeap[ma])\n ma = r;\n // \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n if (ma == i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n Swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.swift/* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n // \u5224\u7a7a\u5904\u7406\n if h.isEmpty() {\n fmt.Println(\"error\")\n return nil\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n h.swap(0, h.size()-1)\n // \u5220\u9664\u8282\u70b9\n val := h.data[len(h.data)-1]\n h.data = h.data[:len(h.data)-1]\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n h.siftDown(0)\n\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\n for true {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n l, r, max := h.left(i), h.right(i), i\n if l < h.size() && h.data[l].(int) > h.data[max].(int) {\n max = l\n }\n if r < h.size() && h.data[r].(int) > h.data[max].(int) {\n max = r\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if max == i {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n h.swap(i, max)\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = max\n }\n}\n
my_heap.js/* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -> Int {\n // \u5224\u7a7a\u5904\u7406\n if isEmpty() {\n fatalError(\"\u5806\u4e3a\u7a7a\")\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(i: 0, j: size() - 1)\n // \u5220\u9664\u8282\u70b9\n let val = maxHeap.remove(at: size() - 1)\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(i: 0)\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\n var i = i\n while true {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = left(i: i)\n let r = right(i: i)\n var ma = i\n if l < size(), maxHeap[l] > maxHeap[ma] {\n ma = l\n }\n if r < size(), maxHeap[r] > maxHeap[ma] {\n ma = r\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(i: i, j: ma)\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n }\n}\n
my_heap.ts/* \u5143\u7d20\u51fa\u5806 */\npop() {\n // \u5224\u7a7a\u5904\u7406\n if (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n this.#swap(0, this.size() - 1);\n // \u5220\u9664\u8282\u70b9\n const val = this.#maxHeap.pop();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n this.#siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n const l = this.#left(i),\n r = this.#right(i);\n let ma = i;\n if (l < this.size() && this.#maxHeap[l] > this.#maxHeap[ma]) ma = l;\n if (r < this.size() && this.#maxHeap[r] > this.#maxHeap[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma === i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n this.#swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.dart/* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n // \u5224\u7a7a\u5904\u7406\n if (this.isEmpty()) throw new RangeError('Heap is empty.');\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n this.swap(0, this.size() - 1);\n // \u5220\u9664\u8282\u70b9\n const val = this.maxHeap.pop();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n this.siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n const l = this.left(i),\n r = this.right(i);\n let ma = i;\n if (l < this.size() && this.maxHeap[l] > this.maxHeap[ma]) ma = l;\n if (r < this.size() && this.maxHeap[r] > this.maxHeap[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma === i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n this.swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.rs/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n // \u5224\u7a7a\u5904\u7406\n if (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n _swap(0, size() - 1);\n // \u5220\u9664\u8282\u70b9\n int val = _maxHeap.removeLast();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = _left(i);\n int r = _right(i);\n int ma = i;\n if (l < size() && _maxHeap[l] > _maxHeap[ma]) ma = l;\n if (r < size() && _maxHeap[r] > _maxHeap[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n _swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.c/* \u5143\u7d20\u51fa\u5806 */\nfn pop(&mut self) -> i32 {\n // \u5224\u7a7a\u5904\u7406\n if self.is_empty() {\n panic!(\"index out of bounds\");\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n self.swap(0, self.size() - 1);\n // \u5220\u9664\u8282\u70b9\n let val = self.max_heap.remove(self.size() - 1);\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n self.sift_down(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(&mut self, mut i: usize) {\n loop {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n if l < self.size() && self.max_heap[l] > self.max_heap[ma] {\n ma = l;\n }\n if r < self.size() && self.max_heap[r] > self.max_heap[ma] {\n ma = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n self.swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n
my_heap.zig/* \u5143\u7d20\u51fa\u5806 */\nint pop(MaxHeap *maxHeap) {\n // \u5224\u7a7a\u5904\u7406\n if (isEmpty(maxHeap)) {\n printf(\"heap is empty!\");\n return INT_MAX;\n }\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(maxHeap, 0, size(maxHeap) - 1);\n // \u5220\u9664\u8282\u70b9\n int val = maxHeap->data[maxHeap->size - 1];\n maxHeap->size--;\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n siftDown(maxHeap, 0);\n\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n int l = left(maxHeap, i);\n int r = right(maxHeap, i);\n int max = i;\n if (l < size(maxHeap) && maxHeap->data[l] > maxHeap->data[max]) {\n max = l;\n }\n if (r < size(maxHeap) && maxHeap->data[r] > maxHeap->data[max]) {\n max = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (max == i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(maxHeap, i, max);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = max;\n }\n}\n
"},{"location":"chapter_heap/heap/#813","title":"8.1.3 \u00a0 \u5806\u5e38\u89c1\u5e94\u7528","text":"// \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n // \u5224\u65ad\u5904\u7406\n if (self.isEmpty()) unreachable;\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n try self.swap(0, self.size() - 1);\n // \u5220\u9664\u8282\u70b9\n var val = self.max_heap.?.pop();\n // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n try self.siftDown(0);\n // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n return val;\n} \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\n var i = i_;\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n var l = left(i);\n var r = right(i);\n var ma = i;\n if (l < self.size() and self.max_heap.?.items[l] > self.max_heap.?.items[ma]) ma = l;\n if (r < self.size() and self.max_heap.?.items[r] > self.max_heap.?.items[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n try self.swap(i, ma);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n- \u4f18\u5148\u961f\u5217\uff1a\u5806\u901a\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \uff0c\u800c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002
- \u5806\u6392\u5e8f\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5b83\u4eec\u5efa\u7acb\u4e00\u4e2a\u5806\uff0c\u7136\u540e\u4e0d\u65ad\u5730\u6267\u884c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u6570\u636e\u3002\u7136\u800c\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u4e00\u79cd\u66f4\u4f18\u96c5\u7684\u65b9\u5f0f\u5b9e\u73b0\u5806\u6392\u5e8f\uff0c\u8be6\u89c1\u201c\u5806\u6392\u5e8f\u201d\u7ae0\u8282\u3002
- \u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1a\u8fd9\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7b97\u6cd5\u95ee\u9898\uff0c\u540c\u65f6\u4e5f\u662f\u4e00\u79cd\u5178\u578b\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u62e9\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u9500\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002
- \u5806\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002
- \u4f18\u5148\u961f\u5217\u7684\u5b9a\u4e49\u662f\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u6765\u5b9e\u73b0\u3002
- \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\)\u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u548c\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002
- \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002
- \u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u7ef4\u62a4\u5806\u7684\u6027\u8d28\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u7528\u5230\u3002
- \u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
- Top-K \u662f\u4e00\u4e2a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u6570\u636e\u7ed3\u6784\u9ad8\u6548\u89e3\u51b3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002
\u6570\u636e\u7ed3\u6784\u7684\u201c\u5806\u201d\u4e0e\u5185\u5b58\u7ba1\u7406\u7684\u201c\u5806\u201d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\u5417\uff1f
\u4e24\u8005\u4e0d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u5806\u3002\u8ba1\u7b97\u673a\u7cfb\u7edf\u5185\u5b58\u4e2d\u7684\u5806\u662f\u52a8\u6001\u5185\u5b58\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5e8f\u5728\u8fd0\u884c\u65f6\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u5b58\u50a8\u6570\u636e\u3002\u7a0b\u5e8f\u53ef\u4ee5\u8bf7\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u5185\u5b58\uff0c\u7528\u4e8e\u5b58\u50a8\u5982\u5bf9\u8c61\u548c\u6570\u7ec4\u7b49\u590d\u6742\u7ed3\u6784\u3002\u5f53\u8fd9\u4e9b\u6570\u636e\u4e0d\u518d\u9700\u8981\u65f6\uff0c\u7a0b\u5e8f\u9700\u8981\u91ca\u653e\u8fd9\u4e9b\u5185\u5b58\uff0c\u4ee5\u9632\u6b62\u5185\u5b58\u6cc4\u6f0f\u3002\u76f8\u8f83\u4e8e\u6808\u5185\u5b58\uff0c\u5806\u5185\u5b58\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8c28\u614e\uff0c\u4f7f\u7528\u4e0d\u5f53\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5185\u5b58\u6cc4\u6f0f\u548c\u91ce\u6307\u9488\u7b49\u95ee\u9898\u3002
"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-K \u95ee\u9898","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u65e0\u5e8f\u6570\u7ec4
nums\uff0c\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u524d \\(k\\) \u5927\u7684\u5143\u7d20\u3002\u5bf9\u4e8e\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u5148\u4ecb\u7ecd\u4e24\u79cd\u601d\u8def\u6bd4\u8f83\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7ecd\u6548\u7387\u66f4\u9ad8\u7684\u5806\u89e3\u6cd5\u3002
"},{"location":"chapter_heap/top_k/#831","title":"8.3.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u904d\u5386\u9009\u62e9","text":"\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u56fe 8-6 \u6240\u793a\u7684 \\(k\\) \u8f6e\u904d\u5386\uff0c\u5206\u522b\u5728\u6bcf\u8f6e\u4e2d\u63d0\u53d6\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(\\dots\\)\u3001\\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\) \u3002
\u6b64\u65b9\u6cd5\u53ea\u9002\u7528\u4e8e \\(k \\ll n\\) \u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u5f53 \\(k\\) \u4e0e \\(n\\) \u6bd4\u8f83\u63a5\u8fd1\u65f6\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411\u4e8e \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u65f6\u3002
\u56fe 8-6 \u00a0 \u904d\u5386\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20
Tip
\u5f53 \\(k = n\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5b8c\u6574\u7684\u6709\u5e8f\u5e8f\u5217\uff0c\u6b64\u65f6\u7b49\u4ef7\u4e8e\u201c\u9009\u62e9\u6392\u5e8f\u201d\u7b97\u6cd5\u3002
"},{"location":"chapter_heap/top_k/#832","title":"8.3.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"\u5982\u56fe 8-7 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u5bf9\u6570\u7ec4
nums\u8fdb\u884c\u6392\u5e8f\uff0c\u518d\u8fd4\u56de\u6700\u53f3\u8fb9\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u663e\u7136\uff0c\u8be5\u65b9\u6cd5\u201c\u8d85\u989d\u201d\u5b8c\u6210\u4efb\u52a1\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u53ea\u9700\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002
\u56fe 8-7 \u00a0 \u6392\u5e8f\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20
"},{"location":"chapter_heap/top_k/#833","title":"8.3.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806","text":"\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u5806\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3 Top-K \u95ee\u9898\uff0c\u6d41\u7a0b\u5982\u56fe 8-8 \u6240\u793a\u3002
- \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\u6700\u5c0f\u3002
- \u5148\u5c06\u6570\u7ec4\u7684\u524d \\(k\\) \u4e2a\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u3002
- \u4ece\u7b2c \\(k + 1\\) \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\uff0c\u5e76\u5c06\u5f53\u524d\u5143\u7d20\u5165\u5806\u3002
- \u904d\u5386\u5b8c\u6210\u540e\uff0c\u5806\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002
\u56fe 8-8 \u00a0 \u57fa\u4e8e\u5806\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20
\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig top_k.py
top_k.cppdef top_k_heap(nums: list[int], k: int) -> list[int]:\n \"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\n # \u521d\u59cb\u5316\u5c0f\u9876\u5806\n heap = []\n # \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for i in range(k):\n heapq.heappush(heap, nums[i])\n # \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for i in range(k, len(nums)):\n # \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if nums[i] > heap[0]:\n heapq.heappop(heap)\n heapq.heappush(heap, nums[i])\n return heap\n
top_k.java/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue<int, vector<int>, greater<int>> topKHeap(vector<int> &nums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n priority_queue<int, vector<int>, greater<int>> heap;\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (int i = 0; i < k; i++) {\n heap.push(nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < nums.size(); i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > heap.top()) {\n heap.pop();\n heap.push(nums[i]);\n }\n }\n return heap;\n}\n
top_k.cs/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue<Integer> topKHeap(int[] nums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n Queue<Integer> heap = new PriorityQueue<Integer>();\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (int i = 0; i < k; i++) {\n heap.offer(nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < nums.length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > heap.peek()) {\n heap.poll();\n heap.offer(nums[i]);\n }\n }\n return heap;\n}\n
top_k.go/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nPriorityQueue<int, int> TopKHeap(int[] nums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n PriorityQueue<int, int> heap = new();\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (int i = 0; i < k; i++) {\n heap.Enqueue(nums[i], nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < nums.Length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > heap.Peek()) {\n heap.Dequeue();\n heap.Enqueue(nums[i], nums[i]);\n }\n }\n return heap;\n}\n
top_k.swift/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n h := &minHeap{}\n heap.Init(h)\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for i := 0; i < k; i++ {\n heap.Push(h, nums[i])\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for i := k; i < len(nums); i++ {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if nums[i] > h.Top().(int) {\n heap.Pop(h)\n heap.Push(h, nums[i])\n }\n }\n return h\n}\n
top_k.js/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -> [Int] {\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5e76\u5c06\u524d k \u4e2a\u5143\u7d20\u5efa\u5806\n var heap = Heap(nums.prefix(k))\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for i in stride(from: k, to: nums.count, by: 1) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if nums[i] > heap.min()! {\n _ = heap.removeMin()\n heap.insert(nums[i])\n }\n }\n return heap.unordered\n}\n
top_k.ts/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap, val) {\n // \u5143\u7d20\u53d6\u53cd\n maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return maxHeap.getMaxHeap().map((num) => -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n const maxHeap = new MaxHeap([]);\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (let i = 0; i < k; i++) {\n pushMinHeap(maxHeap, nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (let i = k; i < nums.length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > peekMinHeap(maxHeap)) {\n popMinHeap(maxHeap);\n pushMinHeap(maxHeap, nums[i]);\n }\n }\n // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n return getMinHeap(maxHeap);\n}\n
top_k.dart/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n // \u5143\u7d20\u53d6\u53cd\n maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n // \u5143\u7d20\u53d6\u53cd\n return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n // \u5143\u7d20\u53d6\u53cd\n return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n // \u5143\u7d20\u53d6\u53cd\n return maxHeap.getMaxHeap().map((num: number) => -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n const maxHeap = new MaxHeap([]);\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (let i = 0; i < k; i++) {\n pushMinHeap(maxHeap, nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (let i = k; i < nums.length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > peekMinHeap(maxHeap)) {\n popMinHeap(maxHeap);\n pushMinHeap(maxHeap, nums[i]);\n }\n }\n // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n return getMinHeap(maxHeap);\n}\n
top_k.rs/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nMinHeap topKHeap(List<int> nums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\uff0c\u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n MinHeap heap = MinHeap(nums.sublist(0, k));\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < nums.length; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > heap.peek()) {\n heap.pop();\n heap.push(nums[i]);\n }\n }\n return heap;\n}\n
top_k.c/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfn top_k_heap(nums: Vec<i32>, k: usize) -> BinaryHeap<Reverse<i32>> {\n // BinaryHeap \u662f\u5927\u9876\u5806\uff0c\u4f7f\u7528 Reverse \u5c06\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u5b9e\u73b0\u5c0f\u9876\u5806\n let mut heap = BinaryHeap::<Reverse<i32>>::new();\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for &num in nums.iter().take(k) {\n heap.push(Reverse(num));\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for &num in nums.iter().skip(k) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if num > heap.peek().unwrap().0 {\n heap.pop();\n heap.push(Reverse(num));\n }\n }\n heap\n}\n
top_k.zig/* \u5143\u7d20\u5165\u5806 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n // \u5143\u7d20\u53d6\u53cd\n push(maxHeap, -val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nint popMinHeap(MaxHeap *maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return -pop(maxHeap);\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n // \u5143\u7d20\u53d6\u53cd\n return -peek(maxHeap);\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n int *res = (int *)malloc(maxHeap->size * sizeof(int));\n for (int i = 0; i < maxHeap->size; i++) {\n res[i] = -maxHeap->data[i];\n }\n return res;\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n int *res = (int *)malloc(maxHeap->size * sizeof(int));\n for (int i = 0; i < maxHeap->size; i++) {\n res[i] = -maxHeap->data[i];\n }\n return res;\n}\n\n// \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\u7684\u51fd\u6570\nint *topKHeap(int *nums, int sizeNums, int k) {\n // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n int *empty = (int *)malloc(0);\n MaxHeap *maxHeap = newMaxHeap(empty, 0);\n // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n for (int i = 0; i < k; i++) {\n pushMinHeap(maxHeap, nums[i]);\n }\n // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n for (int i = k; i < sizeNums; i++) {\n // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n if (nums[i] > peekMinHeap(maxHeap)) {\n popMinHeap(maxHeap);\n pushMinHeap(maxHeap, nums[i]);\n }\n }\n int *res = getMinHeap(maxHeap);\n // \u91ca\u653e\u5185\u5b58\n delMaxHeap(maxHeap);\n return res;\n}\n[class]{}-[func]{topKHeap}\n\u603b\u5171\u6267\u884c\u4e86 \\(n\\) \u8f6e\u5165\u5806\u548c\u51fa\u5806\uff0c\u5806\u7684\u6700\u5927\u957f\u5ea6\u4e3a \\(k\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002\u8be5\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u5f53 \\(k\\) \u8f83\u5c0f\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \uff1b\u5f53 \\(k\\) \u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(O(n \\log n)\\) \u3002
\u53e6\u5916\uff0c\u8be5\u65b9\u6cd5\u9002\u7528\u4e8e\u52a8\u6001\u6570\u636e\u6d41\u7684\u4f7f\u7528\u573a\u666f\u3002\u5728\u4e0d\u65ad\u52a0\u5165\u6570\u636e\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6301\u7eed\u7ef4\u62a4\u5806\u5185\u7684\u5143\u7d20\uff0c\u4ece\u800c\u5b9e\u73b0\u6700\u5927 \\(k\\) \u4e2a\u5143\u7d20\u7684\u52a8\u6001\u66f4\u65b0\u3002
"},{"location":"chapter_introduction/","title":"\u7b2c 1 \u7ae0 \u00a0 \u521d\u8bc6\u7b97\u6cd5","text":"Abstract
\u4e00\u4f4d\u5c11\u5973\u7fe9\u7fe9\u8d77\u821e\uff0c\u4e0e\u6570\u636e\u4ea4\u7ec7\u5728\u4e00\u8d77\uff0c\u88d9\u6446\u4e0a\u98d8\u626c\u7740\u7b97\u6cd5\u7684\u65cb\u5f8b\u3002
\u5979\u9080\u8bf7\u4f60\u5171\u821e\uff0c\u8bf7\u7d27\u8ddf\u5979\u7684\u6b65\u4f10\uff0c\u8e0f\u5165\u5145\u6ee1\u903b\u8f91\u4e0e\u7f8e\u611f\u7684\u7b97\u6cd5\u4e16\u754c\u3002
"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728
- 1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48
- 1.3 \u00a0 \u5c0f\u7ed3
\u5f53\u6211\u4eec\u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\u65f6\uff0c\u5f88\u81ea\u7136\u5730\u4f1a\u60f3\u5230\u6570\u5b66\u3002\u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7b97\u6cd5\u5e76\u4e0d\u6d89\u53ca\u590d\u6742\u6570\u5b66\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8d56\u57fa\u672c\u903b\u8f91\uff0c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002
\u5728\u6b63\u5f0f\u63a2\u8ba8\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u4e2a\u6709\u8da3\u7684\u4e8b\u5b9e\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u5e76\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\u6211\u5c06\u4e3e\u51e0\u4e2a\u5177\u4f53\u7684\u4f8b\u5b50\u6765\u8bc1\u5b9e\u8fd9\u4e00\u70b9\u3002
\u4f8b\u4e00\uff1a\u67e5\u9605\u5b57\u5178\u3002\u5728\u5b57\u5178\u91cc\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u5bf9\u5e94\u4e00\u4e2a\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u67e5\u627e\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u4f1a\u6309\u7167\u56fe 1-1 \u6240\u793a\u7684\u65b9\u5f0f\u5b9e\u73b0\u3002
- \u7ffb\u5f00\u5b57\u5178\u7ea6\u4e00\u534a\u7684\u9875\u6570\uff0c\u67e5\u770b\u8be5\u9875\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff0c\u5047\u8bbe\u9996\u5b57\u6bcd\u4e3a \\(m\\) \u3002
- \u7531\u4e8e\u5728\u62fc\u97f3\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u4e8e \\(m\\) \u4e4b\u540e\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u7f29\u5c0f\u5230\u540e\u534a\u90e8\u5206\u3002
- \u4e0d\u65ad\u91cd\u590d\u6b65\u9aa4
1.\u548c \u6b65\u9aa42.\uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u4e3a\u6b62\u3002
\u56fe 1-1 \u00a0 \u67e5\u5b57\u5178\u6b65\u9aa4
\u67e5\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u5fc5\u5907\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u8457\u540d\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u7b97\u6cd5\u3002\u4ece\u6570\u636e\u7ed3\u6784\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b57\u5178\u89c6\u4e3a\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u201c\u6570\u7ec4\u201d\uff1b\u4ece\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u201c\u4e8c\u5206\u67e5\u627e\u201d\u3002
\u4f8b\u4e8c\uff1a\u6574\u7406\u6251\u514b\u3002\u6211\u4eec\u5728\u6253\u724c\u65f6\uff0c\u6bcf\u5c40\u90fd\u9700\u8981\u6574\u7406\u624b\u4e2d\u7684\u6251\u514b\u724c\uff0c\u4f7f\u5176\u4ece\u5c0f\u5230\u5927\u6392\u5217\uff0c\u5b9e\u73b0\u6d41\u7a0b\u5982\u56fe 1-2 \u6240\u793a\u3002
- \u5c06\u6251\u514b\u724c\u5212\u5206\u4e3a\u201c\u6709\u5e8f\u201d\u548c\u201c\u65e0\u5e8f\u201d\u4e24\u90e8\u5206\uff0c\u5e76\u5047\u8bbe\u521d\u59cb\u72b6\u6001\u4e0b\u6700\u5de6 1 \u5f20\u6251\u514b\u724c\u5df2\u7ecf\u6709\u5e8f\u3002
- \u5728\u65e0\u5e8f\u90e8\u5206\u62bd\u51fa\u4e00\u5f20\u6251\u514b\u724c\uff0c\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\uff1b\u5b8c\u6210\u540e\u6700\u5de6 2 \u5f20\u6251\u514b\u5df2\u7ecf\u6709\u5e8f\u3002
- \u4e0d\u65ad\u5faa\u73af\u6b65\u9aa4
2.\uff0c\u6bcf\u4e00\u8f6e\u5c06\u4e00\u5f20\u6251\u514b\u724c\u4ece\u65e0\u5e8f\u90e8\u5206\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\uff0c\u76f4\u81f3\u6240\u6709\u6251\u514b\u724c\u90fd\u6709\u5e8f\u3002
\u56fe 1-2 \u00a0 \u6251\u514b\u6392\u5e8f\u6b65\u9aa4
\u4e0a\u8ff0\u6574\u7406\u6251\u514b\u724c\u7684\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u63d2\u5165\u6392\u5e8f\u201d\u7b97\u6cd5\uff0c\u5b83\u5728\u5904\u7406\u5c0f\u578b\u6570\u636e\u96c6\u65f6\u975e\u5e38\u9ad8\u6548\u3002\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u7684\u6392\u5e8f\u5e93\u51fd\u6570\u4e2d\u90fd\u6709\u63d2\u5165\u6392\u5e8f\u7684\u8eab\u5f71\u3002
\u4f8b\u4e09\uff1a\u8d27\u5e01\u627e\u96f6\u3002\u5047\u8bbe\u6211\u4eec\u5728\u8d85\u5e02\u8d2d\u4e70\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7ed9\u4e86\u6536\u94f6\u5458 \\(100\\) \u5143\uff0c\u5219\u6536\u94f6\u5458\u9700\u8981\u627e\u6211\u4eec \\(31\\) \u5143\u3002\u4ed6\u4f1a\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u5982\u56fe 1-3 \u6240\u793a\u7684\u601d\u8003\u3002
- \u53ef\u9009\u9879\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8d27\u5e01\uff0c\u5305\u62ec \\(1\\) \u5143\u3001\\(5\\) \u5143\u3001\\(10\\) \u5143\u3001\\(20\\) \u5143\u3002
- \u4ece\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u4f59 \\(31 - 20 = 11\\) \u5143\u3002
- \u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u4f59 \\(11 - 10 = 1\\) \u5143\u3002
- \u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u4f59 \\(1 - 1 = 0\\) \u5143\u3002
- \u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u4e3a \\(20 + 10 + 1 = 31\\) \u5143\u3002
\u56fe 1-3 \u00a0 \u8d27\u5e01\u627e\u96f6\u8fc7\u7a0b
\u5728\u4ee5\u4e0a\u6b65\u9aa4\u4e2d\uff0c\u6211\u4eec\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\uff08\u5c3d\u53ef\u80fd\u7528\u5927\u9762\u989d\u7684\u8d27\u5e01\uff09\uff0c\u6700\u7ec8\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u8d2a\u5fc3\u201d\u7b97\u6cd5\u3002
\u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u4f7f\u5f97\u6211\u4eec\u80fd\u591f\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u540c\u65f6\u7f16\u5199\u4ee3\u7801\u8c03\u7528 CPU \u548c GPU \u6267\u884c\u7b97\u6cd5\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u8f6c\u79fb\u5230\u8ba1\u7b97\u673a\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u51b3\u5404\u79cd\u590d\u6742\u95ee\u9898\u3002
Tip
\u5982\u679c\u4f60\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u548c\u4e8c\u5206\u67e5\u627e\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u8bf7\u7ee7\u7eed\u5f80\u4e0b\u9605\u8bfb\uff0c\u672c\u4e66\u5c06\u5f15\u5bfc\u4f60\u8fc8\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002
"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 \u5c0f\u7ed3","text":"- \u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u65e0\u5904\u4e0d\u5728\uff0c\u5e76\u4e0d\u662f\u9065\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8bc6\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u51b3\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u95ee\u9898\u3002
- \u67e5\u9605\u5b57\u5178\u7684\u539f\u7406\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u4f53\u73b0\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002
- \u6574\u7406\u6251\u514b\u7684\u8fc7\u7a0b\u4e0e\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u975e\u5e38\u7c7b\u4f3c\u3002\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u9002\u5408\u6392\u5e8f\u5c0f\u578b\u6570\u636e\u96c6\u3002
- \u8d27\u5e01\u627e\u96f6\u7684\u6b65\u9aa4\u672c\u8d28\u4e0a\u662f\u8d2a\u5fc3\u7b97\u6cd5\uff0c\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\u3002
- \u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u800c\u6570\u636e\u7ed3\u6784\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002
- \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7d27\u5bc6\u76f8\u8fde\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002
- \u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728\uff0c\u79ef\u6728\u4ee3\u8868\u6570\u636e\uff0c\u79ef\u6728\u7684\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u7b49\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6b65\u9aa4\u5219\u5bf9\u5e94\u7b97\u6cd5\u3002
\u300c\u7b97\u6cd5 algorithm\u300d\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002
- \u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002
- \u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u591f\u5728\u6709\u9650\u6b65\u9aa4\u3001\u65f6\u95f4\u548c\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002
- \u5404\u6b65\u9aa4\u90fd\u6709\u786e\u5b9a\u7684\u542b\u4e49\uff0c\u5728\u76f8\u540c\u7684\u8f93\u5165\u548c\u8fd0\u884c\u6761\u4ef6\u4e0b\uff0c\u8f93\u51fa\u59cb\u7ec8\u76f8\u540c\u3002
\u300c\u6570\u636e\u7ed3\u6784 data structure\u300d\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\uff0c\u5177\u6709\u4ee5\u4e0b\u8bbe\u8ba1\u76ee\u6807\u3002
- \u7a7a\u95f4\u5360\u7528\u5c3d\u91cf\u5c11\uff0c\u4ee5\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002
- \u6570\u636e\u64cd\u4f5c\u5c3d\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u76d6\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002
- \u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002
\u6570\u636e\u7ed3\u6784\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\u3002\u5982\u679c\u60f3\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u63d0\u5347\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u534f\u3002\u4e0b\u9762\u4e3e\u4e24\u4e2a\u4f8b\u5b50\u3002
- \u94fe\u8868\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5728\u6570\u636e\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u8bbf\u95ee\u901f\u5ea6\u3002
- \u56fe\u76f8\u8f83\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u4e30\u5bcc\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u5927\u7684\u5185\u5b58\u7a7a\u95f4\u3002
\u5982\u56fe 1-4 \u6240\u793a\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u9ad8\u5ea6\u76f8\u5173\u3001\u7d27\u5bc6\u7ed3\u5408\uff0c\u5177\u4f53\u8868\u73b0\u5728\u4ee5\u4e0b\u4e09\u4e2a\u65b9\u9762\u3002
- \u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u64cd\u4f5c\u6570\u636e\u7684\u65b9\u6cd5\u3002
- \u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u672c\u8eab\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u7ed3\u5408\u7b97\u6cd5\u624d\u80fd\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002
- \u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u4f46\u6267\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\uff0c\u9009\u62e9\u5408\u9002\u7684\u6570\u636e\u7ed3\u6784\u662f\u5173\u952e\u3002
\u56fe 1-4 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb
\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u72b9\u5982\u56fe 1-5 \u6240\u793a\u7684\u62fc\u88c5\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u5305\u542b\u8bb8\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u8fd8\u9644\u6709\u8be6\u7ec6\u7684\u7ec4\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7ec4\u88c5\u51fa\u7cbe\u7f8e\u7684\u79ef\u6728\u6a21\u578b\u3002
\u56fe 1-5 \u00a0 \u62fc\u88c5\u79ef\u6728
\u4e24\u8005\u7684\u8be6\u7ec6\u5bf9\u5e94\u5173\u7cfb\u5982\u8868 1-1 \u6240\u793a\u3002
\u8868 1-1 \u00a0 \u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728
\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 \u62fc\u88c5\u79ef\u6728 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u662f\u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u4e66\u5f97\u4ee5\u63d0\u4f9b\u57fa\u4e8e\u591a\u79cd\u7f16\u7a0b\u8bed\u8a00\u7684\u5b9e\u73b0\u3002
\u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0
\u5728\u5b9e\u9645\u8ba8\u8bba\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u201c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u201d\u7b80\u79f0\u4e3a\u201c\u7b97\u6cd5\u201d\u3002\u6bd4\u5982\u4f17\u6240\u5468\u77e5\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u67e5\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u65b9\u9762\u7684\u77e5\u8bc6\u3002
"},{"location":"chapter_preface/","title":"\u7b2c 0 \u7ae0 \u00a0 \u524d\u8a00","text":"Abstract
\u7b97\u6cd5\u72b9\u5982\u7f8e\u5999\u7684\u4ea4\u54cd\u4e50\uff0c\u6bcf\u4e00\u884c\u4ee3\u7801\u90fd\u50cf\u97f5\u5f8b\u822c\u6d41\u6dcc\u3002
\u613f\u8fd9\u672c\u4e66\u5728\u4f60\u7684\u8111\u6d77\u4e2d\u8f7b\u8f7b\u54cd\u8d77\uff0c\u7559\u4e0b\u72ec\u7279\u800c\u6df1\u523b\u7684\u65cb\u5f8b\u3002
"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 0.1 \u00a0 \u5173\u4e8e\u672c\u4e66
- 0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66
- 0.3 \u00a0 \u5c0f\u7ed3
\u672c\u9879\u76ee\u65e8\u5728\u521b\u5efa\u4e00\u672c\u5f00\u6e90\u3001\u514d\u8d39\u3001\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u6559\u7a0b\u3002
- \u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u3002
- \u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Python\u3001C++\u3001Java\u3001C#\u3001Go\u3001Swift\u3001JavaScript\u3001TypeScript\u3001Dart\u3001Rust\u3001C\u3001Zig \u7b49\u8bed\u8a00\u3002
- \u9f13\u52b1\u8bfb\u8005\u5728\u7ae0\u8282\u8ba8\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u901a\u5e38\u53ef\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\u3002
\u82e5\u4f60\u662f\u7b97\u6cd5\u521d\u5b66\u8005\uff0c\u4ece\u672a\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u4e00\u4e9b\u5237\u9898\u7ecf\u9a8c\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8ba4\u8bc6\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u672c\u4e66\u6b63\u662f\u4e3a\u4f60\u91cf\u8eab\u5b9a\u5236\u7684\uff01
\u5982\u679c\u4f60\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u5237\u9898\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u52a9\u4f60\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002
\u82e5\u4f60\u662f\u7b97\u6cd5\u5927\u795e\uff0c\u6211\u4eec\u671f\u5f85\u6536\u5230\u4f60\u7684\u5b9d\u8d35\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002
\u524d\u7f6e\u6761\u4ef6
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"},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2 \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"\u672c\u4e66\u7684\u4e3b\u8981\u5185\u5bb9\u5982\u56fe 0-1 \u6240\u793a\u3002
- \u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u4e0e\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002
- \u6570\u636e\u7ed3\u6784\uff1a\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u6570\u636e\u7ed3\u6784\u7684\u5206\u7c7b\u65b9\u6cd5\u3002\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002
- \u7b97\u6cd5\uff1a\u641c\u7d22\u3001\u6392\u5e8f\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u6548\u7387\u3001\u5e94\u7528\u573a\u666f\u3001\u89e3\u9898\u6b65\u9aa4\u548c\u793a\u4f8b\u95ee\u9898\u7b49\u3002
\u56fe 0-1 \u00a0 \u672c\u4e66\u4e3b\u8981\u5185\u5bb9
"},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3 \u00a0 \u81f4\u8c22","text":"\u5728\u672c\u4e66\u7684\u521b\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a
- \u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u4e2d\u4f60\u9f13\u52b1\u6211\u201c\u5feb\u884c\u52a8\u8d77\u6765\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\uff1b
- \u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u4e3a\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8bb8\u591a\u5b9d\u8d35\u5efa\u8bae\uff0c\u4f7f\u5f97\u672c\u4e66\u66f4\u9002\u5408\u65b0\u624b\u9605\u8bfb\uff1b
- \u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e00\u4e2a\u5bcc\u6709\u521b\u610f\u7684\u540d\u5b57\uff0c\u5524\u8d77\u5927\u5bb6\u5199\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801 \"Hello World!\" \u7684\u7f8e\u597d\u56de\u5fc6\uff1b
- \u611f\u8c22\u6821\u94e8\u5728\u77e5\u8bc6\u4ea7\u6743\u65b9\u9762\u63d0\u4f9b\u7684\u4e13\u4e1a\u5e2e\u52a9\uff0c\u8fd9\u5bf9\u672c\u5f00\u6e90\u4e66\u7684\u53d1\u5c55\u8d77\u5230\u4e86\u91cd\u8981\u7684\u4f5c\u7528\uff1b
- \u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c logo \uff0c\u5e76\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\uff1b
- \u611f\u8c22 @squidfunk \u63d0\u4f9b\u7684\u5199\u4f5c\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u4ed6\u5f00\u53d1\u7684\u5f00\u6e90\u6587\u6863\u4e3b\u9898 Material-for-MkDocs \u3002
\u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u5173\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u8fd9\u4e9b\u4f5c\u54c1\u4e3a\u672c\u4e66\u63d0\u4f9b\u4e86\u4f18\u79c0\u7684\u8303\u672c\uff0c\u786e\u4fdd\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u51c6\u786e\u6027\u4e0e\u54c1\u8d28\u3002\u5728\u6b64\u611f\u8c22\u6240\u6709\u8001\u5e08\u548c\u524d\u8f88\u4eec\u7684\u6770\u51fa\u8d21\u732e\uff01
\u672c\u4e66\u5021\u5bfc\u624b\u8111\u5e76\u7528\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u4e00\u70b9\u4e0a\u6df1\u53d7\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u7684\u542f\u53d1\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8bfb\u8005\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u4f18\u79c0\u7684\u8457\u4f5c\u3002
\u8877\u5fc3\u611f\u8c22\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u4eec\u4e00\u76f4\u4ee5\u6765\u7684\u652f\u6301\u4e0e\u9f13\u52b1\uff0c\u8ba9\u6211\u6709\u673a\u4f1a\u505a\u8fd9\u4ef6\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002
"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"Tip
\u4e3a\u4e86\u83b7\u5f97\u6700\u4f73\u7684\u9605\u8bfb\u4f53\u9a8c\uff0c\u5efa\u8bae\u4f60\u901a\u8bfb\u672c\u8282\u5185\u5bb9\u3002
"},{"location":"chapter_preface/suggestions/#021","title":"0.2.1 \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"- \u6807\u9898\u540e\u6807\u6ce8
*\u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u56f0\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002 - \u91cd\u8981\u4e13\u6709\u540d\u8bcd\u53ca\u5176\u82f1\u6587\u7ffb\u8bd1\u4f1a\u7528
\u300c \u300d\u62ec\u53f7\u6807\u6ce8\uff0c\u4f8b\u5982\u300c\u6570\u7ec4 array\u300d\u3002\u5efa\u8bae\u8bb0\u4f4f\u5b83\u4eec\uff0c\u4ee5\u4fbf\u9605\u8bfb\u6587\u732e\u3002 - \u4e13\u6709\u540d\u8bcd\u548c\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528
\u201c\u5f15\u53f7\u201d\u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002 - \u91cd\u8981\u540d\u8bcd\u3001\u91cd\u70b9\u5185\u5bb9\u548c\u603b\u7ed3\u6027\u8bed\u53e5\u4f1a\u88ab \u52a0\u7c97 \uff0c\u8fd9\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002
- \u5f53\u6d89\u53ca\u7f16\u7a0b\u8bed\u8a00\u4e4b\u95f4\u4e0d\u4e00\u81f4\u7684\u540d\u8bcd\u65f6\uff0c\u672c\u4e66\u5747\u4ee5 Python \u4e3a\u51c6\uff0c\u4f8b\u5982\u4f7f\u7528 \\(\\text{None}\\) \u6765\u8868\u793a\u201c\u7a7a\u201d\u3002
- \u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002
\"\"\"\u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\"\"\"\n\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
"},{"location":"chapter_preface/suggestions/#022","title":"0.2.2 \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"// \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n// \u591a\u884c\n// \u6ce8\u91ca\n\u76f8\u8f83\u4e8e\u6587\u5b57\uff0c\u89c6\u9891\u548c\u56fe\u7247\u5177\u6709\u66f4\u9ad8\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\uff0c\u66f4\u6613\u4e8e\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u5c06\u4e3b\u8981\u901a\u8fc7\u52a8\u753b\u548c\u56fe\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5219\u4f5c\u4e3a\u52a8\u753b\u548c\u56fe\u7247\u7684\u89e3\u91ca\u4e0e\u8865\u5145\u3002
\u5982\u679c\u4f60\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u5982\u56fe 0-2 \u6240\u793a\u7684\u52a8\u753b\u6216\u56fe\u89e3\uff0c\u8bf7\u4ee5\u56fe\u4e3a\u4e3b\u3001\u4ee5\u6587\u5b57\u4e3a\u8f85\uff0c\u7efc\u5408\u4e24\u8005\u6765\u7406\u89e3\u5185\u5bb9\u3002
\u56fe 0-2 \u00a0 \u52a8\u753b\u56fe\u89e3\u793a\u4f8b
"},{"location":"chapter_preface/suggestions/#023","title":"0.2.3 \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"\u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728 GitHub \u4ed3\u5e93\u3002\u5982\u56fe 0-3 \u6240\u793a\uff0c\u6e90\u4ee3\u7801\u9644\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u4e00\u952e\u8fd0\u884c\u3002
\u5982\u679c\u65f6\u95f4\u5141\u8bb8\uff0c\u5efa\u8bae\u4f60\u53c2\u7167\u4ee3\u7801\u81ea\u884c\u6572\u4e00\u904d\u3002\u5982\u679c\u5b66\u4e60\u65f6\u95f4\u6709\u9650\uff0c\u8bf7\u81f3\u5c11\u901a\u8bfb\u5e76\u8fd0\u884c\u6240\u6709\u4ee3\u7801\u3002
\u4e0e\u9605\u8bfb\u4ee3\u7801\u76f8\u6bd4\uff0c\u7f16\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u66f4\u591a\u6536\u83b7\u3002\u52a8\u624b\u5b66\uff0c\u624d\u662f\u771f\u7684\u5b66\u3002
\u56fe 0-3 \u00a0 \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b
\u8fd0\u884c\u4ee3\u7801\u7684\u524d\u7f6e\u5de5\u4f5c\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\u3002
\u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u8bf7\u53c2\u7167\u9644\u5f55\u6240\u793a\u7684\u6559\u7a0b\u8fdb\u884c\u5b89\u88c5\uff0c\u5982\u679c\u5df2\u5b89\u88c5\uff0c\u5219\u53ef\u8df3\u8fc7\u6b64\u6b65\u9aa4\u3002
\u7b2c\u4e8c\u6b65\uff1a\u514b\u9686\u6216\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u5e93\u3002\u524d\u5f80 GitHub \u4ed3\u5e93\u3002
\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u4ed3\u5e93\uff1a
git clone https://github.com/krahets/hello-algo.git\n\u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u5728\u56fe 0-4 \u6240\u793a\u7684\u4f4d\u7f6e\uff0c\u70b9\u51fb\u201cDownload ZIP\u201d\u6309\u94ae\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u7136\u540e\u5728\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002
\u56fe 0-4 \u00a0 \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801
\u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u5982\u56fe 0-5 \u6240\u793a\uff0c\u5bf9\u4e8e\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\u7684\u4ee3\u7801\u5757\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u4ed3\u5e93\u7684
codes\u6587\u4ef6\u5939\u5185\u627e\u5230\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u5c06\u5e2e\u52a9\u4f60\u8282\u7701\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u8ba9\u4f60\u80fd\u591f\u4e13\u6ce8\u4e8e\u5b66\u4e60\u5185\u5bb9\u3002\u56fe 0-5 \u00a0 \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6
"},{"location":"chapter_preface/suggestions/#024","title":"0.2.4 \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u8f7b\u6613\u8df3\u8fc7\u90a3\u4e9b\u6ca1\u5b66\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u63d0\u51fa\u4f60\u7684\u95ee\u9898\uff0c\u6211\u548c\u5c0f\u4f19\u4f34\u4eec\u5c06\u7aed\u8bda\u4e3a\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\u53ef\u5728\u4e24\u5929\u5185\u56de\u590d\u3002
\u5982\u56fe 0-6 \u6240\u793a\uff0c\u6bcf\u4e2a\u7ae0\u8282\u7684\u5e95\u90e8\u90fd\u914d\u6709\u8bc4\u8bba\u533a\u3002\u5e0c\u671b\u4f60\u80fd\u591a\u5173\u6ce8\u8bc4\u8bba\u533a\u7684\u5185\u5bb9\u3002\u4e00\u65b9\u9762\uff0c\u4f60\u53ef\u4ee5\u4e86\u89e3\u5927\u5bb6\u9047\u5230\u7684\u95ee\u9898\uff0c\u4ece\u800c\u67e5\u6f0f\u8865\u7f3a\uff0c\u6fc0\u53d1\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u671f\u5f85\u4f60\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u4f19\u4f34\u7684\u95ee\u9898\uff0c\u5206\u4eab\u4f60\u7684\u89c1\u89e3\uff0c\u5e2e\u52a9\u4ed6\u4eba\u8fdb\u6b65\u3002
\u56fe 0-6 \u00a0 \u8bc4\u8bba\u533a\u793a\u4f8b
"},{"location":"chapter_preface/suggestions/#025","title":"0.2.5 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"\u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5212\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\u3002
- \u9636\u6bb5\u4e00\uff1a\u7b97\u6cd5\u5165\u95e8\u3002\u6211\u4eec\u9700\u8981\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u548c\u7528\u6cd5\uff0c\u5b66\u4e60\u4e0d\u540c\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u7684\u5185\u5bb9\u3002
- \u9636\u6bb5\u4e8c\uff1a\u5237\u7b97\u6cd5\u9898\u3002\u5efa\u8bae\u4ece\u70ed\u95e8\u9898\u76ee\u5f00\u5237\uff0c\u5982\u201c\u5251\u6307 Offer\u201d\u548c\u201cLeetCode Hot 100\u201d\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521d\u6b21\u5237\u9898\u65f6\uff0c\u201c\u77e5\u8bc6\u9057\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u4e2a\u6311\u6218\uff0c\u4f46\u8bf7\u653e\u5fc3\uff0c\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u201c\u827e\u5bbe\u6d69\u65af\u9057\u5fd8\u66f2\u7ebf\u201d\u6765\u590d\u4e60\u9898\u76ee\uff0c\u901a\u5e38\u5728\u8fdb\u884c 3\uff5e5 \u8f6e\u7684\u91cd\u590d\u540e\uff0c\u5c31\u80fd\u5c06\u5176\u7262\u8bb0\u5728\u5fc3\u3002
- \u9636\u6bb5\u4e09\uff1a\u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u548c\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65ad\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u91c7\u7528\u8fdb\u9636\u5237\u9898\u7b56\u7565\uff0c\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u7684\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u627e\u5230\u3002
\u5982\u56fe 0-7 \u6240\u793a\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u6db5\u76d6\u201c\u9636\u6bb5\u4e00\u201d\uff0c\u65e8\u5728\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u5f00\u9636\u6bb5\u4e8c\u548c\u9636\u6bb5\u4e09\u7684\u5b66\u4e60\u3002
\u56fe 0-7 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf
"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 \u5c0f\u7ed3","text":"- \u672c\u4e66\u7684\u4e3b\u8981\u53d7\u4f17\u662f\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5982\u679c\u5df2\u6709\u4e00\u5b9a\u57fa\u7840\uff0c\u672c\u4e66\u80fd\u5e2e\u52a9\u4f60\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u4e66\u4e2d\u6e90\u4ee3\u7801\u4e5f\u53ef\u4f5c\u4e3a\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u4f7f\u7528\u3002
- \u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5305\u62ec\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002
- \u5bf9\u4e8e\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u81f3\u5173\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002
- \u4e66\u4e2d\u7684\u52a8\u753b\u548c\u56fe\u89e3\u901a\u5e38\u7528\u4e8e\u4ecb\u7ecd\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u3002\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5e94\u7ed9\u4e88\u8fd9\u4e9b\u5185\u5bb9\u66f4\u591a\u5173\u6ce8\u3002
- \u5b9e\u8df5\u4e43\u5b66\u4e60\u7f16\u7a0b\u4e4b\u6700\u4f73\u9014\u5f84\u3002\u5f3a\u70c8\u5efa\u8bae\u8fd0\u884c\u6e90\u4ee3\u7801\u5e76\u4eb2\u81ea\u6572\u4ee3\u7801\u3002
- \u672c\u4e66\u7f51\u9875\u7248\u7684\u6bcf\u4e2a\u7ae0\u8282\u90fd\u8bbe\u6709\u8ba8\u8bba\u533a\uff0c\u6b22\u8fce\u968f\u65f6\u5206\u4eab\u4f60\u7684\u7591\u60d1\u4e0e\u89c1\u89e3\u3002
[1] Thomas H. Cormen, et al. Introduction to Algorithms (3rd Edition).
[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1st Edition).
[3] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08C \u8bed\u8a00\u7248\uff09.
[4] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.
[5] \u9a6c\u514b \u827e\u4f26 \u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.
[6] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.
[7] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.
[8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6th Edition).
[9] Aston Zhang, et al. Dive into Deep Learning.
"},{"location":"chapter_searching/","title":"\u7b2c 10 \u7ae0 \u00a0 \u641c\u7d22","text":"Abstract
\u641c\u7d22\u662f\u4e00\u573a\u672a\u77e5\u7684\u5192\u9669\uff0c\u6211\u4eec\u6216\u8bb8\u9700\u8981\u8d70\u904d\u795e\u79d8\u7a7a\u95f4\u7684\u6bcf\u4e2a\u89d2\u843d\uff0c\u53c8\u6216\u8bb8\u53ef\u4ee5\u5feb\u901f\u9501\u5b9a\u76ee\u6807\u3002
\u5728\u8fd9\u573a\u5bfb\u89c5\u4e4b\u65c5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u63a2\u7d22\u90fd\u53ef\u80fd\u5f97\u5230\u4e00\u4e2a\u672a\u66fe\u6599\u60f3\u7684\u7b54\u6848\u3002
"},{"location":"chapter_searching/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 10.1 \u00a0 \u4e8c\u5206\u67e5\u627e
- 10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9
- 10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c
- 10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565
- 10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5
- 10.6 \u00a0 \u5c0f\u7ed3
\u300c\u4e8c\u5206\u67e5\u627e binary search\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u9ad8\u6548\u641c\u7d22\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u8303\u56f4\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u4e3a\u6b62\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4
nums\uff0c\u5143\u7d20\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u4e14\u4e0d\u91cd\u590d\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20target\u5728\u8be5\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 10-1 \u6240\u793a\u3002\u56fe 10-1 \u00a0 \u4e8c\u5206\u67e5\u627e\u793a\u4f8b\u6570\u636e
\u5982\u56fe 10-2 \u6240\u793a\uff0c\u6211\u4eec\u5148\u521d\u59cb\u5316\u6307\u9488 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u7d22\u533a\u95f4 \\([0, n - 1]\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u4e2d\u62ec\u53f7\u8868\u793a\u95ed\u533a\u95f4\uff0c\u5176\u5305\u542b\u8fb9\u754c\u503c\u672c\u8eab\u3002
\u63a5\u4e0b\u6765\uff0c\u5faa\u73af\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002
- \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\: \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002
- \u5224\u65ad
nums[m]\u548ctarget\u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e09\u79cd\u60c5\u51b5\u3002- \u5f53
nums[m] < target\u65f6\uff0c\u8bf4\u660etarget\u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(i = m + 1\\) \u3002 - \u5f53
nums[m] > target\u65f6\uff0c\u8bf4\u660etarget\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(j = m - 1\\) \u3002 - \u5f53
nums[m] = target\u65f6\uff0c\u8bf4\u660e\u627e\u5230target\uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \u3002
- \u5f53
\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20\uff0c\u641c\u7d22\u533a\u95f4\u6700\u7ec8\u4f1a\u7f29\u5c0f\u4e3a\u7a7a\u3002\u6b64\u65f6\u8fd4\u56de \\(-1\\) \u3002
<1><2><3><4><5><6><7>\u56fe 10-2 \u00a0 \u4e8c\u5206\u67e5\u627e\u6d41\u7a0b
\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e \\(i\\) \u548c \\(j\\) \u90fd\u662f
int\u7c7b\u578b\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u4f1a\u8d85\u51faint\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u4e3a\u4e86\u907f\u514d\u5927\u6570\u8d8a\u754c\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u6765\u8ba1\u7b97\u4e2d\u70b9\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search.py
binary_search.cppdef binary_search(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\"\"\"\n # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n i, j = 0, len(nums) - 1\n # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while i <= j:\n # \u7406\u8bba\u4e0a Python \u7684\u6570\u5b57\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\uff0c\u65e0\u987b\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\n m = (i + j) // 2 # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target:\n i = m + 1 # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n elif nums[m] > target:\n j = m - 1 # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n else:\n return m # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return -1 # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
binary_search.java/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector<int> &nums, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = nums.size() - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.cs/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = nums.length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.go/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint BinarySearch(int[] nums, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = nums.Length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.swift/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n i, j := 0, len(nums)-1\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n for i <= j {\n m := i + (j-i)/2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1\n } else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1\n}\n
binary_search.js/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -> Int {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n var i = 0\n var j = nums.count - 1\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while i <= j {\n let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1\n } else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1\n}\n
binary_search.ts/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n let i = 0,\n j = nums.length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n const m = parseInt(i + (j - i) / 2);\n if (nums[m] < target)\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target)\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else return m; // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.dart/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n let i = 0,\n j = nums.length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n const m = Math.floor(i + (j - i) / 2);\n if (nums[m] < target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n } else if (nums[m] > target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n }\n return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
binary_search.rs/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(List<int> nums, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = nums.length - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n } else if (nums[m] > target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.c/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfn binary_search(nums: &[i32], target: i32) -> i32 {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n let mut i = 0;\n let mut j = nums.len() as i32 - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while i <= j {\n let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m as usize] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n } else if nums[m as usize] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n } \n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.zig/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int *nums, int len, int target) {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n int i = 0, j = len - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n// \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n var i: usize = 0;\n var j: usize = nums.items.len - 1;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n var m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums.items[m] < target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1;\n } else if (nums.items[m] > target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1;\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return @intCast(m);\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(\\log n)\\) \uff1a\u5728\u4e8c\u5206\u5faa\u73af\u4e2d\uff0c\u533a\u95f4\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\uff0c\u5faa\u73af\u6b21\u6570\u4e3a \\(\\log_2 n\\) \u3002
\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002
"},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1 \u00a0 \u533a\u95f4\u8868\u793a\u65b9\u6cd5","text":"\u9664\u4e86\u4e0a\u8ff0\u53cc\u95ed\u533a\u95f4\u5916\uff0c\u5e38\u89c1\u7684\u533a\u95f4\u8868\u793a\u8fd8\u6709\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u533a\u95f4\uff0c\u5b9a\u4e49\u4e3a \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8be5\u8868\u793a\u4e0b\uff0c\u533a\u95f4 \\([i, j)\\) \u5728 \\(i = j\\) \u65f6\u4e3a\u7a7a\u3002
\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u8be5\u8868\u793a\u5b9e\u73b0\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search.py
binary_search.cppdef binary_search_lcro(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\"\"\"\n # \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n i, j = 0, len(nums)\n # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while i < j:\n m = (i + j) // 2 # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target:\n i = m + 1 # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n elif nums[m] > target:\n j = m # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n else:\n return m # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return -1 # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
binary_search.java/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(vector<int> &nums, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = nums.size();\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.cs/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = nums.length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.go/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint BinarySearchLCRO(int[] nums, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = nums.Length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.swift/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n i, j := 0, len(nums)\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n for i < j {\n m := i + (j-i)/2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1\n } else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1\n}\n
binary_search.js/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -> Int {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n var i = 0\n var j = nums.count\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while i < j {\n let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1\n } else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1\n}\n
binary_search.ts/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums, target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n let i = 0,\n j = nums.length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n const m = parseInt(i + (j - i) / 2);\n if (nums[m] < target)\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target)\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n else return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.dart/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n let i = 0,\n j = nums.length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n const m = Math.floor(i + (j - i) / 2);\n if (nums[m] < target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n } else if (nums[m] > target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n }\n return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
binary_search.rs/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(List<int> nums, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = nums.length;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n } else if (nums[m] > target) {\n // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n } else {\n // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.c/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfn binary_search_lcro(nums: &[i32], target: i32) -> i32 {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n let mut i = 0;\n let mut j = nums.len() as i32;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while i < j {\n let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m as usize] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n } else if nums[m as usize] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m - 1;\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n } \n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n
binary_search.zig/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n int i = 0, j = len;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i < j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n else if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return m;\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n// \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n var i: usize = 0;\n var j: usize = nums.items.len;\n // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n while (i <= j) {\n var m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums.items[m] < target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n i = m + 1;\n } else if (nums.items[m] > target) { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n j = m;\n } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n return @intCast(m);\n }\n }\n // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n return -1;\n}\n\u5982\u56fe 10-3 \u6240\u793a\uff0c\u5728\u4e24\u79cd\u533a\u95f4\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u5faa\u73af\u6761\u4ef6\u548c\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002
\u7531\u4e8e\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e2d\u7684\u5de6\u53f3\u8fb9\u754c\u90fd\u88ab\u5b9a\u4e49\u4e3a\u95ed\u533a\u95f4\uff0c\u56e0\u6b64\u901a\u8fc7\u6307\u9488 \\(i\\) \u548c\u6307\u9488 \\(j\\) \u7f29\u5c0f\u533a\u95f4\u7684\u64cd\u4f5c\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\uff0c\u56e0\u6b64\u4e00\u822c\u5efa\u8bae\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002
\u56fe 10-3 \u00a0 \u4e24\u79cd\u533a\u95f4\u5b9a\u4e49
"},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"\u4e8c\u5206\u67e5\u627e\u5728\u65f6\u95f4\u548c\u7a7a\u95f4\u65b9\u9762\u90fd\u6709\u8f83\u597d\u7684\u6027\u80fd\u3002
- \u4e8c\u5206\u67e5\u627e\u7684\u65f6\u95f4\u6548\u7387\u9ad8\u3002\u5728\u5927\u6570\u636e\u91cf\u4e0b\uff0c\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002
- \u4e8c\u5206\u67e5\u627e\u65e0\u987b\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u8f83\u4e8e\u9700\u8981\u501f\u52a9\u989d\u5916\u7a7a\u95f4\u7684\u641c\u7d22\u7b97\u6cd5\uff08\u4f8b\u5982\u54c8\u5e0c\u67e5\u627e\uff09\uff0c\u4e8c\u5206\u67e5\u627e\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002
\u7136\u800c\uff0c\u4e8c\u5206\u67e5\u627e\u5e76\u975e\u9002\u7528\u4e8e\u6240\u6709\u60c5\u51b5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002
- \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u82e5\u8f93\u5165\u6570\u636e\u65e0\u5e8f\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u507f\u5931\u3002\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u9ad8\u3002\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4fdd\u6301\u6570\u7ec4\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002
- \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u4e8c\u5206\u67e5\u627e\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u6267\u884c\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u7684\u6548\u7387\u8f83\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5e94\u7528\u5728\u94fe\u8868\u6216\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
- \u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u6027\u80fd\u66f4\u4f73\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5f53\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4
nums\uff0c\u5176\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a\u5143\u7d20target\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u56de\u5fc6\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u7684\u65b9\u6cd5\uff0c\u641c\u7d22\u5b8c\u6210\u540e \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a
target\uff0c\u56e0\u6b64\u67e5\u627e\u63d2\u5165\u70b9\u672c\u8d28\u4e0a\u662f\u5728\u67e5\u627e\u6700\u5de6\u4e00\u4e2atarget\u7684\u7d22\u5f15\u3002\u8003\u8651\u901a\u8fc7\u67e5\u627e\u63d2\u5165\u70b9\u7684\u51fd\u6570\u5b9e\u73b0\u67e5\u627e\u5de6\u8fb9\u754c\u3002\u8bf7\u6ce8\u610f\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u4e0d\u5305\u542b
target\uff0c\u8fd9\u79cd\u60c5\u51b5\u53ef\u80fd\u5bfc\u81f4\u4ee5\u4e0b\u4e24\u79cd\u7ed3\u679c\u3002- \u63d2\u5165\u70b9\u7684\u7d22\u5f15 \\(i\\) \u8d8a\u754c\u3002
- \u5143\u7d20
nums[i]\u4e0etarget\u4e0d\u76f8\u7b49\u3002
\u5f53\u9047\u5230\u4ee5\u4e0a\u4e24\u79cd\u60c5\u51b5\u65f6\uff0c\u76f4\u63a5\u8fd4\u56de \\(-1\\) \u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_edge.py
binary_search_edge.cppdef binary_search_left_edge(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target\"\"\"\n # \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n i = binary_search_insertion(nums, target)\n # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if i == len(nums) or nums[i] != target:\n return -1\n # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i\n
binary_search_edge.java/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(vector<int> &nums, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == nums.size() || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.cs/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int[] nums, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binary_search_insertion.binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == nums.length || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.go/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint BinarySearchLeftEdge(int[] nums, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == nums.Length || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.swift/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n i := binarySearchInsertion(nums, target)\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if i == len(nums) || nums[i] != target {\n return -1\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i\n}\n
binary_search_edge.js/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -> Int {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n let i = binarySearchInsertion(nums: nums, target: target)\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if i == nums.count || nums[i] != target {\n return -1\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i\n}\n
binary_search_edge.ts/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums, target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n const i = binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i === nums.length || nums[i] !== target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.dart/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums: Array<number>, target: number): number {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n const i = binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i === nums.length || nums[i] !== target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.rs/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(List<int> nums, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binarySearchInsertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == nums.length || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
binary_search_edge.c/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfn binary_search_left_edge(nums: &[i32], target: i32) -> i32 {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n let i = binary_search_insertion(nums, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if i == nums.len() as i32 || nums[i as usize] != target {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n i\n}\n
binary_search_edge.zig/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n int i = binarySearchInsertion(nums, numSize, target);\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (i == numSize || nums[i] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n return i;\n}\n
"},{"location":"chapter_searching/binary_search_edge/#1032","title":"10.3.2 \u00a0 \u67e5\u627e\u53f3\u8fb9\u754c","text":"[class]{}-[func]{binarySearchLeftEdge}\n\u90a3\u4e48\u5982\u4f55\u67e5\u627e\u6700\u53f3\u4e00\u4e2a
target\u5462\uff1f\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\u662f\u4fee\u6539\u4ee3\u7801\uff0c\u66ff\u6362\u5728nums[m] == target\u60c5\u51b5\u4e0b\u7684\u6307\u9488\u6536\u7f29\u64cd\u4f5c\u3002\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002\u4e0b\u9762\u6211\u4eec\u4ecb\u7ecd\u4e24\u79cd\u66f4\u52a0\u53d6\u5de7\u7684\u65b9\u6cd5\u3002
"},{"location":"chapter_searching/binary_search_edge/#1","title":"1. \u00a0 \u590d\u7528\u67e5\u627e\u5de6\u8fb9\u754c","text":"\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u67e5\u627e\u6700\u5de6\u5143\u7d20\u7684\u51fd\u6570\u6765\u67e5\u627e\u6700\u53f3\u5143\u7d20\uff0c\u5177\u4f53\u65b9\u6cd5\u4e3a\uff1a\u5c06\u67e5\u627e\u6700\u53f3\u4e00\u4e2a
target\u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2atarget + 1\u3002\u5982\u56fe 10-7 \u6240\u793a\uff0c\u67e5\u627e\u5b8c\u6210\u540e\uff0c\u6307\u9488 \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a
target + 1\uff08\u5982\u679c\u5b58\u5728\uff09\uff0c\u800c \\(j\\) \u6307\u5411\u6700\u53f3\u4e00\u4e2atarget\uff0c\u56e0\u6b64\u8fd4\u56de \\(j\\) \u5373\u53ef\u3002\u56fe 10-7 \u00a0 \u5c06\u67e5\u627e\u53f3\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5de6\u8fb9\u754c
\u8bf7\u6ce8\u610f\uff0c\u8fd4\u56de\u7684\u63d2\u5165\u70b9\u662f \\(i\\) \uff0c\u56e0\u6b64\u9700\u8981\u5c06\u5176\u51cf \\(1\\) \uff0c\u4ece\u800c\u83b7\u5f97 \\(j\\) \uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_edge.py
binary_search_edge.cppdef binary_search_right_edge(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target\"\"\"\n # \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n i = binary_search_insertion(nums, target + 1)\n # j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n j = i - 1\n # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if j == -1 or nums[j] != target:\n return -1\n # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j\n
binary_search_edge.java/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(vector<int> &nums, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.cs/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int[] nums, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.go/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint BinarySearchRightEdge(int[] nums, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.swift/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums []int, target int) int {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n i := binarySearchInsertion(nums, target+1)\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n j := i - 1\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if j == -1 || nums[j] != target {\n return -1\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j\n}\n
binary_search_edge.js/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -> Int {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n let i = binarySearchInsertion(nums: nums, target: target + 1)\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n let j = i - 1\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if j == -1 || nums[j] != target {\n return -1\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j\n}\n
binary_search_edge.ts/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums, target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n const i = binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n const j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j === -1 || nums[j] !== target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.dart/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums: Array<number>, target: number): number {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n const i = binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n const j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j === -1 || nums[j] !== target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.rs/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(List<int> nums, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binarySearchInsertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
binary_search_edge.c/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfn binary_search_right_edge(nums: &[i32], target: i32) -> i32 {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n let i = binary_search_insertion(nums, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n let j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if j == -1 || nums[j as usize] != target {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n j\n}\n
binary_search_edge.zig/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n int i = binarySearchInsertion(nums, numSize, target + 1);\n // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n int j = i - 1;\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n if (j == -1 || nums[j] != target) {\n return -1;\n }\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n return j;\n}\n
"},{"location":"chapter_searching/binary_search_edge/#2","title":"2. \u00a0 \u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20","text":"[class]{}-[func]{binarySearchRightEdge}\n\u6211\u4eec\u77e5\u9053\uff0c\u5f53\u6570\u7ec4\u4e0d\u5305\u542b
target\u65f6\uff0c\u6700\u7ec8 \\(i\\) \u548c \\(j\\) \u4f1a\u5206\u522b\u6307\u5411\u9996\u4e2a\u5927\u4e8e\u3001\u5c0f\u4e8etarget\u7684\u5143\u7d20\u3002\u56e0\u6b64\uff0c\u5982\u56fe 10-8 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728\u7684\u5143\u7d20\uff0c\u7528\u4e8e\u67e5\u627e\u5de6\u53f3\u8fb9\u754c\u3002
- \u67e5\u627e\u6700\u5de6\u4e00\u4e2a
target\uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627etarget - 0.5\uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(i\\) \u3002 - \u67e5\u627e\u6700\u53f3\u4e00\u4e2a
target\uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627etarget + 0.5\uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(j\\) \u3002
\u56fe 10-8 \u00a0 \u5c06\u67e5\u627e\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20
\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u4ee5\u4e0b\u4e24\u70b9\u503c\u5f97\u6ce8\u610f\u3002
- \u7ed9\u5b9a\u6570\u7ec4\u4e0d\u5305\u542b\u5c0f\u6570\uff0c\u8fd9\u610f\u5473\u7740\u6211\u4eec\u65e0\u987b\u5173\u5fc3\u5982\u4f55\u5904\u7406\u76f8\u7b49\u7684\u60c5\u51b5\u3002
- \u56e0\u4e3a\u8be5\u65b9\u6cd5\u5f15\u5165\u4e86\u5c0f\u6570\uff0c\u6240\u4ee5\u9700\u8981\u5c06\u51fd\u6570\u4e2d\u7684\u53d8\u91cf
target\u6539\u4e3a\u6d6e\u70b9\u6570\u7c7b\u578b\u3002
\u4e8c\u5206\u67e5\u627e\u4e0d\u4ec5\u53ef\u7528\u4e8e\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff0c\u8fd8\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u53d8\u79cd\u95ee\u9898\uff0c\u6bd4\u5982\u641c\u7d22\u76ee\u6807\u5143\u7d20\u7684\u63d2\u5165\u4f4d\u7f6e\u3002
"},{"location":"chapter_searching/binary_search_insertion/#1021","title":"10.2.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"Question
\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4
nums\u548c\u4e00\u4e2a\u5143\u7d20target\uff0c\u6570\u7ec4\u4e0d\u5b58\u5728\u91cd\u590d\u5143\u7d20\u3002\u73b0\u5c06target\u63d2\u5165\u6570\u7ec4nums\u4e2d\uff0c\u5e76\u4fdd\u6301\u5176\u6709\u5e8f\u6027\u3002\u82e5\u6570\u7ec4\u4e2d\u5df2\u5b58\u5728\u5143\u7d20target\uff0c\u5219\u63d2\u5165\u5230\u5176\u5de6\u65b9\u3002\u8bf7\u8fd4\u56de\u63d2\u5165\u540etarget\u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u56fe 10-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u793a\u4f8b\u6570\u636e
\u5982\u679c\u60f3\u590d\u7528\u4e0a\u4e00\u8282\u7684\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\uff0c\u5219\u9700\u8981\u56de\u7b54\u4ee5\u4e0b\u4e24\u4e2a\u95ee\u9898\u3002
\u95ee\u9898\u4e00\uff1a\u5f53\u6570\u7ec4\u4e2d\u5305\u542b
target\u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u662f\u5426\u662f\u8be5\u5143\u7d20\u7684\u7d22\u5f15\uff1f\u9898\u76ee\u8981\u6c42\u5c06
target\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u5de6\u8fb9\uff0c\u8fd9\u610f\u5473\u7740\u65b0\u63d2\u5165\u7684target\u66ff\u6362\u4e86\u539f\u6765target\u7684\u4f4d\u7f6e\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u6570\u7ec4\u5305\u542btarget\u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u5c31\u662f\u8be5target\u7684\u7d22\u5f15\u3002\u95ee\u9898\u4e8c\uff1a\u5f53\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728
target\u65f6\uff0c\u63d2\u5165\u70b9\u662f\u54ea\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1f\u8fdb\u4e00\u6b65\u601d\u8003\u4e8c\u5206\u67e5\u627e\u8fc7\u7a0b\uff1a\u5f53
nums[m] < target\u65f6 \\(i\\) \u79fb\u52a8\uff0c\u8fd9\u610f\u5473\u7740\u6307\u9488 \\(i\\) \u5728\u5411\u5927\u4e8e\u7b49\u4e8etarget\u7684\u5143\u7d20\u9760\u8fd1\u3002\u540c\u7406\uff0c\u6307\u9488 \\(j\\) \u59cb\u7ec8\u5728\u5411\u5c0f\u4e8e\u7b49\u4e8etarget\u7684\u5143\u7d20\u9760\u8fd1\u3002\u56e0\u6b64\u4e8c\u5206\u7ed3\u675f\u65f6\u4e00\u5b9a\u6709\uff1a\\(i\\) \u6307\u5411\u9996\u4e2a\u5927\u4e8e
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_insertion.pytarget\u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8etarget\u7684\u5143\u7d20\u3002\u6613\u5f97\u5f53\u6570\u7ec4\u4e0d\u5305\u542btarget\u65f6\uff0c\u63d2\u5165\u7d22\u5f15\u4e3a \\(i\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
binary_search_insertion.cppdef binary_search_insertion_simple(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n i, j = 0, len(nums) - 1 # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j:\n m = (i + j) // 2 # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target:\n i = m + 1 # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n elif nums[m] > target:\n j = m - 1 # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n else:\n return m # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n
binary_search_insertion.java/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(vector<int> &nums, int target) {\n int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.cs/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int[] nums, int target) {\n int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.go/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.swift/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n i, j := 0, len(nums)-1\n for i <= j {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n m := i + (j-i)/2\n if nums[m] < target {\n // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1\n } else if nums[m] > target {\n // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n } else {\n // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n return m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n}\n
binary_search_insertion.js/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -> Int {\n var i = 0, j = nums.count - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j {\n let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target {\n i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if nums[m] > target {\n j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n}\n
binary_search_insertion.ts/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(nums, target) {\n let i = 0,\n j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.dart/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(\n nums: Array<number>,\n target: number\n): number {\n let i = 0,\n j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.rs/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(List<int> nums, int target) {\n int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.c/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &[i32], target: i32) -> i32 {\n let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j {\n let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m as usize] < target {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if nums[m as usize] > target {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n i\n}\n
binary_search_insertion.zig/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n }\n }\n // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
"},{"location":"chapter_searching/binary_search_insertion/#1022","title":"10.2.2 \u00a0 \u5b58\u5728\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"[class]{}-[func]{binarySearchInsertionSimple}\nQuestion
\u5728\u4e0a\u4e00\u9898\u7684\u57fa\u7840\u4e0a\uff0c\u89c4\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u5176\u4f59\u4e0d\u53d8\u3002
\u5047\u8bbe\u6570\u7ec4\u4e2d\u5b58\u5728\u591a\u4e2a
target\uff0c\u5219\u666e\u901a\u4e8c\u5206\u67e5\u627e\u53ea\u80fd\u8fd4\u56de\u5176\u4e2d\u4e00\u4e2atarget\u7684\u7d22\u5f15\uff0c\u800c\u65e0\u6cd5\u786e\u5b9a\u8be5\u5143\u7d20\u7684\u5de6\u8fb9\u548c\u53f3\u8fb9\u8fd8\u6709\u591a\u5c11target\u3002\u9898\u76ee\u8981\u6c42\u5c06\u76ee\u6807\u5143\u7d20\u63d2\u5165\u5230\u6700\u5de6\u8fb9\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a
target\u7684\u7d22\u5f15\u3002\u521d\u6b65\u8003\u8651\u901a\u8fc7\u56fe 10-5 \u6240\u793a\u7684\u6b65\u9aa4\u5b9e\u73b0\u3002- \u6267\u884c\u4e8c\u5206\u67e5\u627e\uff0c\u5f97\u5230\u4efb\u610f\u4e00\u4e2a
target\u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\(k\\) \u3002 - \u4ece\u7d22\u5f15 \\(k\\) \u5f00\u59cb\uff0c\u5411\u5de6\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u5f53\u627e\u5230\u6700\u5de6\u8fb9\u7684
target\u65f6\u8fd4\u56de\u3002
\u56fe 10-5 \u00a0 \u7ebf\u6027\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9
\u6b64\u65b9\u6cd5\u867d\u7136\u53ef\u7528\uff0c\u4f46\u5176\u5305\u542b\u7ebf\u6027\u67e5\u627e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5f53\u6570\u7ec4\u4e2d\u5b58\u5728\u5f88\u591a\u91cd\u590d\u7684
target\u65f6\uff0c\u8be5\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\u3002\u73b0\u8003\u8651\u62d3\u5c55\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\u3002\u5982\u56fe 10-6 \u6240\u793a\uff0c\u6574\u4f53\u6d41\u7a0b\u4fdd\u6301\u4e0d\u53d8\uff0c\u6bcf\u8f6e\u5148\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65ad
target\u548cnums[m]\u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u51e0\u79cd\u60c5\u51b5\u3002- \u5f53
nums[m] < target\u6216nums[m] > target\u65f6\uff0c\u8bf4\u660e\u8fd8\u6ca1\u6709\u627e\u5230target\uff0c\u56e0\u6b64\u91c7\u7528\u666e\u901a\u4e8c\u5206\u67e5\u627e\u7684\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5411target\u9760\u8fd1\u3002 - \u5f53
nums[m] == target\u65f6\uff0c\u8bf4\u660e\u5c0f\u4e8etarget\u7684\u5143\u7d20\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u91c7\u7528 \\(j = m - 1\\) \u6765\u7f29\u5c0f\u533a\u95f4\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(j\\) \u5411\u5c0f\u4e8etarget\u7684\u5143\u7d20\u9760\u8fd1\u3002
\u5faa\u73af\u5b8c\u6210\u540e\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u8fb9\u7684
<1><2><3><4><5><6><7><8>target\uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8etarget\u7684\u5143\u7d20\uff0c\u56e0\u6b64\u7d22\u5f15 \\(i\\) \u5c31\u662f\u63d2\u5165\u70b9\u3002\u56fe 10-6 \u00a0 \u4e8c\u5206\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9\u7684\u6b65\u9aa4
\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5224\u65ad\u5206\u652f
nums[m] > target\u548cnums[m] == target\u7684\u64cd\u4f5c\u76f8\u540c\uff0c\u56e0\u6b64\u4e24\u8005\u53ef\u4ee5\u5408\u5e76\u3002\u5373\u4fbf\u5982\u6b64\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5c06\u5224\u65ad\u6761\u4ef6\u4fdd\u6301\u5c55\u5f00\uff0c\u56e0\u4e3a\u5176\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3001\u53ef\u8bfb\u6027\u66f4\u597d\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_insertion.py
binary_search_insertion.cppdef binary_search_insertion(nums: list[int], target: int) -> int:\n \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n i, j = 0, len(nums) - 1 # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j:\n m = (i + j) // 2 # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target:\n i = m + 1 # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n elif nums[m] > target:\n j = m - 1 # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n else:\n j = m - 1 # \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n # \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n
binary_search_insertion.java/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(vector<int> &nums, int target) {\n int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.cs/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int[] nums, int target) {\n int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.go/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertion(int[] nums, int target) {\n int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.swift/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums []int, target int) int {\n // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n i, j := 0, len(nums)-1\n for i <= j {\n // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n m := i + (j-i)/2\n if nums[m] < target {\n // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n i = m + 1\n } else if nums[m] > target {\n // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n } else {\n // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n j = m - 1\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n}\n
binary_search_insertion.js/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums: [Int], target: Int) -> Int {\n var i = 0, j = nums.count - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j {\n let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m] < target {\n i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if nums[m] > target {\n j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i\n}\n
binary_search_insertion.ts/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums, target) {\n let i = 0,\n j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.dart/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums: Array<number>, target: number): number {\n let i = 0,\n j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.rs/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(List<int> nums, int target) {\n int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n
binary_search_insertion.c/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &[i32], target: i32) -> i32 {\n let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while i <= j {\n let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if nums[m as usize] < target {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if nums[m as usize] > target {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n i\n}\n
binary_search_insertion.zig/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n while (i <= j) {\n int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n if (nums[m] < target) {\n i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n } else if (nums[m] > target) {\n j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n } else {\n j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n }\n }\n // \u8fd4\u56de\u63d2\u5165\u70b9 i\n return i;\n}\n[class]{}-[func]{binarySearchInsertion}\nTip
\u672c\u8282\u7684\u4ee3\u7801\u90fd\u662f\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5199\u6cd5\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5199\u6cd5\u3002
\u603b\u7684\u6765\u770b\uff0c\u4e8c\u5206\u67e5\u627e\u65e0\u975e\u5c31\u662f\u7ed9\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u522b\u8bbe\u5b9a\u641c\u7d22\u76ee\u6807\uff0c\u76ee\u6807\u53ef\u80fd\u662f\u4e00\u4e2a\u5177\u4f53\u7684\u5143\u7d20\uff08\u4f8b\u5982
target\uff09\uff0c\u4e5f\u53ef\u80fd\u662f\u4e00\u4e2a\u5143\u7d20\u8303\u56f4\uff08\u4f8b\u5982\u5c0f\u4e8etarget\u7684\u5143\u7d20\uff09\u3002\u5728\u4e0d\u65ad\u7684\u5faa\u73af\u4e8c\u5206\u4e2d\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u90fd\u9010\u6e10\u903c\u8fd1\u9884\u5148\u8bbe\u5b9a\u7684\u76ee\u6807\u3002\u6700\u7ec8\uff0c\u5b83\u4eec\u6216\u662f\u6210\u529f\u627e\u5230\u7b54\u6848\uff0c\u6216\u662f\u8d8a\u8fc7\u8fb9\u754c\u540e\u505c\u6b62\u3002
"},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565","text":"\u5728\u7b97\u6cd5\u9898\u4e2d\uff0c\u6211\u4eec\u5e38\u901a\u8fc7\u5c06\u7ebf\u6027\u67e5\u627e\u66ff\u6362\u4e3a\u54c8\u5e0c\u67e5\u627e\u6765\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u7b97\u6cd5\u9898\u6765\u52a0\u6df1\u7406\u89e3\u3002
Question
\u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4
"},{"location":"chapter_searching/replace_linear_by_hashing/#1041","title":"10.4.1 \u00a0 \u7ebf\u6027\u67e5\u627e\uff1a\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4","text":"nums\u548c\u4e00\u4e2a\u76ee\u6807\u5143\u7d20target\uff0c\u8bf7\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u201c\u548c\u201d\u4e3atarget\u7684\u4e24\u4e2a\u5143\u7d20\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u4e2a\u89e3\u5373\u53ef\u3002\u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\u3002\u5982\u56fe 10-9 \u6240\u793a\uff0c\u6211\u4eec\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a
target\uff0c\u82e5\u662f\uff0c\u5219\u8fd4\u56de\u5b83\u4eec\u7684\u7d22\u5f15\u3002\u56fe 10-9 \u00a0 \u7ebf\u6027\u67e5\u627e\u6c42\u89e3\u4e24\u6570\u4e4b\u548c
\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig two_sum.py
two_sum.cppdef two_sum_brute_force(nums: list[int], target: int) -> list[int]:\n \"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\"\"\"\n # \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for i in range(len(nums) - 1):\n for j in range(i + 1, len(nums)):\n if nums[i] + nums[j] == target:\n return [i, j]\n return []\n
two_sum.java/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector<int> twoSumBruteForce(vector<int> &nums, int target) {\n int size = nums.size();\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (int i = 0; i < size - 1; i++) {\n for (int j = i + 1; j < size; j++) {\n if (nums[i] + nums[j] == target)\n return {i, j};\n }\n }\n return {};\n}\n
two_sum.cs/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\n int size = nums.length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (int i = 0; i < size - 1; i++) {\n for (int j = i + 1; j < size; j++) {\n if (nums[i] + nums[j] == target)\n return new int[] { i, j };\n }\n }\n return new int[0];\n}\n
two_sum.go/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] TwoSumBruteForce(int[] nums, int target) {\n int size = nums.Length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (int i = 0; i < size - 1; i++) {\n for (int j = i + 1; j < size; j++) {\n if (nums[i] + nums[j] == target)\n return [i, j];\n }\n }\n return [];\n}\n
two_sum.swift/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\n size := len(nums)\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for i := 0; i < size-1; i++ {\n for j := i + 1; i < size; j++ {\n if nums[i]+nums[j] == target {\n return []int{i, j}\n }\n }\n }\n return nil\n}\n
two_sum.js/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -> [Int] {\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for i in nums.indices.dropLast() {\n for j in nums.indices.dropFirst(i + 1) {\n if nums[i] + nums[j] == target {\n return [i, j]\n }\n }\n }\n return [0]\n}\n
two_sum.ts/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\n const n = nums.length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (let i = 0; i < n; i++) {\n for (let j = i + 1; j < n; j++) {\n if (nums[i] + nums[j] === target) {\n return [i, j];\n }\n }\n }\n return [];\n}\n
two_sum.dart/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n const n = nums.length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (let i = 0; i < n; i++) {\n for (let j = i + 1; j < n; j++) {\n if (nums[i] + nums[j] === target) {\n return [i, j];\n }\n }\n }\n return [];\n}\n
two_sum.rs/* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u679a\u4e3e */\nList<int> twoSumBruteForce(List<int> nums, int target) {\n int size = nums.length;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for (var i = 0; i < size - 1; i++) {\n for (var j = i + 1; j < size; j++) {\n if (nums[i] + nums[j] == target) return [i, j];\n }\n }\n return [0];\n}\n
two_sum.c/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\npub fn two_sum_brute_force(nums: &Vec<i32>, target: i32) -> Option<Vec<i32>> {\n let size = nums.len();\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n for i in 0..size - 1 {\n for j in i + 1..size {\n if nums[i] + nums[j] == target {\n return Some(vec![i as i32, j as i32]);\n }\n }\n }\n None\n}\n
two_sum.zig/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n for (int i = 0; i < numsSize; ++i) {\n for (int j = i + 1; j < numsSize; ++j) {\n if (nums[i] + nums[j] == target) {\n int *res = malloc(sizeof(int) * 2);\n res[0] = i, res[1] = j;\n *returnSize = 2;\n return res;\n }\n }\n }\n *returnSize = 0;\n return NULL;\n}\n// \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\n var size: usize = nums.len;\n var i: usize = 0;\n // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n while (i < size - 1) : (i += 1) {\n var j = i + 1;\n while (j < size) : (j += 1) {\n if (nums[i] + nums[j] == target) {\n return [_]i32{@intCast(i), @intCast(j)};\n }\n }\n }\n return null;\n}\n\u6b64\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002
"},{"location":"chapter_searching/replace_linear_by_hashing/#1042","title":"10.4.2 \u00a0 \u54c8\u5e0c\u67e5\u627e\uff1a\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4","text":"\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u952e\u503c\u5bf9\u5206\u522b\u4e3a\u6570\u7ec4\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u6267\u884c\u56fe 10-10 \u6240\u793a\u7684\u6b65\u9aa4\u3002
- \u5224\u65ad\u6570\u5b57
target - nums[i]\u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u3002 - \u5c06\u952e\u503c\u5bf9
nums[i]\u548c\u7d22\u5f15i\u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\u3002
\u56fe 10-10 \u00a0 \u8f85\u52a9\u54c8\u5e0c\u8868\u6c42\u89e3\u4e24\u6570\u4e4b\u548c
\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff0c\u4ec5\u9700\u5355\u5c42\u5faa\u73af\u5373\u53ef\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig two_sum.py
two_sum.cppdef two_sum_hash_table(nums: list[int], target: int) -> list[int]:\n \"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\"\"\"\n # \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n dic = {}\n # \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for i in range(len(nums)):\n if target - nums[i] in dic:\n return [dic[target - nums[i]], i]\n dic[nums[i]] = i\n return []\n
two_sum.java/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector<int> twoSumHashTable(vector<int> &nums, int target) {\n int size = nums.size();\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n unordered_map<int, int> dic;\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (int i = 0; i < size; i++) {\n if (dic.find(target - nums[i]) != dic.end()) {\n return {dic[target - nums[i]], i};\n }\n dic.emplace(nums[i], i);\n }\n return {};\n}\n
two_sum.cs/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\n int size = nums.length;\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n Map<Integer, Integer> dic = new HashMap<>();\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (int i = 0; i < size; i++) {\n if (dic.containsKey(target - nums[i])) {\n return new int[] { dic.get(target - nums[i]), i };\n }\n dic.put(nums[i], i);\n }\n return new int[0];\n}\n
two_sum.go/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] TwoSumHashTable(int[] nums, int target) {\n int size = nums.Length;\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n Dictionary<int, int> dic = [];\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (int i = 0; i < size; i++) {\n if (dic.ContainsKey(target - nums[i])) {\n return [dic[target - nums[i]], i];\n }\n dic.Add(nums[i], i);\n }\n return [];\n}\n
two_sum.swift/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n hashTable := map[int]int{}\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for idx, val := range nums {\n if preIdx, ok := hashTable[target-val]; ok {\n return []int{preIdx, idx}\n }\n hashTable[val] = idx\n }\n return nil\n}\n
two_sum.js/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -> [Int] {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n var dic: [Int: Int] = [:]\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for i in nums.indices {\n if let j = dic[target - nums[i]] {\n return [j, i]\n }\n dic[nums[i]] = i\n }\n return [0]\n}\n
two_sum.ts/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n let m = {};\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (let i = 0; i < nums.length; i++) {\n if (m[target - nums[i]] !== undefined) {\n return [m[target - nums[i]], i];\n } else {\n m[nums[i]] = i;\n }\n }\n return [];\n}\n
two_sum.dart/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n let m: Map<number, number> = new Map();\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (let i = 0; i < nums.length; i++) {\n let index = m.get(target - nums[i]);\n if (index !== undefined) {\n return [index, i];\n } else {\n m.set(nums[i], i);\n }\n }\n return [];\n}\n
two_sum.rs/* \u65b9\u6cd5\u4e8c\uff1a \u8f85\u52a9\u54c8\u5e0c\u8868 */\nList<int> twoSumHashTable(List<int> nums, int target) {\n int size = nums.length;\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n Map<int, int> dic = HashMap();\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (var i = 0; i < size; i++) {\n if (dic.containsKey(target - nums[i])) {\n return [dic[target - nums[i]]!, i];\n }\n dic.putIfAbsent(nums[i], () => i);\n }\n return [0];\n}\n
two_sum.c/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\npub fn two_sum_hash_table(nums: &Vec<i32>, target: i32) -> Option<Vec<i32>> {\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n let mut dic = HashMap::new();\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n for (i, num) in nums.iter().enumerate() {\n match dic.get(&(target - num)) {\n Some(v) => return Some(vec![*v as i32, i as i32]),\n None => dic.insert(num, i as i32)\n };\n }\n None\n}\n
two_sum.zig/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n int key;\n int val;\n UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u54c8\u5e0c\u8868\u67e5\u8be2 */\nHashTable *find(HashTable *h, int key) {\n HashTable *tmp;\n HASH_FIND_INT(h, &key, tmp);\n return tmp;\n}\n\n/* \u54c8\u5e0c\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(HashTable *h, int key, int val) {\n HashTable *t = find(h, key);\n if (t == NULL) {\n HashTable *tmp = malloc(sizeof(HashTable));\n tmp->key = key, tmp->val = val;\n HASH_ADD_INT(h, key, tmp);\n } else {\n t->val = val;\n }\n}\n\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n HashTable *hashtable = NULL;\n for (int i = 0; i < numsSize; i++) {\n HashTable *t = find(hashtable, target - nums[i]);\n if (t != NULL) {\n int *res = malloc(sizeof(int) * 2);\n res[0] = t->val, res[1] = i;\n *returnSize = 2;\n return res;\n }\n insert(hashtable, nums[i], i);\n }\n *returnSize = 0;\n return NULL;\n}\n// \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\n var size: usize = nums.len;\n // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n var dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\n defer dic.deinit();\n var i: usize = 0;\n // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n while (i < size) : (i += 1) {\n if (dic.contains(target - nums[i])) {\n return [_]i32{dic.get(target - nums[i]).?, @intCast(i)};\n }\n try dic.put(nums[i], @intCast(i));\n }\n return null;\n}\n\u6b64\u65b9\u6cd5\u901a\u8fc7\u54c8\u5e0c\u67e5\u627e\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u8fd0\u884c\u6548\u7387\u3002
\u7531\u4e8e\u9700\u8981\u7ef4\u62a4\u4e00\u4e2a\u989d\u5916\u7684\u54c8\u5e0c\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u8be5\u65b9\u6cd5\u7684\u6574\u4f53\u65f6\u7a7a\u6548\u7387\u66f4\u4e3a\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u9898\u7684\u6700\u4f18\u89e3\u6cd5\u3002
"},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5","text":"\u300c\u641c\u7d22\u7b97\u6cd5 searching algorithm\u300d\u7528\u4e8e\u5728\u6570\u636e\u7ed3\u6784\uff08\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u6216\u56fe\uff09\u4e2d\u641c\u7d22\u4e00\u4e2a\u6216\u4e00\u7ec4\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5143\u7d20\u3002
\u641c\u7d22\u7b97\u6cd5\u53ef\u6839\u636e\u5b9e\u73b0\u601d\u8def\u5206\u4e3a\u4ee5\u4e0b\u4e24\u7c7b\u3002
- \u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\uff0c\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u904d\u5386\u7b49\u3002
- \u5229\u7528\u6570\u636e\u7ec4\u7ec7\u7ed3\u6784\u6216\u6570\u636e\u5305\u542b\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u5b9e\u73b0\u9ad8\u6548\u5143\u7d20\u67e5\u627e\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u548c\u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u7b49\u3002
\u4e0d\u96be\u53d1\u73b0\uff0c\u8fd9\u4e9b\u77e5\u8bc6\u70b9\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u56e0\u6b64\u641c\u7d22\u7b97\u6cd5\u5bf9\u4e8e\u6211\u4eec\u6765\u8bf4\u5e76\u4e0d\u964c\u751f\u3002\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4ece\u66f4\u52a0\u7cfb\u7edf\u7684\u89c6\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5ba1\u89c6\u641c\u7d22\u7b97\u6cd5\u3002
"},{"location":"chapter_searching/searching_algorithm_revisited/#1051","title":"10.5.1 \u00a0 \u66b4\u529b\u641c\u7d22","text":"\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u7684\u6bcf\u4e2a\u5143\u7d20\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002
- \u201c\u7ebf\u6027\u641c\u7d22\u201d\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\u7b49\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u9010\u4e2a\u8bbf\u95ee\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u5230\u8fbe\u53e6\u4e00\u7aef\u4ecd\u6ca1\u6709\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002
- \u201c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u548c\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u662f\u56fe\u548c\u6811\u7684\u4e24\u79cd\u904d\u5386\u7b56\u7565\u3002\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\u9010\u5c42\u641c\u7d22\uff0c\u7531\u8fd1\u53ca\u8fdc\u5730\u8bbf\u95ee\u5404\u4e2a\u8282\u70b9\u3002\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\uff0c\u6cbf\u7740\u4e00\u6761\u8def\u5f84\u8d70\u5230\u5934\uff0c\u518d\u56de\u6eaf\u5e76\u5c1d\u8bd5\u5176\u4ed6\u8def\u5f84\uff0c\u76f4\u5230\u904d\u5386\u5b8c\u6574\u4e2a\u6570\u636e\u7ed3\u6784\u3002
\u66b4\u529b\u641c\u7d22\u7684\u4f18\u70b9\u662f\u7b80\u5355\u4e14\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u505a\u9884\u5904\u7406\u548c\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\u3002
\u7136\u800c\uff0c\u6b64\u7c7b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u5143\u7d20\u6570\u91cf\uff0c\u56e0\u6b64\u5728\u6570\u636e\u91cf\u8f83\u5927\u7684\u60c5\u51b5\u4e0b\u6027\u80fd\u8f83\u5dee\u3002
"},{"location":"chapter_searching/searching_algorithm_revisited/#1052","title":"10.5.2 \u00a0 \u81ea\u9002\u5e94\u641c\u7d22","text":"\u81ea\u9002\u5e94\u641c\u7d22\u5229\u7528\u6570\u636e\u7684\u7279\u6709\u5c5e\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u6765\u4f18\u5316\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ece\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002
- \u201c\u4e8c\u5206\u67e5\u627e\u201d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\u5b9e\u73b0\u9ad8\u6548\u67e5\u627e\uff0c\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002
- \u201c\u54c8\u5e0c\u67e5\u627e\u201d\u5229\u7528\u54c8\u5e0c\u8868\u5c06\u641c\u7d22\u6570\u636e\u548c\u76ee\u6807\u6570\u636e\u5efa\u7acb\u4e3a\u952e\u503c\u5bf9\u6620\u5c04\uff0c\u4ece\u800c\u5b9e\u73b0\u67e5\u8be2\u64cd\u4f5c\u3002
- \u201c\u6811\u67e5\u627e\u201d\u5728\u7279\u5b9a\u7684\u6811\u7ed3\u6784\uff08\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\uff09\u4e2d\uff0c\u57fa\u4e8e\u6bd4\u8f83\u8282\u70b9\u503c\u6765\u5feb\u901f\u6392\u9664\u8282\u70b9\uff0c\u4ece\u800c\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002
\u6b64\u7c7b\u7b97\u6cd5\u7684\u4f18\u70b9\u662f\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002
\u7136\u800c\uff0c\u4f7f\u7528\u8fd9\u4e9b\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u67e5\u627e\u9700\u8981\u9884\u5148\u5bf9\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\uff0c\u54c8\u5e0c\u67e5\u627e\u548c\u6811\u67e5\u627e\u90fd\u9700\u8981\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7ef4\u62a4\u8fd9\u4e9b\u6570\u636e\u7ed3\u6784\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u5f00\u9500\u3002
Tip
\u81ea\u9002\u5e94\u641c\u7d22\u7b97\u6cd5\u5e38\u88ab\u79f0\u4e3a\u67e5\u627e\u7b97\u6cd5\uff0c\u4e3b\u8981\u7528\u4e8e\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u68c0\u7d22\u76ee\u6807\u5143\u7d20\u3002
"},{"location":"chapter_searching/searching_algorithm_revisited/#1053","title":"10.5.3 \u00a0 \u641c\u7d22\u65b9\u6cd5\u9009\u53d6","text":"\u7ed9\u5b9a\u5927\u5c0f\u4e3a \\(n\\) \u7684\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u7ebf\u6027\u641c\u7d22\u3001\u4e8c\u5206\u67e5\u627e\u3001\u6811\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u7b49\u591a\u79cd\u65b9\u6cd5\u4ece\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\u3002\u5404\u4e2a\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u56fe 10-11 \u6240\u793a\u3002
\u56fe 10-11 \u00a0 \u591a\u79cd\u641c\u7d22\u7b56\u7565
\u4e0a\u8ff0\u51e0\u79cd\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u4e0e\u7279\u6027\u5982\u8868 10-1 \u6240\u793a\u3002
\u8868 10-1 \u00a0 \u67e5\u627e\u7b97\u6cd5\u6548\u7387\u5bf9\u6bd4
\u7ebf\u6027\u641c\u7d22 \u4e8c\u5206\u67e5\u627e \u6811\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u989d\u5916\u7a7a\u95f4 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u6570\u636e\u9884\u5904\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6811 \\(O(n \\log n)\\) \u5efa\u54c8\u5e0c\u8868 \\(O(n)\\) \u6570\u636e\u662f\u5426\u6709\u5e8f \u65e0\u5e8f \u6709\u5e8f \u6709\u5e8f \u65e0\u5e8f\u641c\u7d22\u7b97\u6cd5\u7684\u9009\u62e9\u8fd8\u53d6\u51b3\u4e8e\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u4e0e\u66f4\u65b0\u9891\u7387\u7b49\u3002
\u7ebf\u6027\u641c\u7d22
- \u901a\u7528\u6027\u8f83\u597d\uff0c\u65e0\u987b\u4efb\u4f55\u6570\u636e\u9884\u5904\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u4eec\u4ec5\u9700\u67e5\u8be2\u4e00\u6b21\u6570\u636e\uff0c\u90a3\u4e48\u5176\u4ed6\u4e09\u79cd\u65b9\u6cd5\u7684\u6570\u636e\u9884\u5904\u7406\u7684\u65f6\u95f4\u6bd4\u7ebf\u6027\u641c\u7d22\u7684\u65f6\u95f4\u8fd8\u8981\u66f4\u957f\u3002
- \u9002\u7528\u4e8e\u4f53\u91cf\u8f83\u5c0f\u7684\u6570\u636e\uff0c\u6b64\u60c5\u51b5\u4e0b\u65f6\u95f4\u590d\u6742\u5ea6\u5bf9\u6548\u7387\u5f71\u54cd\u8f83\u5c0f\u3002
- \u9002\u7528\u4e8e\u6570\u636e\u66f4\u65b0\u9891\u7387\u8f83\u9ad8\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u8be5\u65b9\u6cd5\u4e0d\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u4efb\u4f55\u989d\u5916\u7ef4\u62a4\u3002
\u4e8c\u5206\u67e5\u627e
- \u9002\u7528\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u6548\u7387\u8868\u73b0\u7a33\u5b9a\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
- \u6570\u636e\u91cf\u4e0d\u80fd\u8fc7\u5927\uff0c\u56e0\u4e3a\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u3002
- \u4e0d\u9002\u7528\u4e8e\u9ad8\u9891\u589e\u5220\u6570\u636e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u7ef4\u62a4\u6709\u5e8f\u6570\u7ec4\u7684\u5f00\u9500\u8f83\u5927\u3002
\u54c8\u5e0c\u67e5\u627e
- \u9002\u5408\u5bf9\u67e5\u8be2\u6027\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u573a\u666f\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
- \u4e0d\u9002\u5408\u9700\u8981\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u65e0\u6cd5\u7ef4\u62a4\u6570\u636e\u7684\u6709\u5e8f\u6027\u3002
- \u5bf9\u54c8\u5e0c\u51fd\u6570\u548c\u54c8\u5e0c\u51b2\u7a81\u5904\u7406\u7b56\u7565\u7684\u4f9d\u8d56\u6027\u8f83\u9ad8\uff0c\u5177\u6709\u8f83\u5927\u7684\u6027\u80fd\u52a3\u5316\u98ce\u9669\u3002
- \u4e0d\u9002\u5408\u6570\u636e\u91cf\u8fc7\u5927\u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u9700\u8981\u989d\u5916\u7a7a\u95f4\u6765\u6700\u5927\u7a0b\u5ea6\u5730\u51cf\u5c11\u51b2\u7a81\uff0c\u4ece\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8be2\u6027\u80fd\u3002
\u6811\u67e5\u627e
- \u9002\u7528\u4e8e\u6d77\u91cf\u6570\u636e\uff0c\u56e0\u4e3a\u6811\u8282\u70b9\u5728\u5185\u5b58\u4e2d\u662f\u5206\u6563\u5b58\u50a8\u7684\u3002
- \u9002\u5408\u9700\u8981\u7ef4\u62a4\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\u3002
- \u5728\u6301\u7eed\u589e\u5220\u8282\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4ea7\u751f\u503e\u659c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
- \u82e5\u4f7f\u7528 AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u5219\u5404\u9879\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a33\u5b9a\u8fd0\u884c\uff0c\u4f46\u7ef4\u62a4\u6811\u5e73\u8861\u7684\u64cd\u4f5c\u4f1a\u589e\u52a0\u989d\u5916\u5f00\u9500\u3002
- \u4e8c\u5206\u67e5\u627e\u4f9d\u8d56\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u9010\u6b65\u7f29\u51cf\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u8fdb\u884c\u67e5\u627e\u3002\u5b83\u8981\u6c42\u8f93\u5165\u6570\u636e\u6709\u5e8f\uff0c\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
- \u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u6570\u636e\u3002\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\uff0c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u548c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u9002\u7528\u4e8e\u56fe\u548c\u6811\u3002\u6b64\u7c7b\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\uff0c\u4f46\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u8f83\u9ad8\u3002
- \u54c8\u5e0c\u67e5\u627e\u3001\u6811\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u5c5e\u4e8e\u9ad8\u6548\u641c\u7d22\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002\u6b64\u7c7b\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u3002
- \u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u548c\u66f4\u65b0\u9891\u7387\u7b49\u56e0\u7d20\u8fdb\u884c\u5177\u4f53\u5206\u6790\uff0c\u4ece\u800c\u9009\u62e9\u5408\u9002\u7684\u641c\u7d22\u65b9\u6cd5\u3002
- \u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u5c0f\u578b\u6216\u9891\u7e41\u66f4\u65b0\u7684\u6570\u636e\uff1b\u4e8c\u5206\u67e5\u627e\u9002\u7528\u4e8e\u5927\u578b\u3001\u6392\u5e8f\u7684\u6570\u636e\uff1b\u54c8\u5e0c\u67e5\u627e\u9002\u7528\u4e8e\u5bf9\u67e5\u8be2\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u4e14\u65e0\u987b\u8303\u56f4\u67e5\u8be2\u7684\u6570\u636e\uff1b\u6811\u67e5\u627e\u9002\u7528\u4e8e\u9700\u8981\u7ef4\u62a4\u987a\u5e8f\u548c\u652f\u6301\u8303\u56f4\u67e5\u8be2\u7684\u5927\u578b\u52a8\u6001\u6570\u636e\u3002
- \u7528\u54c8\u5e0c\u67e5\u627e\u66ff\u6362\u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u8fd0\u884c\u65f6\u95f4\u7684\u7b56\u7565\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002
Abstract
\u6392\u5e8f\u72b9\u5982\u4e00\u628a\u5c06\u6df7\u4e71\u53d8\u4e3a\u79e9\u5e8f\u7684\u9b54\u6cd5\u94a5\u5319\uff0c\u4f7f\u6211\u4eec\u80fd\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u7406\u89e3\u4e0e\u5904\u7406\u6570\u636e\u3002
\u65e0\u8bba\u662f\u7b80\u5355\u7684\u5347\u5e8f\uff0c\u8fd8\u662f\u590d\u6742\u7684\u5206\u7c7b\u6392\u5217\uff0c\u6392\u5e8f\u90fd\u5411\u6211\u4eec\u5c55\u793a\u4e86\u6570\u636e\u7684\u548c\u8c10\u7f8e\u611f\u3002
"},{"location":"chapter_sorting/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5
- 11.2 \u00a0 \u9009\u62e9\u6392\u5e8f
- 11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f
- 11.4 \u00a0 \u63d2\u5165\u6392\u5e8f
- 11.5 \u00a0 \u5feb\u901f\u6392\u5e8f
- 11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f
- 11.7 \u00a0 \u5806\u6392\u5e8f
- 11.8 \u00a0 \u6876\u6392\u5e8f
- 11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f
- 11.10 \u00a0 \u57fa\u6570\u6392\u5e8f
- 11.11 \u00a0 \u5c0f\u7ed3
\u300c\u5192\u6ce1\u6392\u5e8f bubble sort\u300d\u901a\u8fc7\u8fde\u7eed\u5730\u6bd4\u8f83\u4e0e\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u5b9e\u73b0\u6392\u5e8f\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u50cf\u6c14\u6ce1\u4ece\u5e95\u90e8\u5347\u5230\u9876\u90e8\u4e00\u6837\uff0c\u56e0\u6b64\u5f97\u540d\u5192\u6ce1\u6392\u5e8f\u3002
\u5982\u56fe 11-4 \u6240\u793a\uff0c\u5192\u6ce1\u8fc7\u7a0b\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6765\u6a21\u62df\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u6bd4\u8f83\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 > \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u6362\u4e8c\u8005\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6700\u5927\u7684\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u5230\u6570\u7ec4\u7684\u6700\u53f3\u7aef\u3002
<1><2><3><4><5><6><7>\u56fe 11-4 \u00a0 \u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6a21\u62df\u5192\u6ce1
"},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6b65\u9aa4\u5982\u56fe 11-5 \u6240\u793a\u3002
- \u9996\u5148\uff0c\u5bf9 \\(n\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\uff0c
- \u63a5\u4e0b\u6765\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002
- \u4ee5\u6b64\u7c7b\u63a8\uff0c\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u201c\u5192\u6ce1\u201d\u540e\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002
- \u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002
\u56fe 11-5 \u00a0 \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b
\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig bubble_sort.py
bubble_sort.cppdef bubble_sort(nums: list[int]):\n \"\"\"\u5192\u6ce1\u6392\u5e8f\"\"\"\n n = len(nums)\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in range(n - 1, 0, -1):\n # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j in range(i):\n if nums[j] > nums[j + 1]:\n # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n nums[j], nums[j + 1] = nums[j + 1], nums[j]\n
bubble_sort.java/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector<int> &nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.size() - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n swap(nums[j], nums[j + 1]);\n }\n }\n }\n}\n
bubble_sort.cs/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.go/* \u5192\u6ce1\u6392\u5e8f */\nvoid BubbleSort(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.Length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n }\n }\n }\n}\n
bubble_sort.swift/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := len(nums) - 1; i > 0; i-- {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j := 0; j < i; j++ {\n if nums[j] > nums[j+1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n nums[j], nums[j+1] = nums[j+1], nums[j]\n }\n }\n }\n}\n
bubble_sort.js/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in stride(from: 0, to: i, by: 1) {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n }\n }\n }\n}\n
bubble_sort.ts/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.dart/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.rs/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(List<int> nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.c/* \u5192\u6ce1\u6392\u5e8f */\nfn bubble_sort(nums: &mut [i32]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in (1..nums.len()).rev() {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0..i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n
bubble_sort.zig/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = size - 1; i > 0; i--) {\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n int temp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = temp;\n }\n }\n }\n}\n
"},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2 \u00a0 \u6548\u7387\u4f18\u5316","text":"// \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n var i: usize = nums.len - 1;\n while (i > 0) : (i -= 1) {\n var j: usize = 0;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n while (j < i) : (j += 1) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n var tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n }\n }\n }\n}\n\u6211\u4eec\u53d1\u73b0\uff0c\u5982\u679c\u67d0\u8f6e\u201c\u5192\u6ce1\u201d\u4e2d\u6ca1\u6709\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d
flag\u6765\u76d1\u6d4b\u8fd9\u79cd\u60c5\u51b5\uff0c\u4e00\u65e6\u51fa\u73b0\u5c31\u7acb\u5373\u8fd4\u56de\u3002\u7ecf\u8fc7\u4f18\u5316\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u4f46\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u53ef\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig bubble_sort.py
bubble_sort.cppdef bubble_sort_with_flag(nums: list[int]):\n \"\"\"\u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\"\"\"\n n = len(nums)\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in range(n - 1, 0, -1):\n flag = False # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j in range(i):\n if nums[j] > nums[j + 1]:\n # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n nums[j], nums[j + 1] = nums[j + 1], nums[j]\n flag = True # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n if not flag:\n break # \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n
bubble_sort.java/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector<int> &nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.size() - 1; i > 0; i--) {\n bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n swap(nums[j], nums[j + 1]);\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag)\n break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.cs/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n boolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag)\n break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.go/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid BubbleSortWithFlag(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.Length - 1; i > 0; i--) {\n bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.swift/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := len(nums) - 1; i > 0; i-- {\n flag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for j := 0; j < i; j++ {\n if nums[j] > nums[j+1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n nums[j], nums[j+1] = nums[j+1], nums[j]\n flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if flag == false { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n break\n }\n }\n}\n
bubble_sort.js/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n var flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n for j in stride(from: 0, to: i, by: 1) {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j]\n nums[j] = nums[j + 1]\n nums[j + 1] = tmp\n flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if !flag { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n break\n }\n }\n}\n
bubble_sort.ts/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.dart/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (let i = nums.length - 1; i > 0; i--) {\n let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (let j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.rs/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(List<int> nums) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = nums.length - 1; i > 0; i--) {\n bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n int tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.c/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nfn bubble_sort_with_flag(nums: &mut [i32]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i in (1..nums.len()).rev() {\n let mut flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n for j in 0..i {\n if nums[j] > nums[j + 1] {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n let tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n }\n }\n if !flag {break}; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n
bubble_sort.zig/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for (int i = size - 1; i > 0; i--) {\n bool flag = false;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n for (int j = 0; j < i; j++) {\n if (nums[j] > nums[j + 1]) {\n int temp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = temp;\n flag = true;\n }\n }\n if (!flag)\n break;\n }\n}\n
"},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"// \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n var i: usize = nums.len - 1;\n while (i > 0) : (i -= 1) {\n var flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n var j: usize = 0;\n // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n while (j < i) : (j += 1) {\n if (nums[j] > nums[j + 1]) {\n // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n var tmp = nums[j];\n nums[j] = nums[j + 1];\n nums[j + 1] = tmp;\n flag = true;\n }\n }\n if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n }\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5404\u8f6e\u201c\u5192\u6ce1\u201d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4f9d\u6b21\u4e3a \\(n - 1\\)\u3001\\(n - 2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u603b\u548c\u4e3a \\((n - 1) n / 2\\) \u3002\u5728\u5f15\u5165
flag\u4f18\u5316\u540e\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(n)\\) \u3002 - \u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u6362\u3002
\u524d\u8ff0\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e\u201c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u51e0\u79cd\u201c\u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u7ebf\u6027\u9636\u3002
\u300c\u6876\u6392\u5e8f bucket sort\u300d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u987a\u5e8f\u7684\u6876\uff0c\u6bcf\u4e2a\u6876\u5bf9\u5e94\u4e00\u4e2a\u6570\u636e\u8303\u56f4\uff0c\u5c06\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff1b\u7136\u540e\uff0c\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u5206\u522b\u6267\u884c\u6392\u5e8f\uff1b\u6700\u7ec8\u6309\u7167\u6876\u7684\u987a\u5e8f\u5c06\u6240\u6709\u6570\u636e\u5408\u5e76\u3002
"},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u8003\u8651\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u5176\u5143\u7d20\u662f\u8303\u56f4 \\([0, 1)\\) \u5185\u7684\u6d6e\u70b9\u6570\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-13 \u6240\u793a\u3002
- \u521d\u59cb\u5316 \\(k\\) \u4e2a\u6876\uff0c\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\u3002
- \u5bf9\u6bcf\u4e2a\u6876\u5206\u522b\u6267\u884c\u6392\u5e8f\uff08\u8fd9\u91cc\u91c7\u7528\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff09\u3002
- \u6309\u7167\u6876\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u5408\u5e76\u7ed3\u679c\u3002
\u56fe 11-13 \u00a0 \u6876\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b
\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig bucket_sort.py
bucket_sort.cppdef bucket_sort(nums: list[float]):\n \"\"\"\u6876\u6392\u5e8f\"\"\"\n # \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n k = len(nums) // 2\n buckets = [[] for _ in range(k)]\n # 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for num in nums:\n # \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n i = int(num * k)\n # \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].append(num)\n # 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for bucket in buckets:\n # \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.sort()\n # 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n i = 0\n for bucket in buckets:\n for num in bucket:\n nums[i] = num\n i += 1\n
bucket_sort.java/* \u6876\u6392\u5e8f */\nvoid bucketSort(vector<float> &nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = nums.size() / 2;\n vector<vector<float>> buckets(k);\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (float num : nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int i = num * k;\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n buckets[i].push_back(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (vector<float> &bucket : buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n sort(bucket.begin(), bucket.end());\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n int i = 0;\n for (vector<float> &bucket : buckets) {\n for (float num : bucket) {\n nums[i++] = num;\n }\n }\n}\n
bucket_sort.cs/* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = nums.length / 2;\n List<List<Float>> buckets = new ArrayList<>();\n for (int i = 0; i < k; i++) {\n buckets.add(new ArrayList<>());\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (float num : nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int i = (int) (num * k);\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets.get(i).add(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (List<Float> bucket : buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n Collections.sort(bucket);\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n int i = 0;\n for (List<Float> bucket : buckets) {\n for (float num : bucket) {\n nums[i++] = num;\n }\n }\n}\n
bucket_sort.go/* \u6876\u6392\u5e8f */\nvoid BucketSort(float[] nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = nums.Length / 2;\n List<List<float>> buckets = [];\n for (int i = 0; i < k; i++) {\n buckets.Add([]);\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n foreach (float num in nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int i = (int)(num * k);\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].Add(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n foreach (List<float> bucket in buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.Sort();\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n int j = 0;\n foreach (List<float> bucket in buckets) {\n foreach (float num in bucket) {\n nums[j++] = num;\n }\n }\n}\n
bucket_sort.swift/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n k := len(nums) / 2\n buckets := make([][]float64, k)\n for i := 0; i < k; i++ {\n buckets[i] = make([]float64, 0)\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for _, num := range nums {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n i := int(num * float64(k))\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i] = append(buckets[i], num)\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for i := 0; i < k; i++ {\n // \u4f7f\u7528\u5185\u7f6e\u5207\u7247\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n sort.Float64s(buckets[i])\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n i := 0\n for _, bucket := range buckets {\n for _, num := range bucket {\n nums[i] = num\n i++\n }\n }\n}\n
bucket_sort.js/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n let k = nums.count / 2\n var buckets = (0 ..< k).map { _ in [Double]() }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for num in nums {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n let i = Int(num * Double(k))\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].append(num)\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for i in buckets.indices {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n buckets[i].sort()\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n var i = nums.startIndex\n for bucket in buckets {\n for num in bucket {\n nums[i] = num\n nums.formIndex(after: &i)\n }\n }\n}\n
bucket_sort.ts/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n const k = nums.length / 2;\n const buckets = [];\n for (let i = 0; i < k; i++) {\n buckets.push([]);\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (const num of nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n const i = Math.floor(num * k);\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].push(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (const bucket of buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.sort((a, b) => a - b);\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n let i = 0;\n for (const bucket of buckets) {\n for (const num of bucket) {\n nums[i++] = num;\n }\n }\n}\n
bucket_sort.dart/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n const k = nums.length / 2;\n const buckets: number[][] = [];\n for (let i = 0; i < k; i++) {\n buckets.push([]);\n }\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (const num of nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n const i = Math.floor(num * k);\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].push(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (const bucket of buckets) {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.sort((a, b) => a - b);\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n let i = 0;\n for (const bucket of buckets) {\n for (const num of bucket) {\n nums[i++] = num;\n }\n }\n}\n
bucket_sort.rs/* \u6876\u6392\u5e8f */\nvoid bucketSort(List<double> nums) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = nums.length ~/ 2;\n List<List<double>> buckets = List.generate(k, (index) => []);\n\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (double _num in nums) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 _num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int i = (_num * k).toInt();\n // \u5c06 _num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n buckets[i].add(_num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (List<double> bucket in buckets) {\n bucket.sort();\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n int i = 0;\n for (List<double> bucket in buckets) {\n for (double _num in bucket) {\n nums[i++] = _num;\n }\n }\n}\n
bucket_sort.c/* \u6876\u6392\u5e8f */\nfn bucket_sort(nums: &mut [f64]) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n let k = nums.len() / 2;\n let mut buckets = vec![vec![]; k];\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for &mut num in &mut *nums {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n let i = (num * k as f64) as usize;\n // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n buckets[i].push(num);\n }\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for bucket in &mut buckets {\n // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n }\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n let mut i = 0;\n for bucket in &mut buckets {\n for &mut num in bucket {\n nums[i] = num;\n i += 1;\n }\n }\n}\n
bucket_sort.zig/* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int size) {\n // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n int k = size / 2;\n float **buckets = calloc(k, sizeof(float *));\n for (int i = 0; i < k; i++) {\n // \u6bcf\u4e2a\u6876\u6700\u591a\u53ef\u4ee5\u5206\u914d k \u4e2a\u5143\u7d20\n buckets[i] = calloc(ARRAY_SIZE, sizeof(float));\n }\n\n // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n for (int i = 0; i < size; i++) {\n // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n int bucket_idx = nums[i] * k;\n int j = 0;\n // \u5982\u679c\u6876\u4e2d\u6709\u6570\u636e\u4e14\u6570\u636e\u5c0f\u4e8e\u5f53\u524d\u503c nums[i], \u8981\u5c06\u5176\u653e\u5230\u5f53\u524d\u6876\u7684\u540e\u9762\uff0c\u76f8\u5f53\u4e8e cpp \u4e2d\u7684 push_back\n while (buckets[bucket_idx][j] > 0 && buckets[bucket_idx][j] < nums[i]) {\n j++;\n }\n float temp = nums[i];\n while (j < ARRAY_SIZE && buckets[bucket_idx][j] > 0) {\n swap(&temp, &buckets[bucket_idx][j]);\n j++;\n }\n buckets[bucket_idx][j] = temp;\n }\n\n // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n for (int i = 0; i < k; i++) {\n qsort(buckets[i], ARRAY_SIZE, sizeof(float), compare_float);\n }\n\n // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n for (int i = 0, j = 0; j < k; j++) {\n for (int l = 0; l < ARRAY_SIZE; l++) {\n if (buckets[j][l] > 0) {\n nums[i++] = buckets[j][l];\n }\n }\n }\n\n // \u91ca\u653e\u4e0a\u8ff0\u5206\u914d\u7684\u5185\u5b58\n for (int i = 0; i < k; i++) {\n free(buckets[i]);\n }\n free(buckets);\n}\n
"},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"[class]{}-[func]{bucketSort}\n\u6876\u6392\u5e8f\u9002\u7528\u4e8e\u5904\u7406\u4f53\u91cf\u5f88\u5927\u7684\u6570\u636e\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u5305\u542b 100 \u4e07\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u7a7a\u95f4\u9650\u5236\uff0c\u7cfb\u7edf\u5185\u5b58\u65e0\u6cd5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u3002\u6b64\u65f6\uff0c\u53ef\u4ee5\u5c06\u6570\u636e\u5206\u6210 1000 \u4e2a\u6876\uff0c\u7136\u540e\u5206\u522b\u5bf9\u6bcf\u4e2a\u6876\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u7ed3\u679c\u5408\u5e76\u3002
- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\) \uff1a\u5047\u8bbe\u5143\u7d20\u5728\u5404\u4e2a\u6876\u5185\u5e73\u5747\u5206\u5e03\uff0c\u90a3\u4e48\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \u3002\u5047\u8bbe\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u5219\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u6bd4\u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002\u5408\u5e76\u7ed3\u679c\u65f6\u9700\u8981\u904d\u5386\u6240\u6709\u6876\u548c\u5143\u7d20\uff0c\u82b1\u8d39 \\(O(n + k)\\) \u65f6\u95f4\u3002
- \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u6570\u636e\u88ab\u5206\u914d\u5230\u4e00\u4e2a\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8be5\u6876\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9700\u8981\u501f\u52a9 \\(k\\) \u4e2a\u6876\u548c\u603b\u5171 \\(n\\) \u4e2a\u5143\u7d20\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u6876\u6392\u5e8f\u662f\u5426\u7a33\u5b9a\u53d6\u51b3\u4e8e\u6392\u5e8f\u6876\u5185\u5143\u7d20\u7684\u7b97\u6cd5\u662f\u5426\u7a33\u5b9a\u3002
\u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7406\u8bba\u4e0a\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \uff0c\u5173\u952e\u5728\u4e8e\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u56e0\u4e3a\u5b9e\u9645\u6570\u636e\u5f80\u5f80\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5c06\u6dd8\u5b9d\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 10 \u4e2a\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u4ef7\u683c\u5206\u5e03\u4e0d\u5747\uff0c\u4f4e\u4e8e 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u4e8e 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c06\u4ef7\u683c\u533a\u95f4\u5e73\u5747\u5212\u5206\u4e3a 10 \u4e2a\uff0c\u5404\u4e2a\u6876\u4e2d\u7684\u5546\u54c1\u6570\u91cf\u5dee\u8ddd\u4f1a\u975e\u5e38\u5927\u3002
\u4e3a\u5b9e\u73b0\u5e73\u5747\u5206\u914d\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u8bbe\u5b9a\u4e00\u6761\u5927\u81f4\u7684\u5206\u754c\u7ebf\uff0c\u5c06\u6570\u636e\u7c97\u7565\u5730\u5206\u5230 3 \u4e2a\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u6bd5\u540e\uff0c\u518d\u5c06\u5546\u54c1\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6570\u91cf\u5927\u81f4\u76f8\u7b49\u3002
\u5982\u56fe 11-14 \u6240\u793a\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u521b\u5efa\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u76ee\u6807\u662f\u8ba9\u53f6\u8282\u70b9\u7684\u503c\u5c3d\u53ef\u80fd\u5e73\u5747\u3002\u5f53\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f6e\u5c06\u6570\u636e\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u5177\u4f53\u5212\u5206\u65b9\u5f0f\u53ef\u6839\u636e\u6570\u636e\u7279\u70b9\u7075\u6d3b\u9009\u62e9\u3002
\u56fe 11-14 \u00a0 \u9012\u5f52\u5212\u5206\u6876
\u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u6570\u636e\u6982\u7387\u5206\u5e03\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7edf\u8ba1\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u6570\u636e\u7279\u70b9\u91c7\u7528\u67d0\u79cd\u6982\u7387\u6a21\u578b\u8fdb\u884c\u8fd1\u4f3c\u3002
\u5982\u56fe 11-15 \u6240\u793a\uff0c\u6211\u4eec\u5047\u8bbe\u5546\u54c1\u4ef7\u683c\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8bbe\u5b9a\u4ef7\u683c\u533a\u95f4\uff0c\u4ece\u800c\u5c06\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\u3002
\u56fe 11-15 \u00a0 \u6839\u636e\u6982\u7387\u5206\u5e03\u5212\u5206\u6876
"},{"location":"chapter_sorting/counting_sort/","title":"11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"\u300c\u8ba1\u6570\u6392\u5e8f counting sort\u300d\u901a\u8fc7\u7edf\u8ba1\u5143\u7d20\u6570\u91cf\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u901a\u5e38\u5e94\u7528\u4e8e\u6574\u6570\u6570\u7ec4\u3002
"},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1 \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"\u5148\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4
nums\uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8d1f\u6574\u6570\u201d\uff0c\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-16 \u6240\u793a\u3002- \u904d\u5386\u6570\u7ec4\uff0c\u627e\u51fa\u5176\u4e2d\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u7136\u540e\u521b\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4
counter\u3002 - \u501f\u52a9
counter\u7edf\u8ba1nums\u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2dcounter[num]\u5bf9\u5e94\u6570\u5b57num\u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386nums\uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3anum\uff09\uff0c\u6bcf\u8f6e\u5c06counter[num]\u589e\u52a0 \\(1\\) \u5373\u53ef\u3002 - \u7531\u4e8e
counter\u7684\u5404\u4e2a\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386counter\uff0c\u6839\u636e\u5404\u6570\u5b57\u51fa\u73b0\u6b21\u6570\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165nums\u5373\u53ef\u3002
\u56fe 11-16 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b
\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig counting_sort.py
counting_sort.cppdef counting_sort_naive(nums: list[int]):\n \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n # \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n m = 0\n for num in nums:\n m = max(m, num)\n # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n counter = [0] * (m + 1)\n for num in nums:\n counter[num] += 1\n # 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n i = 0\n for num in range(m + 1):\n for _ in range(counter[num]):\n nums[i] = num\n i += 1\n
counting_sort.java/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector<int> &nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int num : nums) {\n m = max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n vector<int> counter(m + 1, 0);\n for (int num : nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int num = 0; num < m + 1; num++) {\n for (int j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.cs/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int num : nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int[] counter = new int[m + 1];\n for (int num : nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int num = 0; num < m + 1; num++) {\n for (int j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.go/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid CountingSortNaive(int[] nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n foreach (int num in nums) {\n m = Math.Max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int[] counter = new int[m + 1];\n foreach (int num in nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int num = 0; num < m + 1; num++) {\n for (int j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.swift/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n m := 0\n for _, num := range nums {\n if num > m {\n m = num\n }\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n counter := make([]int, m+1)\n for _, num := range nums {\n counter[num]++\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n for i, num := 0, 0; num < m+1; num++ {\n for j := 0; j < counter[num]; j++ {\n nums[i] = num\n i++\n }\n }\n}\n
counting_sort.js/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = nums.max()!\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n var counter = Array(repeating: 0, count: m + 1)\n for num in nums {\n counter[num] += 1\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n var i = 0\n for num in stride(from: 0, to: m + 1, by: 1) {\n for _ in stride(from: 0, to: counter[num], by: 1) {\n nums[i] = num\n i += 1\n }\n }\n}\n
counting_sort.ts/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = 0;\n for (const num of nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n const counter = new Array(m + 1).fill(0);\n for (const num of nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n let i = 0;\n for (let num = 0; num < m + 1; num++) {\n for (let j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.dart/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums: number[]): void {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = 0;\n for (const num of nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n const counter: number[] = new Array<number>(m + 1).fill(0);\n for (const num of nums) {\n counter[num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n let i = 0;\n for (let num = 0; num < m + 1; num++) {\n for (let j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n}\n
counting_sort.rs/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(List<int> nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int _num in nums) {\n m = max(m, _num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n List<int> counter = List.filled(m + 1, 0);\n for (int _num in nums) {\n counter[_num]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int _num = 0; _num < m + 1; _num++) {\n for (int j = 0; j < counter[_num]; j++, i++) {\n nums[i] = _num;\n }\n }\n}\n
counting_sort.c/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfn counting_sort_naive(nums: &mut [i32]) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = *nums.into_iter().max().unwrap();\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n let mut counter = vec![0; m as usize + 1];\n for &num in &*nums {\n counter[num as usize] += 1;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n let mut i = 0;\n for num in 0..m + 1 {\n for _ in 0..counter[num as usize] {\n nums[i] = num;\n i += 1;\n }\n }\n}\n
counting_sort.zig/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int nums[], int size) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int i = 0; i < size; i++) {\n if (nums[i] > m) {\n m = nums[i];\n }\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int *counter = calloc(m, sizeof(int));\n for (int i = 0; i < size; i++) {\n counter[nums[i]]++;\n }\n // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n int i = 0;\n for (int num = 0; num < m + 1; num++) {\n for (int j = 0; j < counter[num]; j++, i++) {\n nums[i] = num;\n }\n }\n // 4. \u91ca\u653e\u5185\u5b58\n free(counter);\n}\n[class]{}-[func]{countingSortNaive}\n\u8ba1\u6570\u6392\u5e8f\u4e0e\u6876\u6392\u5e8f\u7684\u8054\u7cfb
\u4ece\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u4e2d\u7684\u8ba1\u6570\u6570\u7ec4
"},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2 \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"counter\u7684\u6bcf\u4e2a\u7d22\u5f15\u89c6\u4e3a\u4e00\u4e2a\u6876\uff0c\u5c06\u7edf\u8ba1\u6570\u91cf\u7684\u8fc7\u7a0b\u770b\u4f5c\u5c06\u5404\u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u672c\u8d28\u4e0a\uff0c\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u6570\u636e\u4e0b\u7684\u4e00\u4e2a\u7279\u4f8b\u3002\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4
3.\u5c31\u5931\u6548\u4e86\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002\u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97
\\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\]counter\u7684\u201c\u524d\u7f00\u548c\u201d\u3002\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15i\u5904\u7684\u524d\u7f00\u548cprefix[i]\u7b49\u4e8e\u6570\u7ec4\u524di\u4e2a\u5143\u7d20\u4e4b\u548c\uff1a\u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u7684\u610f\u4e49\uff0c
prefix[num] - 1\u4ee3\u8868\u5143\u7d20num\u5728\u7ed3\u679c\u6570\u7ec4res\u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u975e\u5e38\u5173\u952e\uff0c\u56e0\u4e3a\u5b83\u544a\u8bc9\u6211\u4eec\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4nums\u7684\u6bcf\u4e2a\u5143\u7d20num\uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002- \u5c06
num\u586b\u5165\u6570\u7ec4res\u7684\u7d22\u5f15prefix[num] - 1\u5904\u3002 - \u4ee4\u524d\u7f00\u548c
prefix[num]\u51cf\u5c0f \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6enum\u7684\u7d22\u5f15\u3002
\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6570\u7ec4
<1><2><3><4><5><6><7><8>res\u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528res\u8986\u76d6\u539f\u6570\u7ec4nums\u5373\u53ef\u3002\u56fe 11-17 \u5c55\u793a\u4e86\u5b8c\u6574\u7684\u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b\u3002\u56fe 11-17 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6b65\u9aa4
\u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig counting_sort.py
counting_sort.cppdef counting_sort(nums: list[int]):\n \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n # \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n m = max(nums)\n # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n counter = [0] * (m + 1)\n for num in nums:\n counter[num] += 1\n # 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n # \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for i in range(m):\n counter[i + 1] += counter[i]\n # 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n # \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n n = len(nums)\n res = [0] * n\n for i in range(n - 1, -1, -1):\n num = nums[i]\n res[counter[num] - 1] = num # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num] -= 1 # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n # \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in range(n):\n nums[i] = res[i]\n
counting_sort.java/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector<int> &nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int num : nums) {\n m = max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n vector<int> counter(m + 1, 0);\n for (int num : nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int n = nums.size();\n vector<int> res(n);\n for (int i = n - 1; i >= 0; i--) {\n int num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n nums = res;\n}\n
counting_sort.cs/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int num : nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int[] counter = new int[m + 1];\n for (int num : nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int n = nums.length;\n int[] res = new int[n];\n for (int i = n - 1; i >= 0; i--) {\n int num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n
counting_sort.go/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid CountingSort(int[] nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n foreach (int num in nums) {\n m = Math.Max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int[] counter = new int[m + 1];\n foreach (int num in nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int n = nums.Length;\n int[] res = new int[n];\n for (int i = n - 1; i >= 0; i--) {\n int num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n
counting_sort.swift/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n m := 0\n for _, num := range nums {\n if num > m {\n m = num\n }\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n counter := make([]int, m+1)\n for _, num := range nums {\n counter[num]++\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for i := 0; i < m; i++ {\n counter[i+1] += counter[i]\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n n := len(nums)\n res := make([]int, n)\n for i := n - 1; i >= 0; i-- {\n num := nums[i]\n // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n res[counter[num]-1] = num\n // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n counter[num]--\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n copy(nums, res)\n}\n
counting_sort.js/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = nums.max()!\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n var counter = Array(repeating: 0, count: m + 1)\n for num in nums {\n counter[num] += 1\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for i in stride(from: 0, to: m, by: 1) {\n counter[i + 1] += counter[i]\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n var res = Array(repeating: 0, count: nums.count)\n for i in stride(from: nums.count - 1, through: 0, by: -1) {\n let num = nums[i]\n res[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in stride(from: 0, to: nums.count, by: 1) {\n nums[i] = res[i]\n }\n}\n
counting_sort.ts/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = 0;\n for (const num of nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n const counter = new Array(m + 1).fill(0);\n for (const num of nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (let i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n const n = nums.length;\n const res = new Array(n);\n for (let i = n - 1; i >= 0; i--) {\n const num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for (let i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n
counting_sort.dart/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = 0;\n for (const num of nums) {\n m = Math.max(m, num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n const counter: number[] = new Array<number>(m + 1).fill(0);\n for (const num of nums) {\n counter[num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (let i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n const n = nums.length;\n const res: number[] = new Array<number>(n);\n for (let i = n - 1; i >= 0; i--) {\n const num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for (let i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n
counting_sort.rs/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(List<int> nums) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int _num in nums) {\n m = max(m, _num);\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n List<int> counter = List.filled(m + 1, 0);\n for (int _num in nums) {\n counter[_num]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[_num]-1 \u662f _num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int n = nums.length;\n List<int> res = List.filled(n, 0);\n for (int i = n - 1; i >= 0; i--) {\n int _num = nums[i];\n res[counter[_num] - 1] = _num; // \u5c06 _num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[_num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e _num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n nums.setAll(0, res);\n}\n
counting_sort.c/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfn counting_sort(nums: &mut [i32]) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n let m = *nums.into_iter().max().unwrap();\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n let mut counter = vec![0; m as usize + 1];\n for &num in &*nums {\n counter[num as usize] += 1;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for i in 0..m as usize {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n let n = nums.len();\n let mut res = vec![0; n];\n for i in (0..n).rev() {\n let num = nums[i];\n res[counter[num as usize] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num as usize] -= 1; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in 0..n {\n nums[i] = res[i];\n }\n}\n
counting_sort.zig/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n int m = 0;\n for (int i = 0; i < size; i++) {\n if (nums[i] > m) {\n m = nums[i];\n }\n }\n // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n int *counter = calloc(m, sizeof(int));\n for (int i = 0; i < size; i++) {\n counter[nums[i]]++;\n }\n // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n for (int i = 0; i < m; i++) {\n counter[i + 1] += counter[i];\n }\n // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n int *res = malloc(sizeof(int) * size);\n for (int i = size - 1; i >= 0; i--) {\n int num = nums[i];\n res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n }\n // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n memcpy(nums, res, size * sizeof(int));\n // 5. \u91ca\u653e\u5185\u5b58\n free(counter);\n}\n
"},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"[class]{}-[func]{countingSort}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386
nums\u548c\u904d\u5386counter\uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u4e8e \\(O(n)\\) \u3002 - \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u501f\u52a9\u4e86\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \u7684\u6570\u7ec4
res\u548ccounter\u3002 - \u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5411
res\u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386nums\u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u7a33\u5b9a\u6392\u5e8f\u3002\u5b9e\u9645\u4e0a\uff0c\u6b63\u5e8f\u904d\u5386nums\u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u662f\u975e\u7a33\u5b9a\u7684\u3002
\u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u4ec5\u901a\u8fc7\u7edf\u8ba1\u6570\u91cf\u5c31\u53ef\u4ee5\u5b9e\u73b0\u9ad8\u6548\u7684\u6392\u5e8f\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\u76f8\u5bf9\u8f83\u4e3a\u4e25\u683c\u3002
\u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u5c06\u5176\u7528\u4e8e\u5176\u4ed6\u7c7b\u578b\u7684\u6570\u636e\uff0c\u9700\u8981\u786e\u4fdd\u8fd9\u4e9b\u6570\u636e\u53ef\u4ee5\u8f6c\u6362\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u5728\u8f6c\u6362\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u3002
\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u4f1a\u5360\u7528\u8fc7\u591a\u7a7a\u95f4\u3002\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002
"},{"location":"chapter_sorting/heap_sort/","title":"11.7 \u00a0 \u5806\u6392\u5e8f","text":"Tip
\u9605\u8bfb\u672c\u8282\u524d\uff0c\u8bf7\u786e\u4fdd\u5df2\u5b66\u5b8c\u201c\u5806\u201c\u7ae0\u8282\u3002
\u300c\u5806\u6392\u5e8f heap sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5806\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\u7684\u9ad8\u6548\u6392\u5e8f\u7b97\u6cd5\u3002\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5df2\u7ecf\u5b66\u8fc7\u7684\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\u5806\u6392\u5e8f\u3002
- \u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5c0f\u9876\u5806\uff0c\u6b64\u65f6\u6700\u5c0f\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002
- \u4e0d\u65ad\u6267\u884c\u51fa\u5806\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8bb0\u5f55\u51fa\u5806\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002
\u4ee5\u4e0a\u65b9\u6cd5\u867d\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u989d\u5916\u6570\u7ec4\u6765\u4fdd\u5b58\u5f39\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f83\u6d6a\u8d39\u7a7a\u95f4\u3002\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u4e00\u79cd\u66f4\u52a0\u4f18\u96c5\u7684\u5b9e\u73b0\u65b9\u5f0f\u3002
"},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5806\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-12 \u6240\u793a\u3002
- \u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5927\u9876\u5806\u3002\u5b8c\u6210\u540e\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002
- \u5c06\u5806\u9876\u5143\u7d20\uff08\u7b2c\u4e00\u4e2a\u5143\u7d20\uff09\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u6700\u540e\u4e00\u4e2a\u5143\u7d20\uff09\u4ea4\u6362\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u5806\u7684\u957f\u5ea6\u51cf \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u52a0 \\(1\\) \u3002
- \u4ece\u5806\u9876\u5143\u7d20\u5f00\u59cb\uff0c\u4ece\u9876\u5230\u5e95\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff08Sift Down\uff09\u3002\u5b8c\u6210\u5806\u5316\u540e\uff0c\u5806\u7684\u6027\u8d28\u5f97\u5230\u4fee\u590d\u3002
- \u5faa\u73af\u6267\u884c\u7b2c
2.\u6b65\u548c\u7b2c3.\u6b65\u3002\u5faa\u73af \\(n - 1\\) \u8f6e\u540e\uff0c\u5373\u53ef\u5b8c\u6210\u6570\u7ec4\u6392\u5e8f\u3002
Tip
\u5b9e\u9645\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c
<1><2><3><4><5><6><7><8><9><10><11><12>2.\u6b65\u548c\u7b2c3.\u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u4e2a\u5f39\u51fa\u5143\u7d20\u7684\u6b65\u9aa4\u3002\u56fe 11-12 \u00a0 \u5806\u6392\u5e8f\u6b65\u9aa4
\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e86\u4e0e\u201c\u5806\u201d\u7ae0\u8282\u76f8\u540c\u7684\u4ece\u9876\u81f3\u5e95\u5806\u5316
PythonC++JavaC#GoSwiftJSTSDartRustCZig heap_sort.pysift_down()\u51fd\u6570\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u5806\u7684\u957f\u5ea6\u4f1a\u968f\u7740\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u51cf\u5c0f\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7ed9sift_down()\u51fd\u6570\u6dfb\u52a0\u4e00\u4e2a\u957f\u5ea6\u53c2\u6570 \\(n\\) \uff0c\u7528\u4e8e\u6307\u5b9a\u5806\u7684\u5f53\u524d\u6709\u6548\u957f\u5ea6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
heap_sort.cppdef sift_down(nums: list[int], n: int, i: int):\n \"\"\"\u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n while True:\n # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n l = 2 * i + 1\n r = 2 * i + 2\n ma = i\n if l < n and nums[l] > nums[ma]:\n ma = l\n if r < n and nums[r] > nums[ma]:\n ma = r\n # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i:\n break\n # \u4ea4\u6362\u4e24\u8282\u70b9\n nums[i], nums[ma] = nums[ma], nums[i]\n # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n\ndef heap_sort(nums: list[int]):\n \"\"\"\u5806\u6392\u5e8f\"\"\"\n # \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in range(len(nums) // 2 - 1, -1, -1):\n sift_down(nums, len(nums), i)\n # \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for i in range(len(nums) - 1, 0, -1):\n # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n nums[0], nums[i] = nums[i], nums[0]\n # \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n sift_down(nums, i, 0)\n
heap_sort.java/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(vector<int> &nums, int n, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma])\n ma = l;\n if (r < n && nums[r] > nums[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n swap(nums[i], nums[ma]);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(vector<int> &nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = nums.size() / 2 - 1; i >= 0; --i) {\n siftDown(nums, nums.size(), i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = nums.size() - 1; i > 0; --i) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n swap(nums[0], nums[i]);\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.cs/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma])\n ma = l;\n if (r < n && nums[r] > nums[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i)\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n int temp = nums[i];\n nums[i] = nums[ma];\n nums[ma] = temp;\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = nums.length / 2 - 1; i >= 0; i--) {\n siftDown(nums, nums.length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = nums.length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n int tmp = nums[0];\n nums[0] = nums[i];\n nums[i] = tmp;\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.go/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int[] nums, int n, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma])\n ma = l;\n if (r < n && nums[r] > nums[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i)\n break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n (nums[ma], nums[i]) = (nums[i], nums[ma]);\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid HeapSort(int[] nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = nums.Length / 2 - 1; i >= 0; i--) {\n SiftDown(nums, nums.Length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = nums.Length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n (nums[i], nums[0]) = (nums[0], nums[i]);\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n SiftDown(nums, i, 0);\n }\n}\n
heap_sort.swift/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\n for true {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n l := 2*i + 1\n r := 2*i + 2\n ma := i\n if l < n && (*nums)[l] > (*nums)[ma] {\n ma = l\n }\n if r < n && (*nums)[r] > (*nums)[ma] {\n ma = r\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i := len(*nums)/2 - 1; i >= 0; i-- {\n siftDown(nums, len(*nums), i)\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for i := len(*nums) - 1; i > 0; i-- {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0)\n }\n}\n
heap_sort.js/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n var i = i\n while true {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = 2 * i + 1\n let r = 2 * i + 2\n var ma = i\n if l < n, nums[l] > nums[ma] {\n ma = l\n }\n if r < n, nums[r] > nums[ma] {\n ma = r\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n nums.swapAt(i, ma)\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n siftDown(nums: &nums, n: nums.count, i: i)\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for i in stride(from: nums.count - 1, to: 0, by: -1) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n nums.swapAt(0, i)\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums: &nums, n: i, i: 0)\n }\n}\n
heap_sort.ts/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums, n, i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = 2 * i + 1;\n let r = 2 * i + 2;\n let ma = i;\n if (l < n && nums[l] > nums[ma]) {\n ma = l;\n }\n if (r < n && nums[r] > nums[ma]) {\n ma = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma === i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n [nums[i], nums[ma]] = [nums[ma], nums[i]];\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (let i = Math.floor(nums.length / 2) - 1; i >= 0; i--) {\n siftDown(nums, nums.length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (let i = nums.length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n [nums[0], nums[i]] = [nums[i], nums[0]];\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.dart/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = 2 * i + 1;\n let r = 2 * i + 2;\n let ma = i;\n if (l < n && nums[l] > nums[ma]) {\n ma = l;\n }\n if (r < n && nums[r] > nums[ma]) {\n ma = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma === i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n [nums[i], nums[ma]] = [nums[ma], nums[i]];\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (let i = Math.floor(nums.length / 2) - 1; i >= 0; i--) {\n siftDown(nums, nums.length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (let i = nums.length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n [nums[0], nums[i]] = [nums[i], nums[0]];\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.rs/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(List<int> nums, int n, int i) {\n while (true) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma]) ma = l;\n if (r < n && nums[r] > nums[ma]) ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) break;\n // \u4ea4\u6362\u4e24\u8282\u70b9\n int temp = nums[i];\n nums[i] = nums[ma];\n nums[ma] = temp;\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(List<int> nums) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = nums.length ~/ 2 - 1; i >= 0; i--) {\n siftDown(nums, nums.length, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = nums.length - 1; i > 0; i--) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n int tmp = nums[0];\n nums[0] = nums[i];\n nums[i] = tmp;\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
heap_sort.c/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(nums: &mut [i32], n: usize, mut i: usize) {\n loop {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n let l = 2 * i + 1;\n let r = 2 * i + 2;\n let mut ma = i;\n if l < n && nums[l] > nums[ma] {\n ma = l;\n }\n if r < n && nums[r] > nums[ma] {\n ma = r;\n }\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if ma == i {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n let temp = nums[i];\n nums[i] = nums[ma];\n nums[ma] = temp;\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nfn heap_sort(nums: &mut [i32]) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for i in (0..=nums.len() / 2 - 1).rev() {\n sift_down(nums, nums.len(), i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for i in (1..=nums.len() - 1).rev() {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n let tmp = nums[0];\n nums[0] = nums[i];\n nums[i] = tmp;\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n sift_down(nums, i, 0);\n }\n}\n
heap_sort.zig/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int nums[], int n, int i) {\n while (1) {\n // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n int l = 2 * i + 1;\n int r = 2 * i + 2;\n int ma = i;\n if (l < n && nums[l] > nums[ma])\n ma = l;\n if (r < n && nums[r] > nums[ma])\n ma = r;\n // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n if (ma == i) {\n break;\n }\n // \u4ea4\u6362\u4e24\u8282\u70b9\n int temp = nums[i];\n nums[i] = nums[ma];\n nums[ma] = temp;\n // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n i = ma;\n }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n for (int i = n / 2 - 1; i >= 0; --i) {\n siftDown(nums, n, i);\n }\n // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n for (int i = n - 1; i > 0; --i) {\n // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n int tmp = nums[0];\n nums[0] = nums[i];\n nums[i] = tmp;\n // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n siftDown(nums, i, 0);\n }\n}\n
"},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"[class]{}-[func]{siftDown}\n\n[class]{}-[func]{heapSort}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5efa\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5171\u5faa\u73af \\(n - 1\\) \u8f6e\u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\u3002\u5143\u7d20\u4ea4\u6362\u548c\u5806\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\u3002
- \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5806\u9876\u5143\u7d20\u548c\u5806\u5e95\u5143\u7d20\u65f6\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002
\u300c\u63d2\u5165\u6392\u5e8f insertion sort\u300d\u662f\u4e00\u79cd\u7b80\u5355\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u4e0e\u624b\u52a8\u6574\u7406\u4e00\u526f\u724c\u7684\u8fc7\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002
\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u5728\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u4e00\u4e2a\u57fa\u51c6\u5143\u7d20\uff0c\u5c06\u8be5\u5143\u7d20\u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f83\u5927\u5c0f\uff0c\u5e76\u5c06\u8be5\u5143\u7d20\u63d2\u5165\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u3002
\u56fe 11-6 \u5c55\u793a\u4e86\u6570\u7ec4\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u6d41\u7a0b\u3002\u8bbe\u57fa\u51c6\u5143\u7d20\u4e3a
base\uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230base\u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u5c06base\u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002\u56fe 11-6 \u00a0 \u5355\u6b21\u63d2\u5165\u64cd\u4f5c
"},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u63d2\u5165\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-7 \u6240\u793a\u3002
- \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6570\u7ec4\u7684\u7b2c 1 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
- \u9009\u53d6\u6570\u7ec4\u7684\u7b2c 2 \u4e2a\u5143\u7d20\u4f5c\u4e3a
base\uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002 - \u9009\u53d6\u7b2c 3 \u4e2a\u5143\u7d20\u4f5c\u4e3a
base\uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 3 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002 - \u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u6700\u540e\u4e00\u8f6e\u4e2d\uff0c\u9009\u53d6\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a
base\uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002
\u56fe 11-7 \u00a0 \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b
\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig insertion_sort.py
insertion_sort.cppdef insertion_sort(nums: list[int]):\n \"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n # \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n for i in range(1, len(nums)):\n base = nums[i]\n j = i - 1\n # \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n while j >= 0 and nums[j] > base:\n nums[j + 1] = nums[j] # \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j -= 1\n nums[j + 1] = base # \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n
insertion_sort.java/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector<int> &nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < nums.size(); i++) {\n int base = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.cs/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < nums.length; i++) {\n int base = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.go/* \u63d2\u5165\u6392\u5e8f */\nvoid InsertionSort(int[] nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < nums.Length; i++) {\n int bas = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > bas) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = bas; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.swift/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n for i := 1; i < len(nums); i++ {\n base := nums[i]\n j := i - 1\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n for j >= 0 && nums[j] > base {\n nums[j+1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--\n }\n nums[j+1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.js/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for i in stride(from: 1, to: nums.count, by: 1) {\n let base = nums[i]\n var j = i - 1\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while j >= 0, nums[j] > base {\n nums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j -= 1\n }\n nums[j + 1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.ts/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (let i = 1; i < nums.length; i++) {\n let base = nums[i],\n j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.dart/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (let i = 1; i < nums.length; i++) {\n const base = nums[i];\n let j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.rs/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List<int> nums) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < nums.length; i++) {\n int base = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j--;\n }\n nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.c/* \u63d2\u5165\u6392\u5e8f */\nfn insertion_sort(nums: &mut [i32]) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for i in 1..nums.len() {\n let (base, mut j) = (nums[i], (i - 1) as i32);\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while j >= 0 && nums[j as usize] > base {\n nums[(j + 1) as usize] = nums[j as usize]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n j -= 1;\n }\n nums[(j + 1) as usize] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n
insertion_sort.zig/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n for (int i = 1; i < size; i++) {\n int base = nums[i], j = i - 1;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 0 && nums[j] > base) {\n // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n nums[j + 1] = nums[j];\n j--;\n }\n // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n nums[j + 1] = base;\n }\n}\n
"},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"// \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n var i: usize = 1;\n while (i < nums.len) : (i += 1) {\n var base = nums[i];\n var j: usize = i;\n // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n while (j >= 1 and nums[j - 1] > base) : (j -= 1) {\n nums[j] = nums[j - 1]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n }\n nums[j] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n }\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u522b\u9700\u8981\u5faa\u73af \\(n - 1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\((n - 1) n / 2\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u6570\u636e\u65f6\uff0c\u63d2\u5165\u64cd\u4f5c\u4f1a\u63d0\u524d\u7ec8\u6b62\u3002\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u63d2\u5165\u6392\u5e8f\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f1a\u5c06\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\uff0c\u4e0d\u4f1a\u6539\u53d8\u5b83\u4eec\u7684\u987a\u5e8f\u3002
\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u800c\u6211\u4eec\u5373\u5c06\u5b66\u4e60\u7684\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u5c3d\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u9ad8\uff0c\u4f46\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u7684\u60c5\u51b5\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002
\u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u7684\u9002\u7528\u60c5\u51b5\u7684\u7ed3\u8bba\u7c7b\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u8fd9\u7c7b \\(O(n \\log n)\\) \u7684\u7b97\u6cd5\u5c5e\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6570\u503c\u6bd4\u8f83\u63a5\u8fd1\uff0c\u590d\u6742\u5ea6\u4e0d\u5360\u4e3b\u5bfc\u5730\u4f4d\uff1b\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u64cd\u4f5c\u6570\u91cf\u8d77\u5230\u51b3\u5b9a\u6027\u4f5c\u7528\u3002
\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\u91c7\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u4e3a\uff1a\u5bf9\u4e8e\u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5bf9\u4e8e\u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002
\u867d\u7136\u5192\u6ce1\u6392\u5e8f\u3001\u9009\u62e9\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u9891\u7387\u663e\u8457\u9ad8\u4e8e\u5192\u6ce1\u6392\u5e8f\u548c\u9009\u62e9\u6392\u5e8f\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002
- \u5192\u6ce1\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u4ea4\u6362\u5b9e\u73b0\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\uff0c\u5171\u6d89\u53ca 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u8d4b\u503c\u5b9e\u73b0\uff0c\u4ec5\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002
- \u9009\u62e9\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002\u5982\u679c\u7ed9\u5b9a\u4e00\u7ec4\u90e8\u5206\u6709\u5e8f\u7684\u6570\u636e\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9009\u62e9\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002
- \u9009\u62e9\u6392\u5e8f\u4e0d\u7a33\u5b9a\uff0c\u65e0\u6cd5\u5e94\u7528\u4e8e\u591a\u7ea7\u6392\u5e8f\u3002
\u300c\u5f52\u5e76\u6392\u5e8f merge sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5305\u542b\u56fe 11-10 \u6240\u793a\u7684\u201c\u5212\u5206\u201d\u548c\u201c\u5408\u5e76\u201d\u9636\u6bb5\u3002
- \u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u6362\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002
- \u5408\u5e76\u9636\u6bb5\uff1a\u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u5212\u5206\uff0c\u5f00\u59cb\u5408\u5e76\uff0c\u6301\u7eed\u5730\u5c06\u5de6\u53f3\u4e24\u4e2a\u8f83\u77ed\u7684\u6709\u5e8f\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u8f83\u957f\u7684\u6709\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u7ed3\u675f\u3002
\u56fe 11-10 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5
"},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u5982\u56fe 11-11 \u6240\u793a\uff0c\u201c\u5212\u5206\u9636\u6bb5\u201d\u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002
- \u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9
mid\uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4[left, mid]\uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4[mid + 1, right]\uff09\u3002 - \u9012\u5f52\u6267\u884c\u6b65\u9aa4
1.\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u3002
\u201c\u5408\u5e76\u9636\u6bb5\u201d\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u5408\u5e76\u9636\u6bb5\u4e2d\u7684\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002
<1><2><3><4><5><6><7><8><9><10>\u56fe 11-11 \u00a0 \u5f52\u5e76\u6392\u5e8f\u6b65\u9aa4
\u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u4e0e\u4e8c\u53c9\u6811\u540e\u5e8f\u904d\u5386\u7684\u9012\u5f52\u987a\u5e8f\u662f\u4e00\u81f4\u7684\u3002
- \u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6811\uff0c\u6700\u540e\u5904\u7406\u6839\u8282\u70b9\u3002
- \u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6570\u7ec4\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6570\u7ec4\uff0c\u6700\u540e\u5904\u7406\u5408\u5e76\u3002
\u5f52\u5e76\u6392\u5e8f\u7684\u5b9e\u73b0\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\u3002\u8bf7\u6ce8\u610f\uff0c
PythonC++JavaC#GoSwiftJSTSDartRustCZig merge_sort.pynums\u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a[left, right]\uff0c\u800ctmp\u7684\u5bf9\u5e94\u533a\u95f4\u4e3a[0, right - left]\u3002
merge_sort.cppdef merge(nums: list[int], left: int, mid: int, right: int):\n \"\"\"\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\"\"\"\n # \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n # \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n tmp = [0] * (right - left + 1)\n # \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n i, j, k = left, mid + 1, 0\n # \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid and j <= right:\n if nums[i] <= nums[j]:\n tmp[k] = nums[i]\n i += 1\n else:\n tmp[k] = nums[j]\n j += 1\n k += 1\n # \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid:\n tmp[k] = nums[i]\n i += 1\n k += 1\n while j <= right:\n tmp[k] = nums[j]\n j += 1\n k += 1\n # \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for k in range(0, len(tmp)):\n nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n \"\"\"\u5f52\u5e76\u6392\u5e8f\"\"\"\n # \u7ec8\u6b62\u6761\u4ef6\n if left >= right:\n return # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n # \u5212\u5206\u9636\u6bb5\n mid = (left + right) // 2 # \u8ba1\u7b97\u4e2d\u70b9\n merge_sort(nums, left, mid) # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n merge_sort(nums, mid + 1, right) # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n # \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right)\n
merge_sort.java/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(vector<int> &nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n vector<int> tmp(right - left + 1);\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j])\n tmp[k++] = nums[i++];\n else\n tmp[k++] = nums[j++];\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.size(); k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector<int> &nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right)\n return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.cs/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int[] nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n int[] tmp = new int[right - left + 1];\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j])\n tmp[k++] = nums[i++];\n else\n tmp[k++] = nums[j++];\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.length; k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right)\n return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.go/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n int[] tmp = new int[right - left + 1];\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j])\n tmp[k++] = nums[i++];\n else\n tmp[k++] = nums[j++];\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.Length; ++k) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid MergeSort(int[] nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n MergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n MergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n Merge(nums, left, mid, right);\n}\n
merge_sort.swift/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums []int, left, mid, right int) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n tmp := make([]int, right-left+1)\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n i, j, k := left, mid+1, 0\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n for i <= mid && j <= right {\n if nums[i] <= nums[j] {\n tmp[k] = nums[i]\n i++\n } else {\n tmp[k] = nums[j]\n j++\n }\n k++\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n for i <= mid {\n tmp[k] = nums[i]\n i++\n k++\n }\n for j <= right {\n tmp[k] = nums[j]\n j++\n k++\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for k := 0; k < len(tmp); k++ {\n nums[left+k] = tmp[k]\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n // \u7ec8\u6b62\u6761\u4ef6\n if left >= right {\n return\n }\n // \u5212\u5206\u9636\u6bb5\n mid := (left + right) / 2\n mergeSort(nums, left, mid)\n mergeSort(nums, mid+1, right)\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right)\n}\n
merge_sort.js/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n var tmp = Array(repeating: 0, count: right - left + 1)\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n var i = left, j = mid + 1, k = 0\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid, j <= right {\n if nums[i] <= nums[j] {\n tmp[k] = nums[i]\n i += 1\n k += 1\n } else {\n tmp[k] = nums[j]\n j += 1\n k += 1\n }\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid {\n tmp[k] = nums[i]\n i += 1\n k += 1\n }\n while j <= right {\n tmp[k] = nums[j]\n j += 1\n k += 1\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for k in tmp.indices {\n nums[left + k] = tmp[k]\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n // \u7ec8\u6b62\u6761\u4ef6\n if left >= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n return\n }\n // \u5212\u5206\u9636\u6bb5\n let mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums: &nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums: &nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums: &nums, left: left, mid: mid, right: right)\n}\n
merge_sort.ts/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums, left, mid, right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n const tmp = new Array(right - left + 1);\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n let i = left,\n j = mid + 1,\n k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j]) {\n tmp[k++] = nums[i++];\n } else {\n tmp[k++] = nums[j++];\n }\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.length; k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.dart/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n const tmp = new Array(right - left + 1);\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n let i = left,\n j = mid + 1,\n k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j]) {\n tmp[k++] = nums[i++];\n } else {\n tmp[k++] = nums[j++];\n }\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.length; k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.rs/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(List<int> nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n List<int> tmp = List.filled(right - left + 1, 0);\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j])\n tmp[k++] = nums[i++];\n else\n tmp[k++] = nums[j++];\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmp.length; k++) {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(List<int> nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.c/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfn merge(nums: &mut [i32], left: usize, mid: usize, right: usize) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n let tmp_size = right - left + 1;\n let mut tmp = vec![0; tmp_size];\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n let (mut i, mut j, mut k) = (left, mid + 1, 0);\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid && j <= right {\n if nums[i] <= nums[j] {\n tmp[k] = nums[j];\n i += 1;\n } else {\n tmp[k] = nums[j];\n j += 1;\n }\n k += 1;\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while i <= mid {\n tmp[k] = nums[i];\n k += 1;\n i += 1;\n }\n while j <= right {\n tmp[k] = nums[j];\n k += 1;\n j += 1;\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for k in 0..tmp_size {\n nums[left + k] = tmp[k];\n }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfn merge_sort(nums: &mut [i32], left: usize, right: usize) {\n // \u7ec8\u6b62\u6761\u4ef6\n if left >= right { return; } // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n let mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n merge_sort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n merge_sort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
merge_sort.zig/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int *nums, int left, int mid, int right) {\n // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n int tmpSize = right - left + 1;\n int *tmp = (int *)malloc(tmpSize * sizeof(int));\n // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n int i = left, j = mid + 1, k = 0;\n // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid && j <= right) {\n if (nums[i] <= nums[j]) {\n tmp[k++] = nums[i++];\n } else {\n tmp[k++] = nums[j++];\n }\n }\n // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n while (i <= mid) {\n tmp[k++] = nums[i++];\n }\n while (j <= right) {\n tmp[k++] = nums[j++];\n }\n // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n for (k = 0; k < tmpSize; ++k) {\n nums[left + k] = tmp[k];\n }\n // \u91ca\u653e\u5185\u5b58\n free(tmp);\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right)\n return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n merge(nums, left, mid, right);\n}\n
"},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"// \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n // \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\n var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n defer mem_arena.deinit();\n const mem_allocator = mem_arena.allocator();\n var tmp = try mem_allocator.alloc(i32, right + 1 - left);\n std.mem.copy(i32, tmp, nums[left..right+1]);\n // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15 \n var leftStart = left - left;\n var leftEnd = mid - left;\n // \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15 \n var rightStart = mid + 1 - left;\n var rightEnd = right - left;\n // i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\n var i = leftStart;\n var j = rightStart;\n // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n var k = left;\n while (k <= right) : (k += 1) {\n // \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n if (i > leftEnd) {\n nums[k] = tmp[j];\n j += 1;\n // \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n } else if (j > rightEnd or tmp[i] <= tmp[j]) {\n nums[k] = tmp[i];\n i += 1;\n // \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n } else {\n nums[k] = tmp[j];\n j += 1;\n }\n }\n}\n\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n // \u7ec8\u6b62\u6761\u4ef6\n if (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n // \u5212\u5206\u9636\u6bb5\n var mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n try mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n try mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n // \u5408\u5e76\u9636\u6bb5\n try merge(nums, left, mid, right);\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5212\u5206\u4ea7\u751f\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002\u5408\u5e76\u64cd\u4f5c\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u53d8\u3002
\u5bf9\u4e8e\u94fe\u8868\uff0c\u5f52\u5e76\u6392\u5e8f\u76f8\u8f83\u4e8e\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\u5177\u6709\u663e\u8457\u4f18\u52bf\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u6392\u5e8f\u4efb\u52a1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(1)\\) \u3002
- \u5212\u5206\u9636\u6bb5\uff1a\u53ef\u4ee5\u4f7f\u7528\u201c\u8fed\u4ee3\u201d\u66ff\u4ee3\u201c\u9012\u5f52\u201d\u6765\u5b9e\u73b0\u94fe\u8868\u5212\u5206\u5de5\u4f5c\uff0c\u4ece\u800c\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\u3002
- \u5408\u5e76\u9636\u6bb5\uff1a\u5728\u94fe\u8868\u4e2d\uff0c\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u4ec5\u9700\u6539\u53d8\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u5b9e\u73b0\uff0c\u56e0\u6b64\u5408\u5e76\u9636\u6bb5\uff08\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\uff09\u65e0\u987b\u521b\u5efa\u989d\u5916\u94fe\u8868\u3002
\u5177\u4f53\u5b9e\u73b0\u7ec6\u8282\u6bd4\u8f83\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u76f8\u5173\u8d44\u6599\u8fdb\u884c\u5b66\u4e60\u3002
"},{"location":"chapter_sorting/quick_sort/","title":"11.5 \u00a0 \u5feb\u901f\u6392\u5e8f","text":"\u300c\u5feb\u901f\u6392\u5e8f quick sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u8fd0\u884c\u9ad8\u6548\uff0c\u5e94\u7528\u5e7f\u6cdb\u3002
\u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5176\u76ee\u6807\u662f\uff1a\u9009\u62e9\u6570\u7ec4\u4e2d\u7684\u67d0\u4e2a\u5143\u7d20\u4f5c\u4e3a\u201c\u57fa\u51c6\u6570\u201d\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u4fa7\uff0c\u800c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u4fa7\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u54e8\u5175\u5212\u5206\u7684\u6d41\u7a0b\u5982\u56fe 11-8 \u6240\u793a\u3002
- \u9009\u53d6\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488
i\u548cj\u5206\u522b\u6307\u5411\u6570\u7ec4\u7684\u4e24\u7aef\u3002 - \u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u4f7f\u7528
i\uff08j\uff09\u5206\u522b\u5bfb\u627e\u7b2c\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u540e\u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\u3002 - \u5faa\u73af\u6267\u884c\u6b65\u9aa4
2.\uff0c\u76f4\u5230i\u548cj\u76f8\u9047\u65f6\u505c\u6b62\uff0c\u6700\u540e\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\u3002
\u56fe 11-8 \u00a0 \u54e8\u5175\u5212\u5206\u6b65\u9aa4
\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u6570\u7ec4\u3001\u57fa\u51c6\u6570\u3001\u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3\u201c\u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u51c6\u6570 \\(\\leq\\) \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u53ea\u9700\u5bf9\u8fd9\u4e24\u4e2a\u5b50\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\u3002
\u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565
\u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06\u4e00\u4e2a\u8f83\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u7b80\u5316\u4e3a\u4e24\u4e2a\u8f83\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py
quick_sort.cppdef partition(self, nums: list[int], left: int, right: int) -> int:\n \"\"\"\u54e8\u5175\u5212\u5206\"\"\"\n # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n i, j = left, right\n while i < j:\n while i < j and nums[j] >= nums[left]:\n j -= 1 # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while i < j and nums[i] <= nums[left]:\n i += 1 # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n # \u5143\u7d20\u4ea4\u6362\n nums[i], nums[j] = nums[j], nums[i]\n # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n nums[i], nums[left] = nums[left], nums[i]\n return i # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
quick_sort.java/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector<int> &nums, int i, int j) {\n int tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector<int> &nums, int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.cs/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\n int tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.go/* \u5143\u7d20\u4ea4\u6362 */\nvoid Swap(int[] nums, int i, int j) {\n (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint Partition(int[] nums, int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n Swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.swift/* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n i, j := left, right\n for i < j {\n for i < j && nums[j] >= nums[left] {\n j-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n for i < j && nums[i] <= nums[left] {\n i++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n // \u5143\u7d20\u4ea4\u6362\n nums[i], nums[j] = nums[j], nums[i]\n }\n // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n nums[i], nums[left] = nums[left], nums[i]\n return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.js/* \u5143\u7d20\u4ea4\u6362 */\nfunc swap(nums: inout [Int], i: Int, j: Int) {\n let tmp = nums[i]\n nums[i] = nums[j]\n nums[j] = tmp\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -> Int {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n var i = left\n var j = right\n while i < j {\n while i < j, nums[j] >= nums[left] {\n j -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while i < j, nums[i] <= nums[left] {\n i += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n swap(nums: &nums, i: i, j: j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums: &nums, i: i, j: left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.ts/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\n let tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let i = left,\n j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while (i < j && nums[i] <= nums[left]) {\n i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n // \u5143\u7d20\u4ea4\u6362\n this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.dart/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\n let tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let i = left,\n j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while (i < j && nums[i] <= nums[left]) {\n i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n // \u5143\u7d20\u4ea4\u6362\n this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.rs/* \u5143\u7d20\u4ea4\u6362 */\nvoid _swap(List<int> nums, int i, int j) {\n int tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint _partition(List<int> nums, int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.c/* \u54e8\u5175\u5212\u5206 */\nfn partition(nums: &mut [i32], left: usize, right: usize) -> usize {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let (mut i, mut j) = (left, right);\n while i < j {\n while i < j && nums[j] >= nums[left] {\n j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while i < j && nums[i] <= nums[left] {\n i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n nums.swap(i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.zig/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int nums[], int i, int j) {\n int tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n j--;\n }\n while (i < j && nums[i] <= nums[left]) {\n // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n i++;\n }\n // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n swap(nums, i, j);\n }\n // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n swap(nums, i, left);\n // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n return i;\n}\n
"},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"// \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\n var tmp = nums[i];\n nums[i] = nums[j];\n nums[j] = tmp;\n}\n\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n var i = left;\n var j = right;\n while (i < j) {\n while (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n\u5feb\u901f\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-9 \u6240\u793a\u3002
- \u9996\u5148\uff0c\u5bf9\u539f\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u3002
- \u7136\u540e\uff0c\u5bf9\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5206\u522b\u9012\u5f52\u6267\u884c\u201c\u54e8\u5175\u5212\u5206\u201d\u3002
- \u6301\u7eed\u9012\u5f52\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\uff0c\u4ece\u800c\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\u3002
\u56fe 11-9 \u00a0 \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b
PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py
quick_sort.cppdef quick_sort(self, nums: list[int], left: int, right: int):\n \"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if left >= right:\n return\n # \u54e8\u5175\u5212\u5206\n pivot = self.partition(nums, left, right)\n # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n self.quick_sort(nums, left, pivot - 1)\n self.quick_sort(nums, pivot + 1, right)\n
quick_sort.java/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector<int> &nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right)\n return;\n // \u54e8\u5175\u5212\u5206\n int pivot = partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n
quick_sort.cs/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right)\n return;\n // \u54e8\u5175\u5212\u5206\n int pivot = partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n
quick_sort.go/* \u5feb\u901f\u6392\u5e8f */\nvoid QuickSort(int[] nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right)\n return;\n // \u54e8\u5175\u5212\u5206\n int pivot = Partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n QuickSort(nums, left, pivot - 1);\n QuickSort(nums, pivot + 1, right);\n}\n
quick_sort.swift/* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if left >= right {\n return\n }\n // \u54e8\u5175\u5212\u5206\n pivot := q.partition(nums, left, right)\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n q.quickSort(nums, left, pivot-1)\n q.quickSort(nums, pivot+1, right)\n}\n
quick_sort.js/* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if left >= right {\n return\n }\n // \u54e8\u5175\u5212\u5206\n let pivot = partition(nums: &nums, left: left, right: right)\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums: &nums, left: left, right: pivot - 1)\n quickSort(nums: &nums, left: pivot + 1, right: right)\n}\n
quick_sort.ts/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) return;\n // \u54e8\u5175\u5212\u5206\n const pivot = this.partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n this.quickSort(nums, left, pivot - 1);\n this.quickSort(nums, pivot + 1, right);\n}\n
quick_sort.dart/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) {\n return;\n }\n // \u54e8\u5175\u5212\u5206\n const pivot = this.partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n this.quickSort(nums, left, pivot - 1);\n this.quickSort(nums, pivot + 1, right);\n}\n
quick_sort.rs/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List<int> nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) return;\n // \u54e8\u5175\u5212\u5206\n int pivot = _partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n
quick_sort.c/* \u5feb\u901f\u6392\u5e8f */\npub fn quick_sort(left: i32, right: i32, nums: &mut [i32]) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if left >= right {\n return;\n }\n // \u54e8\u5175\u5212\u5206\n let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n Self::quick_sort(left, pivot - 1, nums);\n Self::quick_sort(pivot + 1, right, nums);\n}\n
quick_sort.zig/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n j--;\n }\n while (i < j && nums[i] <= nums[left]) {\n // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n i++;\n }\n // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n swap(nums, i, j);\n }\n // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n swap(nums, i, left);\n // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n return i;\n}\n\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u5feb\u901f\u6392\u5e8f\nvoid quickSort(int nums[], int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) {\n return;\n }\n // \u54e8\u5175\u5212\u5206\n int pivot = partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n
"},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"// \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n if (left >= right) return;\n // \u54e8\u5175\u5212\u5206\n var pivot = partition(nums, left, right);\n // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n quickSort(nums, left, pivot - 1);\n quickSort(nums, pivot + 1, right);\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
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\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u4f2a\u968f\u673a\u6570\u201d\u3002\u5982\u679c\u6211\u4eec\u9488\u5bf9\u4f2a\u968f\u673a\u6570\u5e8f\u5217\u6784\u5efa\u4e00\u4e2a\u7279\u5b9a\u7684\u6d4b\u8bd5\u6837\u4f8b\uff0c\u90a3\u4e48\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002
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\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py
quick_sort.cppdef median_three(self, nums: list[int], left: int, mid: int, right: int) -> int:\n \"\"\"\u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\"\"\"\n # \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n # \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if (nums[left] < nums[mid]) ^ (nums[left] < nums[right]):\n return left\n elif (nums[mid] < nums[left]) ^ (nums[mid] < nums[right]):\n return mid\n return right\n\ndef partition(self, nums: list[int], left: int, right: int) -> int:\n \"\"\"\u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\"\"\"\n # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n med = self.median_three(nums, left, (left + right) // 2, right)\n # \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n nums[left], nums[med] = nums[med], nums[left]\n # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n i, j = left, right\n while i < j:\n while i < j and nums[j] >= nums[left]:\n j -= 1 # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while i < j and nums[i] <= nums[left]:\n i += 1 # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n # \u5143\u7d20\u4ea4\u6362\n nums[i], nums[j] = nums[j], nums[i]\n # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n nums[i], nums[left] = nums[left], nums[i]\n return i # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
quick_sort.java/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector<int> &nums, int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector<int> &nums, int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = medianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.cs/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = medianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.go/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint Partition(int[] nums, int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = MedianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n Swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n Swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.swift/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\uff08!= \u5728\u8fd9\u91cc\u8d77\u5230\u5f02\u6216\u7684\u4f5c\u7528\uff09\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\n return left\n } else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\n return mid\n }\n return right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n med := q.medianThree(nums, left, (left+right)/2, right)\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n nums[left], nums[med] = nums[med], nums[left]\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n i, j := left, right\n for i < j {\n for i < j && nums[j] >= nums[left] {\n j-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n for i < j && nums[i] <= nums[left] {\n i++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n //\u5143\u7d20\u4ea4\u6362\n nums[i], nums[j] = nums[j], nums[i]\n }\n //\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n nums[i], nums[left] = nums[left], nums[i]\n return i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.js/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -> Int {\n if (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\n return left\n } else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\n return mid\n } else {\n return right\n }\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -> Int {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n let med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums: &nums, i: left, j: med)\n return partition(nums: &nums, left: left, right: right)\n}\n
quick_sort.ts/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right])) return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n let med = this.medianThree(\n nums,\n left,\n Math.floor((left + right) / 2),\n right\n );\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n this.swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let i = left,\n j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.dart/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(\n nums: number[],\n left: number,\n mid: number,\n right: number\n): number {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if (Number(nums[left] < nums[mid]) ^ Number(nums[left] < nums[right])) {\n return left;\n } else if (\n Number(nums[mid] < nums[left]) ^ Number(nums[mid] < nums[right])\n ) {\n return mid;\n } else {\n return right;\n }\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n let med = this.medianThree(\n nums,\n left,\n Math.floor((left + right) / 2),\n right\n );\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n this.swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let i = left,\n j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) {\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while (i < j && nums[i] <= nums[left]) {\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.rs/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint _medianThree(List<int> nums, int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint _partition(List<int> nums, int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n _swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.c/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfn median_three(nums: &mut [i32], left: usize, mid: usize, right: usize) -> usize {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if (nums[left] < nums[mid]) ^ (nums[left] < nums[right]) {\n return left;\n } else if (nums[mid] < nums[left]) ^ (nums[mid] < nums[right]) {\n return mid;\n } \n right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfn partition(nums: &mut [i32], left: usize, right: usize) -> usize {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n let med = Self::median_three(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n nums.swap(left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n let (mut i, mut j) = (left, right);\n while i < j {\n while i < j && nums[j] >= nums[left] {\n j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n while i < j && nums[i] <= nums[left] {\n i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n }\n nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n nums.swap(i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
quick_sort.zig/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\n return left;\n else if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\n return mid;\n else\n return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */ \nint partitionMedian(int nums[], int left, int right) {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n int med = medianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n int i = left, j = right;\n while (i < j) {\n while (i < j && nums[j] >= nums[left])\n j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j && nums[i] <= nums[left])\n i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
"},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5 \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n if ((nums[left] < nums[mid]) != (nums[left] < nums[right])) {\n return left;\n } else if ((nums[mid] < nums[left]) != (nums[mid] < nums[right])) {\n return mid;\n } else {\n return right;\n }\n}\n\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n var med = medianThree(nums, left, (left + right) / 2, right);\n // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n swap(nums, left, med);\n // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n var i = left;\n var j = right;\n while (i < j) {\n while (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n while (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n }\n swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u5360\u7528\u7a7a\u95f4\u8f83\u591a\u3002\u4ee5\u5b8c\u5168\u6709\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u8bbe\u9012\u5f52\u4e2d\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(m\\) \uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u4ea7\u751f\u957f\u5ea6\u4e3a \\(0\\) \u7684\u5de6\u5b50\u6570\u7ec4\u548c\u957f\u5ea6\u4e3a \\(m - 1\\) \u7684\u53f3\u5b50\u6570\u7ec4\uff0c\u8fd9\u610f\u5473\u7740\u6bcf\u4e00\u5c42\u9012\u5f52\u8c03\u7528\u51cf\u5c11\u7684\u95ee\u9898\u89c4\u6a21\u975e\u5e38\u5c0f\uff08\u53ea\u51cf\u5c11\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4f1a\u8fbe\u5230 \\(n - 1\\) \uff0c\u6b64\u65f6\u9700\u8981\u5360\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002
\u4e3a\u4e86\u9632\u6b62\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u6bd4\u8f83\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\uff0c\u4ec5\u5bf9\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u8fdb\u884c\u9012\u5f52\u3002\u7531\u4e8e\u8f83\u77ed\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(n / 2\\) \uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u80fd\u786e\u4fdd\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u4ece\u800c\u5c06\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py
quick_sort.cppdef quick_sort(self, nums: list[int], left: int, right: int):\n \"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\"\"\"\n # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while left < right:\n # \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n pivot = self.partition(nums, left, right)\n # \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if pivot - left < right - pivot:\n self.quick_sort(nums, left, pivot - 1) # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1 # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n else:\n self.quick_sort(nums, pivot + 1, right) # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1 # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n
quick_sort.java/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector<int> &nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.cs/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.go/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid QuickSort(int[] nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = Partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n QuickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n QuickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.swift/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n for left < right {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n pivot := q.partition(nums, left, right)\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if pivot-left < right-pivot {\n q.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n q.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.js/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n var left = left\n var right = right\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while left < right {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n let pivot = partition(nums: &nums, left: left, right: right)\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left) < (right - pivot) {\n quickSortTailCall(nums: &nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSortTailCall(nums: &nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.ts/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n let pivot = this.partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.dart/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n let pivot = this.partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.rs/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(List<int> nums, int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = _partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.c/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &mut [i32]) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while left < right {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if pivot - left < right - pivot {\n Self::quick_sort(left, pivot - 1, nums); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n Self::quick_sort(pivot + 1, right, nums); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
quick_sort.zig/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\n// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nvoid quickSortTailCall(int nums[], int left, int right) {\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n int pivot = partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSortTailCall(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSortTailCall(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n
"},{"location":"chapter_sorting/radix_sort/","title":"11.10 \u00a0 \u57fa\u6570\u6392\u5e8f","text":"// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\n var left = left_;\n var right = right_;\n // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n while (left < right) {\n // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n var pivot = partition(nums, left, right);\n // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n if (pivot - left < right - pivot) {\n quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n } else {\n quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n }\n }\n}\n\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u8ba1\u6570\u6392\u5e8f\uff0c\u5b83\u9002\u7528\u4e8e\u6570\u636e\u91cf \\(n\\) \u8f83\u5927\u4f46\u6570\u636e\u8303\u56f4 \\(m\\) \u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u5bf9 \\(n = 10^6\\) \u4e2a\u5b66\u53f7\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u5b66\u53f7\u662f\u4e00\u4e2a \\(8\\) \u4f4d\u6570\u5b57\uff0c\u8fd9\u610f\u5473\u7740\u6570\u636e\u8303\u56f4 \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u5185\u5b58\u7a7a\u95f4\uff0c\u800c\u57fa\u6570\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u3002
\u300c\u57fa\u6570\u6392\u5e8f radix sort\u300d\u7684\u6838\u5fc3\u601d\u60f3\u4e0e\u8ba1\u6570\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u901a\u8fc7\u7edf\u8ba1\u4e2a\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u57fa\u6570\u6392\u5e8f\u5229\u7528\u6570\u5b57\u5404\u4f4d\u4e4b\u95f4\u7684\u9012\u8fdb\u5173\u7cfb\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e00\u4f4d\u8fdb\u884c\u6392\u5e8f\uff0c\u4ece\u800c\u5f97\u5230\u6700\u7ec8\u7684\u6392\u5e8f\u7ed3\u679c\u3002
"},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"\u4ee5\u5b66\u53f7\u6570\u636e\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6570\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6570\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-18 \u6240\u793a\u3002
- \u521d\u59cb\u5316\u4f4d\u6570 \\(k = 1\\) \u3002
- \u5bf9\u5b66\u53f7\u7684\u7b2c \\(k\\) \u4f4d\u6267\u884c\u201c\u8ba1\u6570\u6392\u5e8f\u201d\u3002\u5b8c\u6210\u540e\uff0c\u6570\u636e\u4f1a\u6839\u636e\u7b2c \\(k\\) \u4f4d\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u3002
- \u5c06 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u540e\u8fd4\u56de\u6b65\u9aa4
2.\u7ee7\u7eed\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u540e\u7ed3\u675f\u3002
\u56fe 11-18 \u00a0 \u57fa\u6570\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b
\u4e0b\u9762\u5256\u6790\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e00\u4e2a \\(d\\) \u8fdb\u5236\u7684\u6570\u5b57 \\(x\\) \uff0c\u8981\u83b7\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8ba1\u7b97\u516c\u5f0f\uff1a
\\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\]\u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5bf9\u6d6e\u70b9\u6570 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\: d\\) \u8868\u793a\u5bf9 \\(d\\) \u53d6\u4f59\u3002\u5bf9\u4e8e\u5b66\u53f7\u6570\u636e\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002
\u6b64\u5916\uff0c\u6211\u4eec\u9700\u8981\u5c0f\u5e45\u6539\u52a8\u8ba1\u6570\u6392\u5e8f\u4ee3\u7801\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u636e\u6570\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u8fdb\u884c\u6392\u5e8f\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig radix_sort.py
radix_sort.cppdef digit(num: int, exp: int) -> int:\n \"\"\"\u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n # \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n \"\"\"\u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n # \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n counter = [0] * 10\n n = len(nums)\n # \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for i in range(n):\n d = digit(nums[i], exp) # \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d] += 1 # \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n # \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for i in range(1, 10):\n counter[i] += counter[i - 1]\n # \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n res = [0] * n\n for i in range(n - 1, -1, -1):\n d = digit(nums[i], exp)\n j = counter[d] - 1 # \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i] # \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d] -= 1 # \u5c06 d \u7684\u6570\u91cf\u51cf 1\n # \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in range(n):\n nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n \"\"\"\u57fa\u6570\u6392\u5e8f\"\"\"\n # \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n m = max(nums)\n # \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n exp = 1\n while exp <= m:\n # \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n # k = 1 -> exp = 1\n # k = 2 -> exp = 10\n # \u5373 exp = 10^(k-1)\n counting_sort_digit(nums, exp)\n exp *= 10\n
radix_sort.java/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector<int> &nums, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n vector<int> counter(10, 0);\n int n = nums.size();\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < n; i++) {\n int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n vector<int> res(n, 0);\n for (int i = n - 1; i >= 0; i--) {\n int d = digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++)\n nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(vector<int> &nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n int m = *max_element(nums.begin(), nums.end());\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; exp <= m; exp *= 10)\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n}\n
radix_sort.cs/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n int[] counter = new int[10];\n int n = nums.length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < n; i++) {\n int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n int[] res = new int[n];\n for (int i = n - 1; i >= 0; i--) {\n int d = digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++)\n nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n int m = Integer.MIN_VALUE;\n for (int num : nums)\n if (num > m)\n m = num;\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; exp <= m; exp *= 10)\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n}\n
radix_sort.go/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint Digit(int num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid CountingSortDigit(int[] nums, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n int[] counter = new int[10];\n int n = nums.Length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < n; i++) {\n int d = Digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n int[] res = new int[n];\n for (int i = n - 1; i >= 0; i--) {\n int d = Digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid RadixSort(int[] nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n int m = int.MinValue;\n foreach (int num in nums) {\n if (num > m) m = num;\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; exp <= m; exp *= 10) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n CountingSortDigit(nums, exp);\n }\n}\n
radix_sort.swift/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n counter := make([]int, 10)\n n := len(nums)\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for i := 0; i < n; i++ {\n d := digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++ // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for i := 1; i < 10; i++ {\n counter[i] += counter[i-1]\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n res := make([]int, n)\n for i := n - 1; i >= 0; i-- {\n d := digit(nums[i], exp)\n j := counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]-- // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for i := 0; i < n; i++ {\n nums[i] = res[i]\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums []int) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n max := math.MinInt\n for _, num := range nums {\n if num > max {\n max = num\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for exp := 1; max >= exp; exp *= 10 {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp)\n }\n}\n
radix_sort.js/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -> Int {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n var counter = Array(repeating: 0, count: 10)\n let n = nums.count\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for i in nums.indices {\n let d = digit(num: nums[i], exp: exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d] += 1 // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for i in 1 ..< 10 {\n counter[i] += counter[i - 1]\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n var res = Array(repeating: 0, count: n)\n for i in stride(from: n - 1, through: 0, by: -1) {\n let d = digit(num: nums[i], exp: exp)\n let j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d] -= 1 // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in nums.indices {\n nums[i] = res[i]\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n var m = Int.min\n for num in nums {\n if num > m {\n m = num\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for exp in sequence(first: 1, next: { m >= ($0 * 10) ? $0 * 10 : nil }) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums: &nums, exp: exp)\n }\n}\n
radix_sort.ts/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n const counter = new Array(10).fill(0);\n const n = nums.length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (let i = 0; i < n; i++) {\n const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (let i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n const res = new Array(n).fill(0);\n for (let i = n - 1; i >= 0; i--) {\n const d = digit(nums[i], exp);\n const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (let i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n let m = Number.MIN_VALUE;\n for (const num of nums) {\n if (num > m) {\n m = num;\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (let exp = 1; exp <= m; exp *= 10) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n }\n}\n
radix_sort.dart/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n const counter = new Array(10).fill(0);\n const n = nums.length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (let i = 0; i < n; i++) {\n const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (let i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n const res = new Array(n).fill(0);\n for (let i = n - 1; i >= 0; i--) {\n const d = digit(nums[i], exp);\n const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (let i = 0; i < n; i++) {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n let m = Number.MIN_VALUE;\n for (const num of nums) {\n if (num > m) {\n m = num;\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (let exp = 1; exp <= m; exp *= 10) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n }\n}\n
radix_sort.rs/* \u83b7\u53d6\u5143\u7d20 _num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int _num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (_num ~/ exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List<int> nums, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n List<int> counter = List<int>.filled(10, 0);\n int n = nums.length;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < n; i++) {\n int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n List<int> res = List<int>.filled(n, 0);\n for (int i = n - 1; i >= 0; i--) {\n int d = digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < n; i++) nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(List<int> nums) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n // dart \u4e2d int \u7684\u957f\u5ea6\u662f 64 \u4f4d\u7684\n int m = -1 << 63;\n for (int _num in nums) if (_num > m) m = _num;\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; exp <= m; exp *= 10)\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, exp);\n}\n
radix_sort.c/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -> usize {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return ((num / exp) % 10) as usize;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfn counting_sort_digit(nums: &mut [i32], exp: i32) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n let mut counter = [0; 10];\n let n = nums.len();\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for i in 0..n {\n let d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for i in 1..10 {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n let mut res = vec![0; n];\n for i in (0..n).rev() {\n let d = digit(nums[i], exp);\n let j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d] -= 1; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for i in 0..n {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfn radix_sort(nums: &mut [i32]) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n let m = *nums.into_iter().max().unwrap();\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n let mut exp = 1;\n while exp <= m {\n counting_sort_digit(nums, exp);\n exp *= 10;\n }\n}\n
radix_sort.zig/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n int *counter = (int *)malloc((sizeof(int) * 10));\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (int i = 0; i < size; i++) {\n // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n int d = digit(nums[i], exp);\n // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n counter[d]++;\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n for (int i = 1; i < 10; i++) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n int *res = (int *)malloc(sizeof(int) * size);\n for (int i = size - 1; i >= 0; i--) {\n int d = digit(nums[i], exp);\n int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n for (int i = 0; i < size; i++) {\n nums[i] = res[i];\n }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n int max = INT32_MIN;\n for (size_t i = 0; i < size - 1; i++) {\n if (nums[i] > max) {\n max = nums[i];\n }\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n for (int exp = 1; max >= exp; exp *= 10)\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n countingSortDigit(nums, size, exp);\n}\n// \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n return @mod(@divFloor(num, exp), 10);\n}\n\n// \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n // defer mem_arena.deinit();\n const mem_allocator = mem_arena.allocator();\n var counter = try mem_allocator.alloc(usize, 10);\n @memset(counter, 0);\n var n = nums.len;\n // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n for (nums) |num| {\n var d: u32 = @bitCast(digit(num, exp)); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n }\n // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n var i: usize = 1;\n while (i < 10) : (i += 1) {\n counter[i] += counter[i - 1];\n }\n // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n var res = try mem_allocator.alloc(i32, n);\n i = n - 1;\n while (i >= 0) : (i -= 1) {\n var d: u32 = @bitCast(digit(nums[i], exp));\n var j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n counter[d] -= 1; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n if (i == 0) break;\n }\n // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n i = 0;\n while (i < n) : (i += 1) {\n nums[i] = res[i];\n }\n}\n\n// \u57fa\u6570\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n var m: i32 = std.math.minInt(i32);\n for (nums) |num| {\n if (num > m) m = num;\n }\n // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n var exp: i32 = 1;\n while (exp <= m) : (exp *= 10) {\n // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n // k = 1 -> exp = 1\n // k = 2 -> exp = 10\n // \u5373 exp = 10^(k-1)\n try countingSortDigit(nums, exp); \n }\n} \n\u4e3a\u4ec0\u4e48\u4ece\u6700\u4f4e\u4f4d\u5f00\u59cb\u6392\u5e8f\uff1f
\u5728\u8fde\u7eed\u7684\u6392\u5e8f\u8f6e\u6b21\u4e2d\uff0c\u540e\u4e00\u8f6e\u6392\u5e8f\u4f1a\u8986\u76d6\u524d\u4e00\u8f6e\u6392\u5e8f\u7684\u7ed3\u679c\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u5982\u679c\u7b2c\u4e00\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a < b\\) \uff0c\u800c\u7b2c\u4e8c\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a > b\\) \uff0c\u90a3\u4e48\u7b2c\u4e8c\u8f6e\u7684\u7ed3\u679c\u5c06\u53d6\u4ee3\u7b2c\u4e00\u8f6e\u7684\u7ed3\u679c\u3002\u7531\u4e8e\u6570\u5b57\u7684\u9ad8\u4f4d\u4f18\u5148\u7ea7\u9ad8\u4e8e\u4f4e\u4f4d\uff0c\u56e0\u6b64\u5e94\u8be5\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002
"},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"\u76f8\u8f83\u4e8e\u8ba1\u6570\u6392\u5e8f\uff0c\u57fa\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u503c\u8303\u56f4\u8f83\u5927\u7684\u60c5\u51b5\uff0c\u4f46\u524d\u63d0\u662f\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6570\u4e0d\u80fd\u8fc7\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u70b9\u6570\u4e0d\u9002\u5408\u4f7f\u7528\u57fa\u6570\u6392\u5e8f\uff0c\u56e0\u4e3a\u5176\u4f4d\u6570 \\(k\\) \u8fc7\u5927\uff0c\u53ef\u80fd\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002
- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk)\\)\uff1a\u8bbe\u6570\u636e\u91cf\u4e3a \\(n\\)\u3001\u6570\u636e\u4e3a \\(d\\) \u8fdb\u5236\u3001\u6700\u5927\u4f4d\u6570\u4e3a \\(k\\) \uff0c\u5219\u5bf9\u67d0\u4e00\u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u65f6\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5bf9\u8f83\u5c0f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + d)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6570\u6392\u5e8f\u9700\u8981\u501f\u52a9\u957f\u5ea6\u4e3a \\(n\\) \u548c \\(d\\) \u7684\u6570\u7ec4
res\u548ccounter\u3002 - \u7a33\u5b9a\u6392\u5e8f\uff1a\u5f53\u8ba1\u6570\u6392\u5e8f\u7a33\u5b9a\u65f6\uff0c\u57fa\u6570\u6392\u5e8f\u4e5f\u7a33\u5b9a\uff1b\u5f53\u8ba1\u6570\u6392\u5e8f\u4e0d\u7a33\u5b9a\u65f6\uff0c\u57fa\u6570\u6392\u5e8f\u65e0\u6cd5\u4fdd\u8bc1\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\u3002
\u300c\u9009\u62e9\u6392\u5e8f selection sort\u300d\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u7b80\u5355\uff1a\u5f00\u542f\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4ece\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c06\u5176\u653e\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u672b\u5c3e\u3002
\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u9009\u62e9\u6392\u5e8f\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 11-2 \u6240\u793a\u3002
- \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u533a\u95f4\u4e3a \\([0, n-1]\\) \u3002
- \u9009\u53d6\u533a\u95f4 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(0\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 1 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
- \u9009\u53d6\u533a\u95f4 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(1\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
- \u4ee5\u6b64\u7c7b\u63a8\u3002\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u9009\u62e9\u4e0e\u4ea4\u6362\u540e\uff0c\u6570\u7ec4\u524d \\(n - 1\\) \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
- \u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002
\u56fe 11-2 \u00a0 \u9009\u62e9\u6392\u5e8f\u6b65\u9aa4
\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u7528 \\(k\\) \u6765\u8bb0\u5f55\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig selection_sort.py
selection_sort.cppdef selection_sort(nums: list[int]):\n \"\"\"\u9009\u62e9\u6392\u5e8f\"\"\"\n n = len(nums)\n # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for i in range(n - 1):\n # \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n k = i\n for j in range(i + 1, n):\n if nums[j] < nums[k]:\n k = j # \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n # \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n nums[i], nums[k] = nums[k], nums[i]\n
selection_sort.java/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(vector<int> &nums) {\n int n = nums.size();\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k])\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n swap(nums[i], nums[k]);\n }\n}\n
selection_sort.cs/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\n int n = nums.length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k])\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n int temp = nums[i];\n nums[i] = nums[k];\n nums[k] = temp;\n }\n}\n
selection_sort.go/* \u9009\u62e9\u6392\u5e8f */\nvoid SelectionSort(int[] nums) {\n int n = nums.Length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k])\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n (nums[k], nums[i]) = (nums[i], nums[k]);\n }\n}\n
selection_sort.swift/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums []int) {\n n := len(nums)\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for i := 0; i < n-1; i++ {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n k := i\n for j := i + 1; j < n; j++ {\n if nums[j] < nums[k] {\n // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n k = j\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n nums[i], nums[k] = nums[k], nums[i]\n\n }\n}\n
selection_sort.js/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for i in nums.indices.dropLast() {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n var k = i\n for j in nums.indices.dropFirst(i + 1) {\n if nums[j] < nums[k] {\n k = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n nums.swapAt(i, k)\n }\n}\n
selection_sort.ts/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums) {\n let n = nums.length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (let i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n let k = i;\n for (let j = i + 1; j < n; j++) {\n if (nums[j] < nums[k]) {\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n [nums[i], nums[k]] = [nums[k], nums[i]];\n }\n}\n
selection_sort.dart/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\n let n = nums.length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (let i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n let k = i;\n for (let j = i + 1; j < n; j++) {\n if (nums[j] < nums[k]) {\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n [nums[i], nums[k]] = [nums[k], nums[i]];\n }\n}\n
selection_sort.rs/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(List<int> nums) {\n int n = nums.length;\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k]) k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n int temp = nums[i];\n nums[i] = nums[k];\n nums[k] = temp;\n }\n}\n
selection_sort.c/* \u9009\u62e9\u6392\u5e8f */\nfn selection_sort(nums: &mut [i32]) {\n let n = nums.len();\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for i in 0..n-1 {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n let mut k = i;\n for j in i+1..n {\n if nums[j] < nums[k] {\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n nums.swap(i, k);\n }\n}\n
selection_sort.zig/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n for (int i = 0; i < n - 1; i++) {\n // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n int k = i;\n for (int j = i + 1; j < n; j++) {\n if (nums[j] < nums[k])\n k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n }\n // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n int temp = nums[i];\n nums[i] = nums[k];\n nums[k] = temp;\n }\n}\n
"},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"[class]{}-[func]{selectionSort}\n- \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5916\u5faa\u73af\u5171 \\(n - 1\\) \u8f6e\uff0c\u7b2c\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u6700\u540e\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(2\\) \uff0c\u5373\u5404\u8f6e\u5916\u5faa\u73af\u5206\u522b\u5305\u542b \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(3\\)\u3001\\(2\\) \u8f6e\u5185\u5faa\u73af\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
- \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5982\u56fe 11-3 \u6240\u793a\uff0c\u5143\u7d20
nums[i]\u6709\u53ef\u80fd\u88ab\u4ea4\u6362\u81f3\u4e0e\u5176\u76f8\u7b49\u7684\u5143\u7d20\u7684\u53f3\u8fb9\uff0c\u5bfc\u81f4\u4e24\u8005\u7684\u76f8\u5bf9\u987a\u5e8f\u53d1\u751f\u6539\u53d8\u3002
\u56fe 11-3 \u00a0 \u9009\u62e9\u6392\u5e8f\u975e\u7a33\u5b9a\u793a\u4f8b
"},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5","text":"\u300c\u6392\u5e8f\u7b97\u6cd5 sorting algorithm\u300d\u7528\u4e8e\u5bf9\u4e00\u7ec4\u6570\u636e\u6309\u7167\u7279\u5b9a\u987a\u5e8f\u8fdb\u884c\u6392\u5217\u3002\u6392\u5e8f\u7b97\u6cd5\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u56e0\u4e3a\u6709\u5e8f\u6570\u636e\u901a\u5e38\u80fd\u591f\u88ab\u66f4\u9ad8\u6548\u5730\u67e5\u627e\u3001\u5206\u6790\u548c\u5904\u7406\u3002
\u5982\u56fe 11-1 \u6240\u793a\uff0c\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u6570\u636e\u7c7b\u578b\u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7b49\u3002\u6392\u5e8f\u7684\u5224\u65ad\u89c4\u5219\u53ef\u6839\u636e\u9700\u6c42\u8bbe\u5b9a\uff0c\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u6216\u81ea\u5b9a\u4e49\u89c4\u5219\u3002
\u56fe 11-1 \u00a0 \u6570\u636e\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219\u793a\u4f8b
"},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1 \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"\u8fd0\u884c\u6548\u7387\uff1a\u6211\u4eec\u671f\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c3d\u91cf\u4f4e\uff0c\u4e14\u603b\u4f53\u64cd\u4f5c\u6570\u91cf\u8f83\u5c11\uff08\u65f6\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\uff09\u3002\u5bf9\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u8fd0\u884c\u6548\u7387\u663e\u5f97\u5c24\u4e3a\u91cd\u8981\u3002
\u5c31\u5730\u6027\uff1a\u987e\u540d\u601d\u4e49\uff0c\u300c\u539f\u5730\u6392\u5e8f\u300d\u901a\u8fc7\u5728\u539f\u6570\u7ec4\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\uff0c\u65e0\u987b\u501f\u52a9\u989d\u5916\u7684\u8f85\u52a9\u6570\u7ec4\uff0c\u4ece\u800c\u8282\u7701\u5185\u5b58\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u6570\u636e\u642c\u8fd0\u64cd\u4f5c\u8f83\u5c11\uff0c\u8fd0\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002
\u7a33\u5b9a\u6027\uff1a\u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u4e0d\u53d1\u751f\u6539\u53d8\u3002
\u7a33\u5b9a\u6392\u5e8f\u662f\u591a\u7ea7\u6392\u5e8f\u573a\u666f\u7684\u5fc5\u8981\u6761\u4ef6\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1 \u5217\u548c\u7b2c 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u53ef\u80fd\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e27\u5931\uff1a
# \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n ('A', 19)\n ('B', 18)\n ('C', 21)\n ('D', 19)\n ('E', 23)\n\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n ('B', 18)\n ('D', 19)\n ('A', 19)\n ('C', 21)\n ('E', 23)\n\u81ea\u9002\u5e94\u6027\uff1a\u300c\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u53d7\u8f93\u5165\u6570\u636e\u7684\u5f71\u54cd\uff0c\u5373\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5e76\u4e0d\u5b8c\u5168\u76f8\u7b49\u3002
\u81ea\u9002\u5e94\u6027\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u8bc4\u4f30\u3002\u5982\u679c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u5dee\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8bf4\u660e\u6392\u5e8f\u7b97\u6cd5\u5728\u67d0\u4e9b\u6570\u636e\u4e0b\u6027\u80fd\u53ef\u80fd\u52a3\u5316\uff0c\u56e0\u6b64\u88ab\u89c6\u4e3a\u8d1f\u9762\u5c5e\u6027\uff1b\u800c\u5982\u679c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5219\u88ab\u89c6\u4e3a\u6b63\u9762\u5c5e\u6027\u3002
\u662f\u5426\u57fa\u4e8e\u6bd4\u8f83\uff1a\u300c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u300d\u4f9d\u8d56\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff08\\(<\\)\u3001\\(=\\)\u3001\\(>\\)\uff09\u6765\u5224\u65ad\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\uff0c\u4ece\u800c\u6392\u5e8f\u6574\u4e2a\u6570\u7ec4\uff0c\u7406\u8bba\u6700\u4f18\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u800c\u300c\u975e\u6bd4\u8f83\u6392\u5e8f\u300d\u4e0d\u4f7f\u7528\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5bf9\u8f83\u5dee\u3002
"},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2 \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"\u8fd0\u884c\u5feb\u3001\u539f\u5730\u3001\u7a33\u5b9a\u3001\u6b63\u5411\u81ea\u9002\u5e94\u3001\u901a\u7528\u6027\u597d\u3002\u663e\u7136\uff0c\u8fc4\u4eca\u4e3a\u6b62\u5c1a\u672a\u53d1\u73b0\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u65f6\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u7684\u6570\u636e\u7279\u70b9\u548c\u95ee\u9898\u9700\u6c42\u6765\u51b3\u5b9a\u3002
\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5171\u540c\u5b66\u4e60\u5404\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5e76\u57fa\u4e8e\u4e0a\u8ff0\u8bc4\u4ef7\u7ef4\u5ea6\u5bf9\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u4f18\u7f3a\u70b9\u8fdb\u884c\u5206\u6790\u3002
"},{"location":"chapter_sorting/summary/","title":"11.11 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_sorting/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"- \u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(n)\\) \u3002
- \u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u867d\u7136\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u7531\u4e8e\u5355\u5143\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u56e0\u6b64\u5728\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u4efb\u52a1\u4e2d\u975e\u5e38\u53d7\u6b22\u8fce\u3002
- \u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u4ee5\u964d\u4f4e\u8fd9\u79cd\u52a3\u5316\u7684\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u51cf\u5c11\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(\\log n)\\) \u3002
- \u5f52\u5e76\u6392\u5e8f\u5305\u62ec\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u5178\u578b\u5730\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5f52\u5e76\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u521b\u5efa\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002
- \u6876\u6392\u5e8f\u5305\u542b\u4e09\u4e2a\u6b65\u9aa4\uff1a\u6570\u636e\u5206\u6876\u3001\u6876\u5185\u6392\u5e8f\u548c\u5408\u5e76\u7ed3\u679c\u3002\u5b83\u540c\u6837\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9002\u7528\u4e8e\u6570\u636e\u4f53\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u3002\u6876\u6392\u5e8f\u7684\u5173\u952e\u5728\u4e8e\u5bf9\u6570\u636e\u8fdb\u884c\u5e73\u5747\u5206\u914d\u3002
- \u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5b83\u901a\u8fc7\u7edf\u8ba1\u6570\u636e\u51fa\u73b0\u7684\u6b21\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u6709\u9650\u7684\u60c5\u51b5\uff0c\u5e76\u4e14\u8981\u6c42\u6570\u636e\u80fd\u591f\u8f6c\u6362\u4e3a\u6b63\u6574\u6570\u3002
- \u57fa\u6570\u6392\u5e8f\u901a\u8fc7\u9010\u4f4d\u6392\u5e8f\u6765\u5b9e\u73b0\u6570\u636e\u6392\u5e8f\uff0c\u8981\u6c42\u6570\u636e\u80fd\u591f\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u6570\u5b57\u3002
- \u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a33\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7b49\u4f18\u70b9\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e00\u6837\uff0c\u6ca1\u6709\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\u80fd\u591f\u540c\u65f6\u6ee1\u8db3\u6240\u6709\u8fd9\u4e9b\u6761\u4ef6\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u6570\u636e\u7684\u7279\u6027\u6765\u9009\u62e9\u5408\u9002\u7684\u6392\u5e8f\u7b97\u6cd5\u3002
- \u56fe 11-19 \u5bf9\u6bd4\u4e86\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u6548\u7387\u3001\u7a33\u5b9a\u6027\u3001\u5c31\u5730\u6027\u548c\u81ea\u9002\u5e94\u6027\u7b49\u3002
\u56fe 11-19 \u00a0 \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4
"},{"location":"chapter_sorting/summary/#2-q-a","title":"2. \u00a0 Q & A","text":"\u6392\u5e8f\u7b97\u6cd5\u7a33\u5b9a\u6027\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f\u5fc5\u9700\u7684\uff1f
\u5728\u73b0\u5b9e\u4e2d\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u662f\u57fa\u4e8e\u5bf9\u8c61\u7684\u67d0\u4e2a\u5c5e\u6027\u8fdb\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b66\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u4e24\u4e2a\u5c5e\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u591a\u7ea7\u6392\u5e8f\uff1a
\u5148\u6309\u7167\u59d3\u540d\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230
(A, 180) (B, 185) (C, 170) (D, 170)\uff1b\u518d\u5bf9\u8eab\u9ad8\u8fdb\u884c\u6392\u5e8f\u3002\u7531\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e0d\u7a33\u5b9a\uff0c\u56e0\u6b64\u53ef\u80fd\u5f97\u5230(D, 170) (C, 170) (A, 180) (B, 185)\u3002\u53ef\u4ee5\u53d1\u73b0\uff0c\u5b66\u751f D \u548c C \u7684\u4f4d\u7f6e\u53d1\u751f\u4e86\u4ea4\u6362\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u574f\u4e86\uff0c\u800c\u8fd9\u662f\u6211\u4eec\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002
\u54e8\u5175\u5212\u5206\u4e2d\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u7684\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f
\u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002
\u54e8\u5175\u5212\u5206
partition()\u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362nums[left]\u548cnums[i]\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd<=\u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524dnums[left] >= nums[i]\u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728i == j\u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fdnums[j] == nums[i] > nums[left]\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4
[0, 0, 0, 0, 1]\uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a[1, 0, 0, 0, 0]\uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u6b63\u786e\u7684\u3002\u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9
nums[right]\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u5173\u4e8e\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u4e3a\u4ec0\u4e48\u9009\u77ed\u7684\u6570\u7ec4\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff1f
\u9012\u5f52\u6df1\u5ea6\u5c31\u662f\u5f53\u524d\u672a\u8fd4\u56de\u7684\u9012\u5f52\u65b9\u6cd5\u7684\u6570\u91cf\u3002\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u6211\u4eec\u5c06\u539f\u6570\u7ec4\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002\u5728\u5c3e\u9012\u5f52\u4f18\u5316\u540e\uff0c\u5411\u4e0b\u9012\u5f52\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u6700\u5927\u4e3a\u539f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a\u3002\u5047\u8bbe\u6700\u5dee\u60c5\u51b5\uff0c\u4e00\u76f4\u4e3a\u4e00\u534a\u957f\u5ea6\uff0c\u90a3\u4e48\u6700\u7ec8\u7684\u9012\u5f52\u6df1\u5ea6\u5c31\u662f \\(\\log n\\) \u3002
\u56de\u987e\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u4f1a\u8fde\u7eed\u5730\u9012\u5f52\u957f\u5ea6\u8f83\u5927\u7684\u6570\u7ec4\uff0c\u6700\u5dee\u60c5\u51b5\u4e0b\u4e3a \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002\u5c3e\u9012\u5f52\u4f18\u5316\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u51fa\u73b0\u3002
\u5f53\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u65f6\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n^2)\\) \u5417\uff1f\u8be5\u5982\u4f55\u5904\u7406\u8fd9\u79cd\u9000\u5316\u60c5\u51b5\uff1f
\u662f\u7684\u3002\u5bf9\u4e8e\u8fd9\u79cd\u60c5\u51b5\uff0c\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u54e8\u5175\u5212\u5206\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a\u5c0f\u4e8e\u3001\u7b49\u4e8e\u3001\u5927\u4e8e\u57fa\u51c6\u6570\u3002\u4ec5\u5411\u4e0b\u9012\u5f52\u5c0f\u4e8e\u548c\u5927\u4e8e\u7684\u4e24\u90e8\u5206\u3002\u5728\u8be5\u65b9\u6cd5\u4e0b\uff0c\u8f93\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u6570\u7ec4\uff0c\u4ec5\u4e00\u8f6e\u54e8\u5175\u5212\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002
\u6876\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n^2)\\) \uff1f
\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u4e2a\u6876\u4e2d\u3002\u5982\u679c\u6211\u4eec\u91c7\u7528\u4e00\u4e2a \\(O(n^2)\\) \u7b97\u6cd5\u6765\u6392\u5e8f\u8fd9\u4e9b\u5143\u7d20\uff0c\u5219\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002
"},{"location":"chapter_stack_and_queue/","title":"\u7b2c 5 \u7ae0 \u00a0 \u6808\u4e0e\u961f\u5217","text":"Abstract
\u6808\u5982\u540c\u53e0\u732b\u732b\uff0c\u800c\u961f\u5217\u5c31\u50cf\u732b\u732b\u6392\u961f\u3002
\u4e24\u8005\u5206\u522b\u4ee3\u8868\u5148\u5165\u540e\u51fa\u548c\u5148\u5165\u5148\u51fa\u7684\u903b\u8f91\u5173\u7cfb\u3002
"},{"location":"chapter_stack_and_queue/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 5.1 \u00a0 \u6808
- 5.2 \u00a0 \u961f\u5217
- 5.3 \u00a0 \u53cc\u5411\u961f\u5217
- 5.4 \u00a0 \u5c0f\u7ed3
\u5728\u961f\u5217\u4e2d\uff0c\u6211\u4eec\u4ec5\u80fd\u5220\u9664\u5934\u90e8\u5143\u7d20\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u3002\u5982\u56fe 5-7 \u6240\u793a\uff0c\u300c\u53cc\u5411\u961f\u5217 double-ended queue\u300d\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u7075\u6d3b\u6027\uff0c\u5141\u8bb8\u5728\u5934\u90e8\u548c\u5c3e\u90e8\u6267\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002
\u56fe 5-7 \u00a0 \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c
"},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1 \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"\u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-3 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u79f0\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002
\u8868 5-3 \u00a0 \u53cc\u5411\u961f\u5217\u64cd\u4f5c\u6548\u7387
\u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 pushFirst() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) pushLast() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) popFirst() \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) popLast() \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) peekFirst() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) peekLast() \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\)\u540c\u6837\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5df2\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig deque.py
deque.cppfrom collections import deque\n\n# \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217\ndeque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\ndeque.append(2) # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3) # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n\n# \u8bbf\u95ee\u5143\u7d20\nfront: int = deque[0] # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1] # \u961f\u5c3e\u5143\u7d20\n\n# \u5143\u7d20\u51fa\u961f\npop_front: int = deque.popleft() # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop() # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n# \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(deque)\n\n# \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(deque) == 0\n
deque.java/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque<int> deque;\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back(); // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back(); // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n
deque.cs/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque<Integer> deque = new LinkedList<>();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst(); // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast(); // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast(); // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n
deque_test.go/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList<int> deque = new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value; // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast(); // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n
deque.swift/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back() // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front) // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear) // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n
deque.js/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n
deque.ts/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack = deque.pop();\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
deque.dart/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast: number = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack: number = deque.pop() as number;\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
deque.rs/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u4e49\u4e3a\u53cc\u5411\u961f\u5217\nQueue<int> deque = Queue<int>();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.addLast(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.addFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.first; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.last; // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.removeFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.removeLast(); // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.length;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.isEmpty;W\n
deque.c/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nlet mut deque: VecDeque<u32> = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2); // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nif let Some(front) = deque.front() { // \u961f\u9996\u5143\u7d20\n}\nif let Some(rear) = deque.back() { // \u961f\u5c3e\u5143\u7d20\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop_front) = deque.pop_front() { // \u961f\u9996\u5143\u7d20\u51fa\u961f\n}\nif let Some(pop_rear) = deque.pop_back() { // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n
deque.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u53cc\u5411\u961f\u5217\n
"},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2 \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"\n\u53cc\u5411\u961f\u5217\u7684\u5b9e\u73b0\u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53ef\u4ee5\u9009\u62e9\u94fe\u8868\u6216\u6570\u7ec4\u4f5c\u4e3a\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
"},{"location":"chapter_stack_and_queue/deque/#1","title":"1. \u00a0 \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"\u56de\u987e\u4e0a\u4e00\u8282\u5185\u5bb9\uff0c\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\uff0c\u56e0\u4e3a\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u5934\u8282\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0\u65b0\u8282\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\u3002
\u5bf9\u4e8e\u53cc\u5411\u961f\u5217\u800c\u8a00\uff0c\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u201c\u53cc\u5411\u94fe\u8868\u201d\u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
\u5982\u56fe 5-8 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u8282\u70b9\u548c\u5c3e\u8282\u70b9\u89c6\u4e3a\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u540c\u65f6\u5b9e\u73b0\u5728\u4e24\u7aef\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u7684\u529f\u80fd\u3002
LinkedListDequepushLast()pushFirst()popLast()popFirst()\u56fe 5-8 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c
\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_deque.py
linkedlist_deque.cppclass ListNode:\n \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\n\n def __init__(self, val: int):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self.val: int = val\n self.next: ListNode | None = None # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n self.prev: ListNode | None = None # \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\nclass LinkedListDeque:\n \"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._front: ListNode | None = None # \u5934\u8282\u70b9 front\n self._rear: ListNode | None = None # \u5c3e\u8282\u70b9 rear\n self._size: int = 0 # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n return self.size() == 0\n\n def push(self, num: int, is_front: bool):\n \"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\n node = ListNode(num)\n # \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if self.is_empty():\n self._front = self._rear = node\n # \u961f\u9996\u5165\u961f\u64cd\u4f5c\n elif is_front:\n # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n self._front.prev = node\n node.next = self._front\n self._front = node # \u66f4\u65b0\u5934\u8282\u70b9\n # \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else:\n # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n self._rear.next = node\n node.prev = self._rear\n self._rear = node # \u66f4\u65b0\u5c3e\u8282\u70b9\n self._size += 1 # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n\n def push_first(self, num: int):\n \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n self.push(num, True)\n\n def push_last(self, num: int):\n \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n self.push(num, False)\n\n def pop(self, is_front: bool) -> int:\n \"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n # \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if is_front:\n val: int = self._front.val # \u6682\u5b58\u5934\u8282\u70b9\u503c\n # \u5220\u9664\u5934\u8282\u70b9\n fnext: ListNode | None = self._front.next\n if fnext != None:\n fnext.prev = None\n self._front.next = None\n self._front = fnext # \u66f4\u65b0\u5934\u8282\u70b9\n # \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else:\n val: int = self._rear.val # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n # \u5220\u9664\u5c3e\u8282\u70b9\n rprev: ListNode | None = self._rear.prev\n if rprev != None:\n rprev.next = None\n self._rear.prev = None\n self._rear = rprev # \u66f4\u65b0\u5c3e\u8282\u70b9\n self._size -= 1 # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val\n\n def pop_first(self) -> int:\n \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n return self.pop(True)\n\n def pop_last(self) -> int:\n \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n return self.pop(False)\n\n def peek_first(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n return self._front.val\n\n def peek_last(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n return self._rear.val\n\n def to_array(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n node = self._front\n res = [0] * self.size()\n for i in range(self.size()):\n res[i] = node.val\n node = node.next\n return res\n
linkedlist_deque.java/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\n int val; // \u8282\u70b9\u503c\n DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\n DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n }\n};\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n private:\n DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n LinkedListDeque() : front(nullptr), rear(nullptr) {\n }\n\n /* \u6790\u6784\u65b9\u6cd5 */\n ~LinkedListDeque() {\n // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n DoublyListNode *pre, *cur = front;\n while (cur != nullptr) {\n pre = cur;\n cur = cur->next;\n delete pre;\n }\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n void push(int num, bool isFront) {\n DoublyListNode *node = new DoublyListNode(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (isEmpty())\n front = rear = node;\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if (isFront) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n front->prev = node;\n node->next = front;\n front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n rear->next = node;\n node->prev = rear;\n rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n void pushFirst(int num) {\n push(num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n void pushLast(int num) {\n push(num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n int pop(bool isFront) {\n if (isEmpty())\n throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n int val;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (isFront) {\n val = front->val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n DoublyListNode *fNext = front->next;\n if (fNext != nullptr) {\n fNext->prev = nullptr;\n front->next = nullptr;\n delete front;\n }\n front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n } else {\n val = rear->val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n DoublyListNode *rPrev = rear->prev;\n if (rPrev != nullptr) {\n rPrev->next = nullptr;\n rear->prev = nullptr;\n delete rear;\n }\n rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n int popFirst() {\n return pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n int popLast() {\n return pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peekFirst() {\n if (isEmpty())\n throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return front->val;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n int peekLast() {\n if (isEmpty())\n throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return rear->val;\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n vector<int> toVector() {\n DoublyListNode *node = front;\n vector<int> res(size());\n for (int i = 0; i < res.size(); i++) {\n res[i] = node->val;\n node = node->next;\n }\n return res;\n }\n};\n
linkedlist_deque.cs/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n ListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n ListNode(int val) {\n this.val = val;\n prev = next = null;\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n private int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n public LinkedListDeque() {\n front = rear = null;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n public int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n private void push(int num, boolean isFront) {\n ListNode node = new ListNode(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (isEmpty())\n front = rear = node;\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if (isFront) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n front.prev = node;\n node.next = front;\n front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n rear.next = node;\n node.prev = rear;\n rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n public void pushFirst(int num) {\n push(num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n public void pushLast(int num) {\n push(num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n private int pop(boolean isFront) {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n int val;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (isFront) {\n val = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n ListNode fNext = front.next;\n if (fNext != null) {\n fNext.prev = null;\n front.next = null;\n }\n front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n } else {\n val = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n ListNode rPrev = rear.prev;\n if (rPrev != null) {\n rPrev.next = null;\n rear.prev = null;\n }\n rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n public int popFirst() {\n return pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n public int popLast() {\n return pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int peekFirst() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return front.val;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n public int peekLast() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return rear.val;\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n public int[] toArray() {\n ListNode node = front;\n int[] res = new int[size()];\n for (int i = 0; i < res.length; i++) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_deque.go/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(int val) {\n public int val = val; // \u8282\u70b9\u503c\n public ListNode? next = null; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n public ListNode? prev = null; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n ListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\n int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n public LinkedListDeque() {\n front = null;\n rear = null;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n public int Size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return Size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n void Push(int num, bool isFront) {\n ListNode node = new(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (IsEmpty()) {\n front = node;\n rear = node;\n }\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if (isFront) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n front!.prev = node;\n node.next = front;\n front = node; // \u66f4\u65b0\u5934\u8282\u70b9 \n }\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n rear!.next = node;\n node.prev = rear;\n rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n\n queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n public void PushFirst(int num) {\n Push(num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n public void PushLast(int num) {\n Push(num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n int? Pop(bool isFront) {\n if (IsEmpty())\n throw new Exception();\n int? val;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (isFront) {\n val = front?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n ListNode? fNext = front?.next;\n if (fNext != null) {\n fNext.prev = null;\n front!.next = null;\n }\n front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else {\n val = rear?.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n ListNode? rPrev = rear?.prev;\n if (rPrev != null) {\n rPrev.next = null;\n rear!.prev = null;\n }\n rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n\n queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n public int? PopFirst() {\n return Pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n public int? PopLast() {\n return Pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int? PeekFirst() {\n if (IsEmpty())\n throw new Exception();\n return front?.val;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n public int? PeekLast() {\n if (IsEmpty())\n throw new Exception();\n return rear?.val;\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n public int?[] ToArray() {\n ListNode? node = front;\n int?[] res = new int?[Size()];\n for (int i = 0; i < res.Length; i++) {\n res[i] = node?.val;\n node = node?.next;\n }\n\n return res;\n }\n}\n
linkedlist_deque.swift/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n // \u4f7f\u7528\u5185\u7f6e\u5305 list\n data *list.List\n}\n\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\n return &linkedListDeque{\n data: list.New(),\n }\n}\n\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\n s.data.PushFront(value)\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\n s.data.PushBack(value)\n}\n\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Front()\n s.data.Remove(e)\n return e.Value\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Back()\n s.data.Remove(e)\n return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Front()\n return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Back()\n return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\n return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\n return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\n return s.data\n}\n
linkedlist_deque.js/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n var val: Int // \u8282\u70b9\u503c\n var next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n weak var prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n init(val: Int) {\n self.val = val\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n private var front: ListNode? // \u5934\u8282\u70b9 front\n private var rear: ListNode? // \u5c3e\u8282\u70b9 rear\n private var queSize: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n init() {\n queSize = 0\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n func size() -> Int {\n queSize\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n size() == 0\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n private func push(num: Int, isFront: Bool) {\n let node = ListNode(val: num)\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if isEmpty() {\n front = node\n rear = node\n }\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if isFront {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n front?.prev = node\n node.next = front\n front = node // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n rear?.next = node\n node.prev = rear\n rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n func pushFirst(num: Int) {\n push(num: num, isFront: true)\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n func pushLast(num: Int) {\n push(num: num, isFront: false)\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n private func pop(isFront: Bool) -> Int {\n if isEmpty() {\n fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n }\n let val: Int\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if isFront {\n val = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n let fNext = front?.next\n if fNext != nil {\n fNext?.prev = nil\n front?.next = nil\n }\n front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else {\n val = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n let rPrev = rear?.prev\n if rPrev != nil {\n rPrev?.next = nil\n rear?.prev = nil\n }\n rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n queSize -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val\n }\n\n /* \u961f\u9996\u51fa\u961f */\n func popFirst() -> Int {\n pop(isFront: true)\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n func popLast() -> Int {\n pop(isFront: false)\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n func peekFirst() -> Int? {\n isEmpty() ? nil : front?.val\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n func peekLast() -> Int? {\n isEmpty() ? nil : rear?.val\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n func toArray() -> [Int] {\n var node = front\n var res = Array(repeating: 0, count: size())\n for i in res.indices {\n res[i] = node!.val\n node = node?.next\n }\n return res\n }\n}\n
linkedlist_deque.ts/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n val; // \u8282\u70b9\u503c\n\n constructor(val) {\n this.val = val;\n this.next = null;\n this.prev = null;\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n #front; // \u5934\u8282\u70b9 front\n #rear; // \u5c3e\u8282\u70b9 rear\n #queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n constructor() {\n this.#front = null;\n this.#rear = null;\n this.#queSize = 0;\n }\n\n /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n pushLast(val) {\n const node = new ListNode(val);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (this.#queSize === 0) {\n this.#front = node;\n this.#rear = node;\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n this.#rear.next = node;\n node.prev = this.#rear;\n this.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n this.#queSize++;\n }\n\n /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n pushFirst(val) {\n const node = new ListNode(val);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (this.#queSize === 0) {\n this.#front = node;\n this.#rear = node;\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n this.#front.prev = node;\n node.next = this.#front;\n this.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n this.#queSize++;\n }\n\n /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n popLast() {\n if (this.#queSize === 0) {\n return null;\n }\n const value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n let temp = this.#rear.prev;\n if (temp !== null) {\n temp.next = null;\n this.#rear.prev = null;\n }\n this.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n this.#queSize--;\n return value;\n }\n\n /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n popFirst() {\n if (this.#queSize === 0) {\n return null;\n }\n const value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n let temp = this.#front.next;\n if (temp !== null) {\n temp.prev = null;\n this.#front.next = null;\n }\n this.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n this.#queSize--;\n return value;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n peekLast() {\n return this.#queSize === 0 ? null : this.#rear.val;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peekFirst() {\n return this.#queSize === 0 ? null : this.#front.val;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n size() {\n return this.#queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.#queSize === 0;\n }\n\n /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n print() {\n const arr = [];\n let temp = this.#front;\n while (temp !== null) {\n arr.push(temp.val);\n temp = temp.next;\n }\n console.log('[' + arr.join(', ') + ']');\n }\n}\n
linkedlist_deque.dart/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n prev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n next: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n val: number; // \u8282\u70b9\u503c\n\n constructor(val: number) {\n this.val = val;\n this.next = null;\n this.prev = null;\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n private front: ListNode; // \u5934\u8282\u70b9 front\n private rear: ListNode; // \u5c3e\u8282\u70b9 rear\n private queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n constructor() {\n this.front = null;\n this.rear = null;\n this.queSize = 0;\n }\n\n /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n pushLast(val: number): void {\n const node: ListNode = new ListNode(val);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (this.queSize === 0) {\n this.front = node;\n this.rear = node;\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n this.rear.next = node;\n node.prev = this.rear;\n this.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n this.queSize++;\n }\n\n /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n pushFirst(val: number): void {\n const node: ListNode = new ListNode(val);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (this.queSize === 0) {\n this.front = node;\n this.rear = node;\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n this.front.prev = node;\n node.next = this.front;\n this.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n this.queSize++;\n }\n\n /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n popLast(): number {\n if (this.queSize === 0) {\n return null;\n }\n const value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n let temp: ListNode = this.rear.prev;\n if (temp !== null) {\n temp.next = null;\n this.rear.prev = null;\n }\n this.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n this.queSize--;\n return value;\n }\n\n /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n popFirst(): number {\n if (this.queSize === 0) {\n return null;\n }\n const value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n let temp: ListNode = this.front.next;\n if (temp !== null) {\n temp.prev = null;\n this.front.next = null;\n }\n this.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n this.queSize--;\n return value;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n peekLast(): number {\n return this.queSize === 0 ? null : this.rear.val;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peekFirst(): number {\n return this.queSize === 0 ? null : this.front.val;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n size(): number {\n return this.queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.queSize === 0;\n }\n\n /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n print(): void {\n const arr: number[] = [];\n let temp: ListNode = this.front;\n while (temp !== null) {\n arr.push(temp.val);\n temp = temp.next;\n }\n console.log('[' + arr.join(', ') + ']');\n }\n}\n
linkedlist_deque.rs/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n int val; // \u8282\u70b9\u503c\n ListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n ListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n ListNode(this.val, {this.next, this.prev});\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\n late ListNode? _front; // \u5934\u8282\u70b9 _front\n late ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n int _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n LinkedListDeque() {\n this._front = null;\n this._rear = null;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\n int size() {\n return this._queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n void push(int _num, bool isFront) {\n final ListNode node = ListNode(_num);\n if (isEmpty()) {\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n _front = _rear = node;\n } else if (isFront) {\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n _front!.prev = node;\n node.next = _front;\n _front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n } else {\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n _rear!.next = node;\n node.prev = _rear;\n _rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n _queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n void pushFirst(int _num) {\n push(_num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n void pushLast(int _num) {\n push(_num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n int? pop(bool isFront) {\n // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\n if (isEmpty()) {\n return null;\n }\n final int val;\n if (isFront) {\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n val = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n ListNode? fNext = _front!.next;\n if (fNext != null) {\n fNext.prev = null;\n _front!.next = null;\n }\n _front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n } else {\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n val = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n ListNode? rPrev = _rear!.prev;\n if (rPrev != null) {\n rPrev.next = null;\n _rear!.prev = null;\n }\n _rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n _queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n int? popFirst() {\n return pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n int? popLast() {\n return pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int? peekFirst() {\n return _front?.val;\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n int? peekLast() {\n return _rear?.val;\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n List<int> toArray() {\n ListNode? node = _front;\n final List<int> res = [];\n for (int i = 0; i < _queSize; i++) {\n res.add(node!.val);\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_deque.c/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\npub struct ListNode<T> {\n pub val: T, // \u8282\u70b9\u503c\n pub next: Option<Rc<RefCell<ListNode<T>>>>, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n pub prev: Option<Rc<RefCell<ListNode<T>>>>, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n}\n\nimpl<T> ListNode<T> {\n pub fn new(val: T) -> Rc<RefCell<ListNode<T>>> {\n Rc::new(RefCell::new(ListNode {\n val,\n next: None,\n prev: None,\n }))\n }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque<T> {\n front: Option<Rc<RefCell<ListNode<T>>>>, // \u5934\u8282\u70b9 front\n rear: Option<Rc<RefCell<ListNode<T>>>>, // \u5c3e\u8282\u70b9 rear \n que_size: usize, // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl<T: Copy> LinkedListDeque<T> {\n pub fn new() -> Self {\n Self {\n front: None,\n rear: None,\n que_size: 0, \n }\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n pub fn size(&self) -> usize {\n return self.que_size;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n pub fn is_empty(&self) -> bool {\n return self.size() == 0;\n }\n\n /* \u5165\u961f\u64cd\u4f5c */\n pub fn push(&mut self, num: T, is_front: bool) {\n let node = ListNode::new(num);\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n if is_front {\n match self.front.take() {\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n None => {\n self.rear = Some(node.clone());\n self.front = Some(node);\n }\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n Some(old_front) => {\n old_front.borrow_mut().prev = Some(node.clone());\n node.borrow_mut().next = Some(old_front);\n self.front = Some(node); // \u66f4\u65b0\u5934\u8282\u70b9\n }\n }\n } \n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else {\n match self.rear.take() {\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n None => {\n self.front = Some(node.clone());\n self.rear = Some(node);\n }\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n Some(old_rear) => {\n old_rear.borrow_mut().next = Some(node.clone());\n node.borrow_mut().prev = Some(old_rear);\n self.rear = Some(node); // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n }\n }\n self.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n }\n\n /* \u961f\u9996\u5165\u961f */\n pub fn push_first(&mut self, num: T) {\n self.push(num, true);\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n pub fn push_last(&mut self, num: T) {\n self.push(num, false);\n }\n\n /* \u51fa\u961f\u64cd\u4f5c */\n pub fn pop(&mut self, is_front: bool) -> Option<T> {\n // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\n if self.is_empty() { \n return None \n };\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if is_front {\n self.front.take().map(|old_front| {\n match old_front.borrow_mut().next.take() {\n Some(new_front) => {\n new_front.borrow_mut().prev.take();\n self.front = Some(new_front); // \u66f4\u65b0\u5934\u8282\u70b9\n }\n None => {\n self.rear.take();\n }\n }\n self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n })\n\n } \n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else {\n self.rear.take().map(|old_rear| {\n match old_rear.borrow_mut().prev.take() {\n Some(new_rear) => {\n new_rear.borrow_mut().next.take();\n self.rear = Some(new_rear); // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n None => {\n self.front.take();\n }\n }\n self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n Rc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n })\n }\n }\n\n /* \u961f\u9996\u51fa\u961f */\n pub fn pop_first(&mut self) -> Option<T> {\n return self.pop(true);\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n pub fn pop_last(&mut self) -> Option<T> {\n return self.pop(false);\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n pub fn peek_first(&self) -> Option<&Rc<RefCell<ListNode<T>>>> {\n self.front.as_ref()\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n pub fn peek_last(&self) -> Option<&Rc<RefCell<ListNode<T>>>> {\n self.rear.as_ref()\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n pub fn to_array(&self, head: Option<&Rc<RefCell<ListNode<T>>>>) -> Vec<T> {\n if let Some(node) = head {\n let mut nums = self.to_array(node.borrow().next.as_ref());\n nums.insert(0, node.borrow().val);\n return nums;\n }\n return Vec::new();\n }\n}\n
linkedlist_deque.zig/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\ntypedef struct DoublyListNode {\n int val; // \u8282\u70b9\u503c\n struct DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\n struct DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n} DoublyListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nDoublyListNode *newDoublyListNode(int num) {\n DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n new->val = num;\n new->next = NULL;\n new->prev = NULL;\n return new;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(DoublyListNode *node) {\n free(node);\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n int queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n} LinkedListDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListDeque *newLinkedListDeque() {\n LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n deque->front = NULL;\n deque->rear = NULL;\n deque->queSize = 0;\n return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n // \u91ca\u653e\u6240\u6709\u8282\u70b9\n for (int i = 0; i < deque->queSize && deque->front != NULL; i++) {\n DoublyListNode *tmp = deque->front;\n deque->front = deque->front->next;\n free(tmp);\n }\n // \u91ca\u653e deque \u7ed3\u6784\u4f53\n free(deque);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListDeque *deque) {\n return deque->queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListDeque *deque) {\n return (size(deque) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n DoublyListNode *node = newDoublyListNode(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411node\n if (empty(deque)) {\n deque->front = deque->rear = node;\n }\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n else if (isFront) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n deque->front->prev = node;\n node->next = deque->front;\n deque->front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n deque->rear->next = node;\n node->prev = deque->rear;\n deque->rear = node;\n }\n deque->queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n push(deque, num, true);\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(LinkedListDeque *deque, int num) {\n push(deque, num, false);\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\n assert(size(deque) && deque->front);\n return deque->front->val;\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\n assert(size(deque) && deque->rear);\n return deque->rear->val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListDeque *deque, bool isFront) {\n if (empty(deque))\n return -1;\n int val;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (isFront) {\n val = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\n DoublyListNode *fNext = deque->front->next;\n if (fNext) {\n fNext->prev = NULL;\n deque->front->next = NULL;\n delDoublyListNode(deque->front);\n }\n deque->front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n }\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n else {\n val = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n DoublyListNode *rPrev = deque->rear->prev;\n if (rPrev) {\n rPrev->next = NULL;\n deque->rear->prev = NULL;\n delDoublyListNode(deque->rear);\n }\n deque->rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n deque->queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(LinkedListDeque *deque) {\n return pop(deque, true);\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(LinkedListDeque *deque) {\n return pop(deque, false);\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n int *arr = malloc(sizeof(int) * deque->queSize);\n // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n int i;\n DoublyListNode *node;\n for (i = 0, node = deque->front; i < deque->queSize; i++) {\n arr[i] = node->val;\n node = node->next;\n }\n printArray(arr, deque->queSize);\n free(arr);\n}\n
"},{"location":"chapter_stack_and_queue/deque/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"// \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\n return struct {\n const Self = @This();\n\n val: T = undefined, // \u8282\u70b9\u503c\n next: ?*Self = null, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n prev: ?*Self = null, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n\n // Initialize a list node with specific value\n pub fn init(self: *Self, x: i32) void {\n self.val = x;\n self.next = null;\n self.prev = null;\n }\n };\n}\n\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\n return struct {\n const Self = @This();\n\n front: ?*ListNode(T) = null, // \u5934\u8282\u70b9 front\n rear: ?*ListNode(T) = null, // \u5c3e\u8282\u70b9 rear\n que_size: usize = 0, // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.front = null;\n self.rear = null;\n self.que_size = 0;\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.que_size;\n }\n\n // \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.size() == 0;\n }\n\n // \u5165\u961f\u64cd\u4f5c\n pub fn push(self: *Self, num: T, is_front: bool) !void {\n var node = try self.mem_allocator.create(ListNode(T));\n node.init(num);\n // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n if (self.isEmpty()) {\n self.front = node;\n self.rear = node;\n // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n } else if (is_front) {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n self.front.?.prev = node;\n node.next = self.front;\n self.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n } else {\n // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n self.rear.?.next = node;\n node.prev = self.rear;\n self.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n self.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n } \n\n // \u961f\u9996\u5165\u961f\n pub fn pushFirst(self: *Self, num: T) !void {\n try self.push(num, true);\n } \n\n // \u961f\u5c3e\u5165\u961f\n pub fn pushLast(self: *Self, num: T) !void {\n try self.push(num, false);\n } \n\n // \u51fa\u961f\u64cd\u4f5c\n pub fn pop(self: *Self, is_front: bool) T {\n if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n var val: T = undefined;\n // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n if (is_front) {\n val = self.front.?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n // \u5220\u9664\u5934\u8282\u70b9\n var fNext = self.front.?.next;\n if (fNext != null) {\n fNext.?.prev = null;\n self.front.?.next = null;\n }\n self.front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n } else {\n val = self.rear.?.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n // \u5220\u9664\u5c3e\u8282\u70b9\n var rPrev = self.rear.?.prev;\n if (rPrev != null) {\n rPrev.?.next = null;\n self.rear.?.prev = null;\n }\n self.rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n }\n self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n return val;\n } \n\n // \u961f\u9996\u51fa\u961f\n pub fn popFirst(self: *Self) T {\n return self.pop(true);\n } \n\n // \u961f\u5c3e\u51fa\u961f\n pub fn popLast(self: *Self) T {\n return self.pop(false);\n } \n\n // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n pub fn peekFirst(self: *Self) T {\n if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return self.front.?.val;\n } \n\n // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\n pub fn peekLast(self: *Self) T {\n if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return self.rear.?.val;\n }\n\n // \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\n pub fn toArray(self: *Self) ![]T {\n var node = self.front;\n var res = try self.mem_allocator.alloc(T, self.size());\n @memset(res, @as(T, 0));\n var i: usize = 0;\n while (i < res.len) : (i += 1) {\n res[i] = node.?.val;\n node = node.?.next;\n }\n return res;\n }\n };\n}\n\u5982\u56fe 5-9 \u6240\u793a\uff0c\u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002
ArrayDequepushLast()pushFirst()popLast()popFirst()\u56fe 5-9 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c
\u5728\u961f\u5217\u7684\u5b9e\u73b0\u57fa\u7840\u4e0a\uff0c\u4ec5\u9700\u589e\u52a0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u7684\u65b9\u6cd5\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array_deque.py
array_deque.cppclass ArrayDeque:\n \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n def __init__(self, capacity: int):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._nums: list[int] = [0] * capacity\n self._front: int = 0\n self._size: int = 0\n\n def capacity(self) -> int:\n \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n return len(self._nums)\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n return self._size == 0\n\n def index(self, i: int) -> int:\n \"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + self.capacity()) % self.capacity()\n\n def push_first(self, num: int):\n \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n if self._size == self.capacity():\n print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n # \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n self._front = self.index(self._front - 1)\n # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n self._nums[self._front] = num\n self._size += 1\n\n def push_last(self, num: int):\n \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n if self._size == self.capacity():\n print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n # \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n rear = self.index(self._front + self._size)\n # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n self._nums[rear] = num\n self._size += 1\n\n def pop_first(self) -> int:\n \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n num = self.peek_first()\n # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n self._front = self.index(self._front + 1)\n self._size -= 1\n return num\n\n def pop_last(self) -> int:\n \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n num = self.peek_last()\n self._size -= 1\n return num\n\n def peek_first(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n return self._nums[self._front]\n\n def peek_last(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n # \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n last = self.index(self._front + self._size - 1)\n return self._nums[last]\n\n def to_array(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n res = []\n for i in range(self._size):\n res.append(self._nums[self.index(self._front + i)])\n return res\n
array_deque.java/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n private:\n vector<int> nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n ArrayDeque(int capacity) {\n nums.resize(capacity);\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n int capacity() {\n return nums.size();\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return queSize == 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n int index(int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + capacity()) % capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n void pushFirst(int num) {\n if (queSize == capacity()) {\n cout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n front = index(front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n nums[front] = num;\n queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n void pushLast(int num) {\n if (queSize == capacity()) {\n cout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = index(front + queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n int popFirst() {\n int num = peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n front = index(front + 1);\n queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n int popLast() {\n int num = peekLast();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peekFirst() {\n if (isEmpty())\n throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n return nums[front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n int peekLast() {\n if (isEmpty())\n throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n int last = index(front + queSize - 1);\n return nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n vector<int> toVector() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n vector<int> res(queSize);\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[index(j)];\n }\n return res;\n }\n};\n
array_deque.cs/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public ArrayDeque(int capacity) {\n this.nums = new int[capacity];\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n public int capacity() {\n return nums.length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n public int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return queSize == 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n private int index(int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + capacity()) % capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n public void pushFirst(int num) {\n if (queSize == capacity()) {\n System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n front = index(front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n nums[front] = num;\n queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n public void pushLast(int num) {\n if (queSize == capacity()) {\n System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = index(front + queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n public int popFirst() {\n int num = peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n front = index(front + 1);\n queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n public int popLast() {\n int num = peekLast();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int peekFirst() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return nums[front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n public int peekLast() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n int last = index(front + queSize - 1);\n return nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n public int[] toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] res = new int[queSize];\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[index(j)];\n }\n return res;\n }\n}\n
array_deque.go/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public ArrayDeque(int capacity) {\n nums = new int[capacity];\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n int Capacity() {\n return nums.Length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n public int Size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return queSize == 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n int Index(int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + Capacity()) % Capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n public void PushFirst(int num) {\n if (queSize == Capacity()) {\n Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n front = Index(front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n nums[front] = num;\n queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n public void PushLast(int num) {\n if (queSize == Capacity()) {\n Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = Index(front + queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n public int PopFirst() {\n int num = PeekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n front = Index(front + 1);\n queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n public int PopLast() {\n int num = PeekLast();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int PeekFirst() {\n if (IsEmpty()) {\n throw new InvalidOperationException();\n }\n return nums[front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n public int PeekLast() {\n if (IsEmpty()) {\n throw new InvalidOperationException();\n }\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n int last = Index(front + queSize - 1);\n return nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n public int[] ToArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] res = new int[queSize];\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[Index(j)];\n }\n return res;\n }\n}\n
array_deque.swift/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\n nums []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n front int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n queSize int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n queCapacity int // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n return &arrayDeque{\n nums: make([]int, queCapacity),\n queCapacity: queCapacity,\n front: 0,\n queSize: 0,\n }\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\n return q.queSize\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\n return q.queSize == 0\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + q.queCapacity) % q.queCapacity\n}\n\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\n if q.queSize == q.queCapacity {\n fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n q.front = q.index(q.front - 1)\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n q.nums[q.front] = num\n q.queSize++\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\n if q.queSize == q.queCapacity {\n fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n rear := q.index(q.front + q.queSize)\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n q.nums[rear] = num\n q.queSize++\n}\n\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\n num := q.peekFirst()\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n q.front = q.index(q.front + 1)\n q.queSize--\n return num\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\n num := q.peekLast()\n q.queSize--\n return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\n if q.isEmpty() {\n return nil\n }\n return q.nums[q.front]\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\n if q.isEmpty() {\n return nil\n }\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n last := q.index(q.front + q.queSize - 1)\n return q.nums[last]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n res := make([]int, q.queSize)\n for i, j := 0, q.front; i < q.queSize; i++ {\n res[i] = q.nums[q.index(j)]\n j++\n }\n return res\n}\n
array_deque.js/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private var queSize: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init(capacity: Int) {\n nums = Array(repeating: 0, count: capacity)\n front = 0\n queSize = 0\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n func capacity() -> Int {\n nums.count\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n func size() -> Int {\n queSize\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n size() == 0\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n private func index(i: Int) -> Int {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n (i + capacity()) % capacity()\n }\n\n /* \u961f\u9996\u5165\u961f */\n func pushFirst(num: Int) {\n if size() == capacity() {\n print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n front = index(i: front - 1)\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n nums[front] = num\n queSize += 1\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n func pushLast(num: Int) {\n if size() == capacity() {\n print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n let rear = index(i: front + size())\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num\n queSize += 1\n }\n\n /* \u961f\u9996\u51fa\u961f */\n func popFirst() -> Int {\n let num = peekFirst()\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n front = index(i: front + 1)\n queSize -= 1\n return num\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n func popLast() -> Int {\n let num = peekLast()\n queSize -= 1\n return num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n func peekFirst() -> Int {\n if isEmpty() {\n fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n }\n return nums[front]\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n func peekLast() -> Int {\n if isEmpty() {\n fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n }\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n let last = index(i: front + size() - 1)\n return nums[last]\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n func toArray() -> [Int] {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n var res = Array(repeating: 0, count: size())\n for (i, j) in sequence(first: (0, front), next: { $0 < self.size() - 1 ? ($0 + 1, $1 + 1) : nil }) {\n res[i] = nums[index(i: j)]\n }\n return res\n }\n}\n
array_deque.ts/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n #nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n #front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n #queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(capacity) {\n this.#nums = new Array(capacity);\n this.#front = 0;\n this.#queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n capacity() {\n return this.#nums.length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n size() {\n return this.#queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.#queSize === 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n index(i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + this.capacity()) % this.capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n pushFirst(num) {\n if (this.#queSize === this.capacity()) {\n console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n this.#front = this.index(this.#front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n this.#nums[this.#front] = num;\n this.#queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n pushLast(num) {\n if (this.#queSize === this.capacity()) {\n console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n const rear = this.index(this.#front + this.#queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n this.#nums[rear] = num;\n this.#queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n popFirst() {\n const num = this.peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n this.#front = this.index(this.#front + 1);\n this.#queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n popLast() {\n const num = this.peekLast();\n this.#queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peekFirst() {\n if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n return this.#nums[this.#front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n peekLast() {\n if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n const last = this.index(this.#front + this.#queSize - 1);\n return this.#nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const res = [];\n for (let i = 0, j = this.#front; i < this.#queSize; i++, j++) {\n res[i] = this.#nums[this.index(j)];\n }\n return res;\n }\n}\n
array_deque.dart/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(capacity: number) {\n this.nums = new Array(capacity);\n this.front = 0;\n this.queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n capacity(): number {\n return this.nums.length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n size(): number {\n return this.queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.queSize === 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n index(i: number): number {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + this.capacity()) % this.capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n pushFirst(num: number): void {\n if (this.queSize === this.capacity()) {\n console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n this.front = this.index(this.front - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n this.nums[this.front] = num;\n this.queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n pushLast(num: number): void {\n if (this.queSize === this.capacity()) {\n console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n const rear: number = this.index(this.front + this.queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n this.nums[rear] = num;\n this.queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n popFirst(): number {\n const num: number = this.peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n this.front = this.index(this.front + 1);\n this.queSize--;\n return num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n popLast(): number {\n const num: number = this.peekLast();\n this.queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peekFirst(): number {\n if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n return this.nums[this.front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n peekLast(): number {\n if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n const last = this.index(this.front + this.queSize - 1);\n return this.nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n toArray(): number[] {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const res: number[] = [];\n for (let i = 0, j = this.front; i < this.queSize; i++, j++) {\n res[i] = this.nums[this.index(j)];\n }\n return res;\n }\n}\n
array_deque.rs/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n late List<int> _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n late int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n /* \u6784\u9020\u65b9\u6cd5 */\n ArrayDeque(int capacity) {\n this._nums = List.filled(capacity, 0);\n this._front = this._queSize = 0;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n int capacity() {\n return _nums.length;\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return _queSize;\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _queSize == 0;\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n int index(int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return (i + capacity()) % capacity();\n }\n\n /* \u961f\u9996\u5165\u961f */\n void pushFirst(int _num) {\n if (_queSize == capacity()) {\n throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n _front = index(_front - 1);\n // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u9996\n _nums[_front] = _num;\n _queSize++;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n void pushLast(int _num) {\n if (_queSize == capacity()) {\n throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = index(_front + _queSize);\n // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n _nums[rear] = _num;\n _queSize++;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n int popFirst() {\n int _num = peekFirst();\n // \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n _front = index(_front + 1);\n _queSize--;\n return _num;\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n int popLast() {\n int _num = peekLast();\n _queSize--;\n return _num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peekFirst() {\n if (isEmpty()) {\n throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n }\n return _nums[_front];\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n int peekLast() {\n if (isEmpty()) {\n throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n }\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n int last = index(_front + _queSize - 1);\n return _nums[last];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n List<int> toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n List<int> res = List.filled(_queSize, 0);\n for (int i = 0, j = _front; i < _queSize; i++, j++) {\n res[i] = _nums[index(j)];\n }\n return res;\n }\n}\n
array_deque.c/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct ArrayDeque {\n nums: Vec<i32>, // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n front: usize, // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n que_size: usize, // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n}\n\nimpl ArrayDeque {\n /* \u6784\u9020\u65b9\u6cd5 */\n pub fn new(capacity: usize) -> Self {\n Self {\n nums: vec![0; capacity],\n front: 0,\n que_size: 0,\n }\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n pub fn capacity(&self) -> usize {\n self.nums.len()\n }\n\n /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n pub fn size(&self) -> usize {\n self.que_size\n }\n\n /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n pub fn is_empty(&self) -> bool {\n self.que_size == 0\n }\n\n /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n fn index(&self, i: i32) -> usize {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n }\n\n /* \u961f\u9996\u5165\u961f */\n pub fn push_first(&mut self, num: i32) {\n if self.que_size == self.capacity() {\n println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n self.front = self.index(self.front as i32 - 1);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n self.nums[self.front] = num;\n self.que_size += 1;\n }\n\n /* \u961f\u5c3e\u5165\u961f */\n pub fn push_last(&mut self, num: i32) {\n if self.que_size == self.capacity() {\n println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n let rear = self.index(self.front as i32 + self.que_size as i32);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n self.nums[rear] = num;\n self.que_size += 1;\n }\n\n /* \u961f\u9996\u51fa\u961f */\n fn pop_first(&mut self) -> i32 {\n let num = self.peek_first();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n self.front = self.index(self.front as i32 + 1);\n self.que_size -= 1;\n num\n }\n\n /* \u961f\u5c3e\u51fa\u961f */\n fn pop_last(&mut self) -> i32 {\n let num = self.peek_last();\n self.que_size -= 1;\n num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n fn peek_first(&self) -> i32 {\n if self.is_empty() { panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\") };\n self.nums[self.front]\n }\n\n /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n fn peek_last(&self) -> i32 {\n if self.is_empty() { panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\") };\n // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n self.nums[last]\n }\n\n /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n fn to_array(&self) -> Vec<i32> {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n let mut res = vec![0; self.que_size];\n let mut j = self.front;\n for i in 0..self.que_size {\n res[i] = self.nums[self.index(j as i32)];\n j += 1;\n }\n res\n }\n}\n
array_deque.zig/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n int *nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayDeque *newArrayDeque(int capacity) {\n ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n // \u521d\u59cb\u5316\u6570\u7ec4\n deque->queCapacity = capacity;\n deque->nums = (int *)malloc(sizeof(int) * deque->queCapacity);\n deque->front = deque->queSize = 0;\n return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(ArrayDeque *deque) {\n free(deque->nums);\n free(deque);\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\n return deque->queCapacity;\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayDeque *deque) {\n return deque->queSize;\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayDeque *deque) {\n return deque->queSize == 0;\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(ArrayDeque *deque, int num) {\n if (deque->queSize == capacity(deque)) {\n printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n return;\n }\n // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\n deque->front = dequeIndex(deque, deque->front - 1);\n // \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\n deque->nums[deque->front] = num;\n deque->queSize++;\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(ArrayDeque *deque, int num) {\n if (deque->queSize == capacity(deque)) {\n printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n int rear = dequeIndex(deque, deque->front + deque->queSize);\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n deque->nums[rear] = num;\n deque->queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n assert(empty(deque) == 0);\n return deque->nums[deque->front];\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n assert(empty(deque) == 0);\n int last = dequeIndex(deque, deque->front + deque->queSize - 1);\n return deque->nums[last];\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(ArrayDeque *deque) {\n int num = peekFirst(deque);\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n deque->front = dequeIndex(deque, deque->front + 1);\n deque->queSize--;\n return num;\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(ArrayDeque *deque) {\n int num = peekLast(deque);\n deque->queSize--;\n return num;\n}\n
"},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3 \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"[class]{ArrayDeque}-[func]{}\n\u53cc\u5411\u961f\u5217\u517c\u5177\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5b9e\u73b0\u8fd9\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\u573a\u666f\uff0c\u540c\u65f6\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002
\u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1a\u7cfb\u7edf\u5c06\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c
"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 \u961f\u5217","text":"push\u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7pop\u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u7684\u9650\u5236\uff0c\u8f6f\u4ef6\u901a\u5e38\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\u3002\u5f53\u6808\u7684\u957f\u5ea6\u8d85\u8fc7 \\(50\\) \u65f6\uff0c\u8f6f\u4ef6\u9700\u8981\u5728\u6808\u5e95\uff08\u961f\u9996\uff09\u6267\u884c\u5220\u9664\u64cd\u4f5c\u3002\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\u8be5\u529f\u80fd\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u8bf7\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u9075\u5faa\u6808\u7684\u5148\u5165\u540e\u51fa\u539f\u5219\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u80fd\u591f\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u903b\u8f91\u3002\u300c\u961f\u5217 queue\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u4e86\u6392\u961f\u73b0\u8c61\uff0c\u5373\u65b0\u6765\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u5c3e\u90e8\uff0c\u800c\u4f4d\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u9010\u4e2a\u79bb\u5f00\u3002
\u5982\u56fe 5-4 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u961f\u5217\u5934\u90e8\u79f0\u4e3a\u201c\u961f\u9996\u201d\uff0c\u5c3e\u90e8\u79f0\u4e3a\u201c\u961f\u5c3e\u201d\uff0c\u5c06\u628a\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u5165\u961f\u201d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u51fa\u961f\u201d\u3002
\u56fe 5-4 \u00a0 \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219
"},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1 \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"\u961f\u5217\u7684\u5e38\u89c1\u64cd\u4f5c\u5982\u8868 5-2 \u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u79f0\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u6211\u4eec\u5728\u6b64\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002
\u8868 5-2 \u00a0 \u961f\u5217\u64cd\u4f5c\u6548\u7387
\u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) pop() \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) peek() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\)\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u73b0\u6210\u7684\u961f\u5217\u7c7b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig queue.py
queue.cppfrom collections import deque\n\n# \u521d\u59cb\u5316\u961f\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u5f53\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u63a8\u8350\nque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# \u8bbf\u95ee\u961f\u9996\u5143\u7d20\nfront: int = que[0];\n\n# \u5143\u7d20\u51fa\u961f\npop: int = que.popleft()\n\n# \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(que)\n\n# \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(que) == 0\n
queue.java/* \u521d\u59cb\u5316\u961f\u5217 */\nqueue<int> queue;\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n
queue.cs/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<Integer> queue = new LinkedList<>();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n
queue_test.go/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<int> queue = new();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count == 0;\n
queue.swift/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n
queue.js/* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n
queue.ts/* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
queue.dart/* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
queue.rs/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u961f\u5217\u7c7b Qeque \u662f\u53cc\u5411\u961f\u5217\uff0c\u4e5f\u53ef\u4f5c\u4e3a\u961f\u5217\u4f7f\u7528\nQueue<int> queue = Queue();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.first;\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.removeFirst();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.isEmpty;\n
queue.c/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Rust \u4e2d\u4f7f\u7528\u53cc\u5411\u961f\u5217\u4f5c\u4e3a\u666e\u901a\u961f\u5217\u6765\u4f7f\u7528\nlet mut deque: VecDeque<u32> = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nif let Some(front) = deque.front() {\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n
queue.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u961f\u5217\n
"},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2 \u00a0 \u961f\u5217\u5b9e\u73b0","text":"\n\u4e3a\u4e86\u5b9e\u73b0\u961f\u5217\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\u5143\u7d20\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u5143\u7d20\u3002\u94fe\u8868\u548c\u6570\u7ec4\u90fd\u7b26\u5408\u8981\u6c42\u3002
"},{"location":"chapter_stack_and_queue/queue/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"\u5982\u56fe 5-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u201c\u5934\u8282\u70b9\u201d\u548c\u201c\u5c3e\u8282\u70b9\u201d\u5206\u522b\u89c6\u4e3a\u201c\u961f\u9996\u201d\u548c\u201c\u961f\u5c3e\u201d\uff0c\u89c4\u5b9a\u961f\u5c3e\u4ec5\u53ef\u6dfb\u52a0\u8282\u70b9\uff0c\u961f\u9996\u4ec5\u53ef\u5220\u9664\u8282\u70b9\u3002
LinkedListQueuepush()pop()\u56fe 5-5 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c
\u4ee5\u4e0b\u662f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_queue.py
linkedlist_queue.cppclass LinkedListQueue:\n \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._front: ListNode | None = None # \u5934\u8282\u70b9 front\n self._rear: ListNode | None = None # \u5c3e\u8282\u70b9 rear\n self._size: int = 0\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n return not self._front\n\n def push(self, num: int):\n \"\"\"\u5165\u961f\"\"\"\n # \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n node = ListNode(num)\n # \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if self._front is None:\n self._front = node\n self._rear = node\n # \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n else:\n self._rear.next = node\n self._rear = node\n self._size += 1\n\n def pop(self) -> int:\n \"\"\"\u51fa\u961f\"\"\"\n num = self.peek()\n # \u5220\u9664\u5934\u8282\u70b9\n self._front = self._front.next\n self._size -= 1\n return num\n\n def peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n return self._front.val\n\n def to_list(self) -> list[int]:\n \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n queue = []\n temp = self._front\n while temp:\n queue.append(temp.val)\n temp = temp.next\n return queue\n
linkedlist_queue.java/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n private:\n ListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n int queSize;\n\n public:\n LinkedListQueue() {\n front = nullptr;\n rear = nullptr;\n queSize = 0;\n }\n\n ~LinkedListQueue() {\n // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n freeMemoryLinkedList(front);\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return queSize == 0;\n }\n\n /* \u5165\u961f */\n void push(int num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n ListNode *node = new ListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (front == nullptr) {\n front = node;\n rear = node;\n }\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n else {\n rear->next = node;\n rear = node;\n }\n queSize++;\n }\n\n /* \u51fa\u961f */\n int pop() {\n int num = peek();\n // \u5220\u9664\u5934\u8282\u70b9\n ListNode *tmp = front;\n front = front->next;\n // \u91ca\u653e\u5185\u5b58\n delete tmp;\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peek() {\n if (size() == 0)\n throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n return front->val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n vector<int> toVector() {\n ListNode *node = front;\n vector<int> res(size());\n for (int i = 0; i < res.size(); i++) {\n res[i] = node->val;\n node = node->next;\n }\n return res;\n }\n};\n
linkedlist_queue.cs/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n private int queSize = 0;\n\n public LinkedListQueue() {\n front = null;\n rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n public int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f */\n public void push(int num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n ListNode node = new ListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (front == null) {\n front = node;\n rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else {\n rear.next = node;\n rear = node;\n }\n queSize++;\n }\n\n /* \u51fa\u961f */\n public int pop() {\n int num = peek();\n // \u5220\u9664\u5934\u8282\u70b9\n front = front.next;\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int peek() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return front.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] toArray() {\n ListNode node = front;\n int[] res = new int[size()];\n for (int i = 0; i < res.length; i++) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_queue.go/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n ListNode? front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \n int queSize = 0;\n\n public LinkedListQueue() {\n front = null;\n rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n public int Size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return Size() == 0;\n }\n\n /* \u5165\u961f */\n public void Push(int num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n ListNode node = new(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (front == null) {\n front = node;\n rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else if (rear != null) {\n rear.next = node;\n rear = node;\n }\n queSize++;\n }\n\n /* \u51fa\u961f */\n public int Pop() {\n int num = Peek();\n // \u5220\u9664\u5934\u8282\u70b9\n front = front?.next;\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int Peek() {\n if (IsEmpty())\n throw new Exception();\n return front!.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] ToArray() {\n if (front == null)\n return [];\n\n ListNode? node = front;\n int[] res = new int[Size()];\n for (int i = 0; i < res.Length; i++) {\n res[i] = node!.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_queue.swift/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\n data *list.List\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\n return &linkedListQueue{\n data: list.New(),\n }\n}\n\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\n s.data.PushBack(value)\n}\n\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Front()\n s.data.Remove(e)\n return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Front()\n return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\n return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\n return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\n return s.data\n}\n
linkedlist_queue.js/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n private var front: ListNode? // \u5934\u8282\u70b9\n private var rear: ListNode? // \u5c3e\u8282\u70b9\n private var _size = 0\n\n init() {}\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n func size() -> Int {\n _size\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n size() == 0\n }\n\n /* \u5165\u961f */\n func push(num: Int) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n let node = ListNode(x: num)\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if front == nil {\n front = node\n rear = node\n }\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n else {\n rear?.next = node\n rear = node\n }\n _size += 1\n }\n\n /* \u51fa\u961f */\n @discardableResult\n func pop() -> Int {\n let num = peek()\n // \u5220\u9664\u5934\u8282\u70b9\n front = front?.next\n _size -= 1\n return num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n func peek() -> Int {\n if isEmpty() {\n fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n }\n return front!.val\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n func toArray() -> [Int] {\n var node = front\n var res = Array(repeating: 0, count: size())\n for i in res.indices {\n res[i] = node!.val\n node = node?.next\n }\n return res\n }\n}\n
linkedlist_queue.ts/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n #front; // \u5934\u8282\u70b9 #front\n #rear; // \u5c3e\u8282\u70b9 #rear\n #queSize = 0;\n\n constructor() {\n this.#front = null;\n this.#rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n get size() {\n return this.#queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.size === 0;\n }\n\n /* \u5165\u961f */\n push(num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n const node = new ListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (!this.#front) {\n this.#front = node;\n this.#rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else {\n this.#rear.next = node;\n this.#rear = node;\n }\n this.#queSize++;\n }\n\n /* \u51fa\u961f */\n pop() {\n const num = this.peek();\n // \u5220\u9664\u5934\u8282\u70b9\n this.#front = this.#front.next;\n this.#queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peek() {\n if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n return this.#front.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n toArray() {\n let node = this.#front;\n const res = new Array(this.size);\n for (let i = 0; i < res.length; i++) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_queue.dart/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n private front: ListNode | null; // \u5934\u8282\u70b9 front\n private rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\n private queSize: number = 0;\n\n constructor() {\n this.front = null;\n this.rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n get size(): number {\n return this.queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.size === 0;\n }\n\n /* \u5165\u961f */\n push(num: number): void {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n const node = new ListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (!this.front) {\n this.front = node;\n this.rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else {\n this.rear!.next = node;\n this.rear = node;\n }\n this.queSize++;\n }\n\n /* \u51fa\u961f */\n pop(): number {\n const num = this.peek();\n if (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n // \u5220\u9664\u5934\u8282\u70b9\n this.front = this.front.next;\n this.queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peek(): number {\n if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n return this.front!.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n toArray(): number[] {\n let node = this.front;\n const res = new Array<number>(this.size);\n for (let i = 0; i < res.length; i++) {\n res[i] = node!.val;\n node = node!.next;\n }\n return res;\n }\n}\n
linkedlist_queue.rs/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n ListNode? _front; // \u5934\u8282\u70b9 _front\n ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n int _queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n LinkedListQueue() {\n _front = null;\n _rear = null;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return _queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _queSize == 0;\n }\n\n /* \u5165\u961f */\n void push(int _num) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 _num\n final node = ListNode(_num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (_front == null) {\n _front = node;\n _rear = node;\n } else {\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n _rear!.next = node;\n _rear = node;\n }\n _queSize++;\n }\n\n /* \u51fa\u961f */\n int pop() {\n final int _num = peek();\n // \u5220\u9664\u5934\u8282\u70b9\n _front = _front!.next;\n _queSize--;\n return _num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peek() {\n if (_queSize == 0) {\n throw Exception('\u961f\u5217\u4e3a\u7a7a');\n }\n return _front!.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n List<int> toArray() {\n ListNode? node = _front;\n final List<int> queue = [];\n while (node != null) {\n queue.add(node.val);\n node = node.next;\n }\n return queue;\n }\n}\n
linkedlist_queue.c/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue<T> {\n front: Option<Rc<RefCell<ListNode<T>>>>, // \u5934\u8282\u70b9 front\n rear: Option<Rc<RefCell<ListNode<T>>>>, // \u5c3e\u8282\u70b9 rear \n que_size: usize, // \u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl<T: Copy> LinkedListQueue<T> {\n pub fn new() -> Self {\n Self {\n front: None,\n rear: None,\n que_size: 0, \n }\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n pub fn size(&self) -> usize {\n return self.que_size;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n pub fn is_empty(&self) -> bool {\n return self.size() == 0;\n }\n\n /* \u5165\u961f */\n pub fn push(&mut self, num: T) {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n let new_rear = ListNode::new(num);\n match self.rear.take() {\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n Some(old_rear) => {\n old_rear.borrow_mut().next = Some(new_rear.clone());\n self.rear = Some(new_rear);\n }\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n None => {\n self.front = Some(new_rear.clone());\n self.rear = Some(new_rear);\n }\n }\n self.que_size += 1;\n }\n\n /* \u51fa\u961f */\n pub fn pop(&mut self) -> Option<T> {\n self.front.take().map(|old_front| {\n match old_front.borrow_mut().next.take() {\n Some(new_front) => {\n self.front = Some(new_front);\n }\n None => {\n self.rear.take();\n }\n }\n self.que_size -= 1;\n Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n })\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n pub fn peek(&self) -> Option<&Rc<RefCell<ListNode<T>>>> {\n self.front.as_ref()\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n pub fn to_array(&self, head: Option<&Rc<RefCell<ListNode<T>>>>) -> Vec<T> {\n if let Some(node) = head {\n let mut nums = self.to_array(node.borrow().next.as_ref());\n nums.insert(0, node.borrow().val);\n return nums;\n }\n return Vec::new();\n }\n}\n
linkedlist_queue.zig/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n ListNode *front, *rear;\n int queSize;\n} LinkedListQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListQueue *newLinkedListQueue() {\n LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n queue->front = NULL;\n queue->rear = NULL;\n queue->queSize = 0;\n return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n // \u91ca\u653e\u6240\u6709\u8282\u70b9\n for (int i = 0; i < queue->queSize && queue->front != NULL; i++) {\n ListNode *tmp = queue->front;\n queue->front = queue->front->next;\n free(tmp);\n }\n // \u91ca\u653e queue \u7ed3\u6784\u4f53\n free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListQueue *queue) {\n return queue->queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListQueue *queue) {\n return (size(queue) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListQueue *queue, int num) {\n // \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\n ListNode *node = newListNode(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (queue->front == NULL) {\n queue->front = node;\n queue->rear = node;\n }\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n else {\n queue->rear->next = node;\n queue->rear = node;\n }\n queue->queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\n assert(size(queue) && queue->front);\n return queue->front->val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListQueue *queue) {\n int num = peek(queue);\n ListNode *tmp = queue->front;\n queue->front = queue->front->next;\n free(tmp);\n queue->queSize--;\n return num;\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n int *arr = malloc(sizeof(int) * queue->queSize);\n // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n int i;\n ListNode *node;\n for (i = 0, node = queue->front; i < queue->queSize; i++) {\n arr[i] = node->val;\n node = node->next;\n }\n printArray(arr, queue->queSize);\n free(arr);\n}\n
"},{"location":"chapter_stack_and_queue/queue/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\n return struct {\n const Self = @This();\n\n front: ?*inc.ListNode(T) = null, // \u5934\u8282\u70b9 front\n rear: ?*inc.ListNode(T) = null, // \u5c3e\u8282\u70b9 rear\n que_size: usize = 0, // \u961f\u5217\u7684\u957f\u5ea6\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.front = null;\n self.rear = null;\n self.que_size = 0;\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.que_size;\n }\n\n // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.size() == 0;\n }\n\n // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n pub fn peek(self: *Self) T {\n if (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n return self.front.?.val;\n } \n\n // \u5165\u961f\n pub fn push(self: *Self, num: T) !void {\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n var node = try self.mem_allocator.create(inc.ListNode(T));\n node.init(num);\n // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n if (self.front == null) {\n self.front = node;\n self.rear = node;\n // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n } else {\n self.rear.?.next = node;\n self.rear = node;\n }\n self.que_size += 1;\n } \n\n // \u51fa\u961f\n pub fn pop(self: *Self) T {\n var num = self.peek();\n // \u5220\u9664\u5934\u8282\u70b9\n self.front = self.front.?.next;\n self.que_size -= 1;\n return num;\n } \n\n // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n pub fn toArray(self: *Self) ![]T {\n var node = self.front;\n var res = try self.mem_allocator.alloc(T, self.size());\n @memset(res, @as(T, 0));\n var i: usize = 0;\n while (i < res.len) : (i += 1) {\n res[i] = node.?.val;\n node = node.?.next;\n }\n return res;\n }\n };\n}\n\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002
\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf
front\u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u4e00\u4e2a\u53d8\u91cfsize\u7528\u4e8e\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u5b9a\u4e49rear = front + size\uff0c\u8fd9\u4e2a\u516c\u5f0f\u8ba1\u7b97\u51fa\u7684rear\u6307\u5411\u961f\u5c3e\u5143\u7d20\u4e4b\u540e\u7684\u4e0b\u4e00\u4e2a\u4f4d\u7f6e\u3002\u57fa\u4e8e\u6b64\u8bbe\u8ba1\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a
[front, rear - 1]\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u6cd5\u5982\u56fe 5-6 \u6240\u793a\u3002- \u5165\u961f\u64cd\u4f5c\uff1a\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9
rear\u7d22\u5f15\u5904\uff0c\u5e76\u5c06size\u589e\u52a0 1 \u3002 - \u51fa\u961f\u64cd\u4f5c\uff1a\u53ea\u9700\u5c06
front\u589e\u52a0 1 \uff0c\u5e76\u5c06size\u51cf\u5c11 1 \u3002
\u53ef\u4ee5\u770b\u5230\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u90fd\u53ea\u9700\u8fdb\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002
ArrayQueuepush()pop()\u56fe 5-6 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c
\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u8fdb\u884c\u5165\u961f\u548c\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0c
front\u548crear\u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5f53\u5b83\u4eec\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u65f6\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u89c6\u4e3a\u9996\u5c3e\u76f8\u63a5\u7684\u201c\u73af\u5f62\u6570\u7ec4\u201d\u3002\u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u8ba9
PythonC++JavaC#GoSwiftJSTSDartRustCZig array_queue.pyfront\u6216rear\u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u76f4\u63a5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u201c\u53d6\u4f59\u64cd\u4f5c\u201d\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
array_queue.cppclass ArrayQueue:\n \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n def __init__(self, size: int):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._nums: list[int] = [0] * size # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n self._front: int = 0 # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n self._size: int = 0 # \u961f\u5217\u957f\u5ea6\n\n def capacity(self) -> int:\n \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n return len(self._nums)\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n return self._size == 0\n\n def push(self, num: int):\n \"\"\"\u5165\u961f\"\"\"\n if self._size == self.capacity():\n raise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n # \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n rear: int = (self._front + self._size) % self.capacity()\n # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n self._nums[rear] = num\n self._size += 1\n\n def pop(self) -> int:\n \"\"\"\u51fa\u961f\"\"\"\n num: int = self.peek()\n # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n self._front = (self._front + 1) % self.capacity()\n self._size -= 1\n return num\n\n def peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n return self._nums[self._front]\n\n def to_list(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n res = [0] * self.size()\n j: int = self._front\n for i in range(self.size()):\n res[i] = self._nums[(j % self.capacity())]\n j += 1\n return res\n
array_queue.java/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n private:\n int *nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u961f\u5217\u957f\u5ea6\n int queCapacity; // \u961f\u5217\u5bb9\u91cf\n\n public:\n ArrayQueue(int capacity) {\n // \u521d\u59cb\u5316\u6570\u7ec4\n nums = new int[capacity];\n queCapacity = capacity;\n front = queSize = 0;\n }\n\n ~ArrayQueue() {\n delete[] nums;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n int capacity() {\n return queCapacity;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u961f */\n void push(int num) {\n if (queSize == queCapacity) {\n cout << \"\u961f\u5217\u5df2\u6ee1\" << endl;\n return;\n }\n // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (front + queSize) % queCapacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u51fa\u961f */\n int pop() {\n int num = peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n front = (front + 1) % queCapacity;\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peek() {\n if (isEmpty())\n throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n return nums[front];\n }\n\n /* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n vector<int> toVector() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n vector<int> arr(queSize);\n for (int i = 0, j = front; i < queSize; i++, j++) {\n arr[i] = nums[j % queCapacity];\n }\n return arr;\n }\n};\n
array_queue.cs/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private int queSize; // \u961f\u5217\u957f\u5ea6\n\n public ArrayQueue(int capacity) {\n nums = new int[capacity];\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n public int capacity() {\n return nums.length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n public int size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return queSize == 0;\n }\n\n /* \u5165\u961f */\n public void push(int num) {\n if (queSize == capacity()) {\n System.out.println(\"\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (front + queSize) % capacity();\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u51fa\u961f */\n public int pop() {\n int num = peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n front = (front + 1) % capacity();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int peek() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return nums[front];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4 */\n public int[] toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] res = new int[queSize];\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[j % capacity()];\n }\n return res;\n }\n}\n
array_queue.go/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u961f\u5217\u957f\u5ea6\n\n public ArrayQueue(int capacity) {\n nums = new int[capacity];\n front = queSize = 0;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n int Capacity() {\n return nums.Length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n public int Size() {\n return queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return queSize == 0;\n }\n\n /* \u5165\u961f */\n public void Push(int num) {\n if (queSize == Capacity()) {\n Console.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (front + queSize) % Capacity();\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num;\n queSize++;\n }\n\n /* \u51fa\u961f */\n public int Pop() {\n int num = Peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n front = (front + 1) % Capacity();\n queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n public int Peek() {\n if (IsEmpty())\n throw new Exception();\n return nums[front];\n }\n\n /* \u8fd4\u56de\u6570\u7ec4 */\n public int[] ToArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n int[] res = new int[queSize];\n for (int i = 0, j = front; i < queSize; i++, j++) {\n res[i] = nums[j % this.Capacity()];\n }\n return res;\n }\n}\n
array_queue.swift/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\n nums []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n front int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n queSize int // \u961f\u5217\u957f\u5ea6\n queCapacity int // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n return &arrayQueue{\n nums: make([]int, queCapacity),\n queCapacity: queCapacity,\n front: 0,\n queSize: 0,\n }\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\n return q.queSize\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\n return q.queSize == 0\n}\n\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n // \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\n if q.queSize == q.queCapacity {\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n rear := (q.front + q.queSize) % q.queCapacity\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n q.nums[rear] = num\n q.queSize++\n}\n\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\n num := q.peek()\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n q.front = (q.front + 1) % q.queCapacity\n q.queSize--\n return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\n if q.isEmpty() {\n return nil\n }\n return q.nums[q.front]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\n rear := (q.front + q.queSize)\n if rear >= q.queCapacity {\n rear %= q.queCapacity\n return append(q.nums[q.front:], q.nums[:rear]...)\n }\n return q.nums[q.front:rear]\n}\n
array_queue.js/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private var queSize = 0 // \u961f\u5217\u957f\u5ea6\n\n init(capacity: Int) {\n // \u521d\u59cb\u5316\u6570\u7ec4\n nums = Array(repeating: 0, count: capacity)\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n func capacity() -> Int {\n nums.count\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n func size() -> Int {\n queSize\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n queSize == 0\n }\n\n /* \u5165\u961f */\n func push(num: Int) {\n if size() == capacity() {\n print(\"\u961f\u5217\u5df2\u6ee1\")\n return\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n let rear = (front + queSize) % capacity()\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n nums[rear] = num\n queSize += 1\n }\n\n /* \u51fa\u961f */\n @discardableResult\n func pop() -> Int {\n let num = peek()\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n front = (front + 1) % capacity()\n queSize -= 1\n return num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n func peek() -> Int {\n if isEmpty() {\n fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n }\n return nums[front]\n }\n\n /* \u8fd4\u56de\u6570\u7ec4 */\n func toArray() -> [Int] {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n var res = Array(repeating: 0, count: queSize)\n for (i, j) in sequence(first: (0, front), next: { $0 < self.queSize - 1 ? ($0 + 1, $1 + 1) : nil }) {\n res[i] = nums[j % capacity()]\n }\n return res\n }\n}\n
array_queue.ts/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n #nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n #front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n #queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n constructor(capacity) {\n this.#nums = new Array(capacity);\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n get capacity() {\n return this.#nums.length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n get size() {\n return this.#queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.#queSize === 0;\n }\n\n /* \u5165\u961f */\n push(num) {\n if (this.size === this.capacity) {\n console.log('\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n const rear = (this.#front + this.size) % this.capacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n this.#nums[rear] = num;\n this.#queSize++;\n }\n\n /* \u51fa\u961f */\n pop() {\n const num = this.peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n this.#front = (this.#front + 1) % this.capacity;\n this.#queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peek() {\n if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n return this.#nums[this.#front];\n }\n\n /* \u8fd4\u56de Array */\n toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const arr = new Array(this.size);\n for (let i = 0, j = this.#front; i < this.size; i++, j++) {\n arr[i] = this.#nums[j % this.capacity];\n }\n return arr;\n }\n}\n
array_queue.dart/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n private queSize: number; // \u961f\u5217\u957f\u5ea6\n\n constructor(capacity: number) {\n this.nums = new Array(capacity);\n this.front = this.queSize = 0;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n get capacity(): number {\n return this.nums.length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n get size(): number {\n return this.queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.queSize === 0;\n }\n\n /* \u5165\u961f */\n push(num: number): void {\n if (this.size === this.capacity) {\n console.log('\u961f\u5217\u5df2\u6ee1');\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n const rear = (this.front + this.queSize) % this.capacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n this.nums[rear] = num;\n this.queSize++;\n }\n\n /* \u51fa\u961f */\n pop(): number {\n const num = this.peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n this.front = (this.front + 1) % this.capacity;\n this.queSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n peek(): number {\n if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n return this.nums[this.front];\n }\n\n /* \u8fd4\u56de Array */\n toArray(): number[] {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n const arr = new Array(this.size);\n for (let i = 0, j = this.front; i < this.size; i++, j++) {\n arr[i] = this.nums[j % this.capacity];\n }\n return arr;\n }\n}\n
array_queue.rs/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n late List<int> _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n late int _queSize; // \u961f\u5217\u957f\u5ea6\n\n ArrayQueue(int capacity) {\n _nums = List.filled(capacity, 0);\n _front = _queSize = 0;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n int capaCity() {\n return _nums.length;\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n int size() {\n return _queSize;\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _queSize == 0;\n }\n\n /* \u5165\u961f */\n void push(int _num) {\n if (_queSize == capaCity()) {\n throw Exception(\"\u961f\u5217\u5df2\u6ee1\");\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (_front + _queSize) % capaCity();\n // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n _nums[rear] = _num;\n _queSize++;\n }\n\n /* \u51fa\u961f */\n int pop() {\n int _num = peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n _front = (_front + 1) % capaCity();\n _queSize--;\n return _num;\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n int peek() {\n if (isEmpty()) {\n throw Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n }\n return _nums[_front];\n }\n\n /* \u8fd4\u56de Array */\n List<int> toArray() {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n final List<int> res = List.filled(_queSize, 0);\n for (int i = 0, j = _front; i < _queSize; i++, j++) {\n res[i] = _nums[j % capaCity()];\n }\n return res;\n }\n}\n
array_queue.c/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct ArrayQueue {\n nums: Vec<i32>, // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n front: i32, // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n que_size: i32, // \u961f\u5217\u957f\u5ea6\n que_capacity: i32, // \u961f\u5217\u5bb9\u91cf\n}\n\nimpl ArrayQueue {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new(capacity: i32) -> ArrayQueue {\n ArrayQueue {\n nums: vec![0; capacity as usize],\n front: 0,\n que_size: 0,\n que_capacity: capacity,\n }\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n fn capacity(&self) -> i32 {\n self.que_capacity\n }\n\n /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n fn size(&self) -> i32 {\n self.que_size\n }\n\n /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n fn is_empty(&self) -> bool {\n self.que_size == 0\n }\n\n /* \u5165\u961f */\n fn push(&mut self, num: i32) {\n if self.que_size == self.capacity() {\n println!(\"\u961f\u5217\u5df2\u6ee1\");\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n let rear = (self.front + self.que_size) % self.que_capacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n self.nums[rear as usize] = num;\n self.que_size += 1;\n }\n\n /* \u51fa\u961f */\n fn pop(&mut self) -> i32 {\n let num = self.peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n self.front = (self.front + 1) % self.que_capacity;\n self.que_size -= 1;\n num\n }\n\n /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n fn peek(&self) -> i32 {\n if self.is_empty() {\n panic!(\"index out of bounds\");\n }\n self.nums[self.front as usize]\n }\n\n /* \u8fd4\u56de\u6570\u7ec4 */\n fn to_vector(&self) -> Vec<i32> {\n let cap = self.que_capacity;\n let mut j = self.front;\n let mut arr = vec![0; self.que_size as usize];\n for i in 0..self.que_size {\n arr[i as usize] = self.nums[(j % cap) as usize];\n j += 1;\n }\n arr\n }\n}\n
array_queue.zig/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n int *nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n int queSize; // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayQueue *newArrayQueue(int capacity) {\n ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n // \u521d\u59cb\u5316\u6570\u7ec4\n queue->queCapacity = capacity;\n queue->nums = (int *)malloc(sizeof(int) * queue->queCapacity);\n queue->front = queue->queSize = 0;\n return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(ArrayQueue *queue) {\n free(queue->nums);\n free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\n return queue->queCapacity;\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayQueue *queue) {\n return queue->queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayQueue *queue) {\n return queue->queSize == 0;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\n assert(size(queue) != 0);\n return queue->nums[queue->front];\n}\n\n/* \u5165\u961f */\nvoid push(ArrayQueue *queue, int num) {\n if (size(queue) == capacity(queue)) {\n printf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\n return;\n }\n // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n int rear = (queue->front + queue->queSize) % queue->queCapacity;\n // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n queue->nums[rear] = num;\n queue->queSize++;\n}\n\n/* \u51fa\u961f */\nint pop(ArrayQueue *queue) {\n int num = peek(queue);\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n queue->front = (queue->front + 1) % queue->queCapacity;\n queue->queSize--;\n return num;\n}\n// \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\n return struct {\n const Self = @This();\n\n nums: []T = undefined, // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4 \n cap: usize = 0, // \u961f\u5217\u5bb9\u91cf\n front: usize = 0, // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n queSize: usize = 0, // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.cap = cap;\n self.nums = try self.mem_allocator.alloc(T, self.cap);\n @memset(self.nums, @as(T, 0));\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\n pub fn capacity(self: *Self) usize {\n return self.cap;\n }\n\n // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.queSize;\n }\n\n // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.queSize == 0;\n }\n\n // \u5165\u961f\n pub fn push(self: *Self, num: T) !void {\n if (self.size() == self.capacity()) {\n std.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\n return;\n }\n // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n var rear = (self.front + self.queSize) % self.capacity();\n // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n self.nums[rear] = num;\n self.queSize += 1;\n } \n\n // \u51fa\u961f\n pub fn pop(self: *Self) T {\n var num = self.peek();\n // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n self.front = (self.front + 1) % self.capacity();\n self.queSize -= 1;\n return num;\n } \n\n // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n pub fn peek(self: *Self) T {\n if (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n return self.nums[self.front];\n } \n\n // \u8fd4\u56de\u6570\u7ec4\n pub fn toArray(self: *Self) ![]T {\n // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n var res = try self.mem_allocator.alloc(T, self.size());\n @memset(res, @as(T, 0));\n var i: usize = 0;\n var j: usize = self.front;\n while (i < self.size()) : ({ i += 1; j += 1; }) {\n res[i] = self.nums[j % self.capacity()];\n }\n return res;\n }\n };\n}\n\u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u7136\u5177\u6709\u5c40\u9650\u6027\uff1a\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u95ee\u9898\u4e0d\u96be\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u52a8\u6001\u6570\u7ec4\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002
\u4e24\u79cd\u5b9e\u73b0\u7684\u5bf9\u6bd4\u7ed3\u8bba\u4e0e\u6808\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
"},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3 \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"- \u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c06\u52a0\u5165\u961f\u5217\u4e2d\uff0c\u7cfb\u7edf\u968f\u540e\u4f1a\u6839\u636e\u987a\u5e8f\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u671f\u95f4\uff0c\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u8ba2\u5355\uff0c\u9ad8\u5e76\u53d1\u6210\u4e3a\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u653b\u514b\u7684\u95ee\u9898\u3002
- \u5404\u7c7b\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u529f\u80fd\u7684\u573a\u666f\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\uff0c\u961f\u5217\u5728\u8fd9\u4e9b\u573a\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7ef4\u62a4\u5904\u7406\u987a\u5e8f\u3002
\u300c\u6808 stack\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u903b\u8f91\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002
\u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u60f3\u53d6\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u79fb\u8d70\u3002\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u5404\u79cd\u7c7b\u578b\u7684\u5143\u7d20\uff08\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u8fd9\u79cd\u6570\u636e\u7ed3\u6784\u3002
\u5982\u56fe 5-1 \u6240\u793a\uff0c\u6211\u4eec\u628a\u5806\u53e0\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u201c\u6808\u9876\u201d\uff0c\u5e95\u90e8\u79f0\u4e3a\u201c\u6808\u5e95\u201d\u3002\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u5165\u6808\u201d\uff0c\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u51fa\u6808\u201d\u3002
\u56fe 5-1 \u00a0 \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219
"},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1 \u00a0 \u6808\u5e38\u7528\u64cd\u4f5c","text":"\u6808\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-1 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u4eec\u4ee5\u5e38\u89c1\u7684
push()\u3001pop()\u3001peek()\u547d\u540d\u4e3a\u4f8b\u3002\u8868 5-1 \u00a0 \u6808\u7684\u64cd\u4f5c\u6548\u7387
\u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) pop() \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) peek() \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\)\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u6808\u7c7b\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8bed\u8a00\u53ef\u80fd\u6ca1\u6709\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u8bed\u8a00\u7684\u201c\u6570\u7ec4\u201d\u6216\u201c\u94fe\u8868\u201d\u5f53\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u5728\u7a0b\u5e8f\u903b\u8f91\u4e0a\u5ffd\u7565\u4e0e\u6808\u65e0\u5173\u7684\u64cd\u4f5c\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig stack.py
stack.cpp# \u521d\u59cb\u5316\u6808\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: list[int] = []\n\n# \u5143\u7d20\u5165\u6808\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npeek: int = stack[-1]\n\n# \u5143\u7d20\u51fa\u6808\npop: int = stack.pop()\n\n# \u83b7\u53d6\u6808\u7684\u957f\u5ea6\nsize: int = len(stack)\n\n# \u5224\u65ad\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(stack) == 0\n
stack.java/* \u521d\u59cb\u5316\u6808 */\nstack<int> stack;\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n
stack.cs/* \u521d\u59cb\u5316\u6808 */\nStack<Integer> stack = new Stack<>();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n
stack_test.go/* \u521d\u59cb\u5316\u6808 */\nStack<int> stack = new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count == 0;\n
stack.swift/* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n
stack.js/* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n
stack.ts/* \u521d\u59cb\u5316\u6808 */\n// Javascript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
stack.dart/* \u521d\u59cb\u5316\u6808 */\n// Typescript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
stack.rs/* \u521d\u59cb\u5316\u6808 */\n// Dart \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nList<int> stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.last;\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.removeLast();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.isEmpty;\n
stack.c/* \u521d\u59cb\u5316\u6808 */\n// \u628a Vec \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nlet mut stack: Vec<i32> = Vec::new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet top = stack.last().unwrap();\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.pop().unwrap();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.len();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = stack.is_empty();\n
stack.zig// C \u672a\u63d0\u4f9b\u5185\u7f6e\u6808\n
"},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2 \u00a0 \u6808\u7684\u5b9e\u73b0","text":"\n\u4e3a\u4e86\u6df1\u5165\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u81ea\u5df1\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002
\u6808\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u539f\u5219\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u548c\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u6808\u53ef\u4ee5\u89c6\u4e3a\u4e00\u79cd\u53d7\u9650\u5236\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u5176\u5bf9\u5916\u8868\u73b0\u7684\u903b\u8f91\u7b26\u5408\u6808\u7684\u7279\u6027\u3002
"},{"location":"chapter_stack_and_queue/stack/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u5934\u8282\u70b9\u89c6\u4e3a\u6808\u9876\uff0c\u5c3e\u8282\u70b9\u89c6\u4e3a\u6808\u5e95\u3002
\u5982\u56fe 5-2 \u6240\u793a\uff0c\u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u5143\u7d20\u63d2\u5165\u94fe\u8868\u5934\u90e8\uff0c\u8fd9\u79cd\u8282\u70b9\u63d2\u5165\u65b9\u6cd5\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06\u5934\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002
LinkedListStackpush()pop()\u56fe 5-2 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c
\u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_stack.py
linkedlist_stack.cppclass LinkedListStack:\n \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._peek: ListNode | None = None\n self._size: int = 0\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n return self._size\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n return not self._peek\n\n def push(self, val: int):\n \"\"\"\u5165\u6808\"\"\"\n node = ListNode(val)\n node.next = self._peek\n self._peek = node\n self._size += 1\n\n def pop(self) -> int:\n \"\"\"\u51fa\u6808\"\"\"\n num = self.peek()\n self._peek = self._peek.next\n self._size -= 1\n return num\n\n def peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u6808\u4e3a\u7a7a\")\n return self._peek.val\n\n def to_list(self) -> list[int]:\n \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n arr = []\n node = self._peek\n while node:\n arr.append(node.val)\n node = node.next\n arr.reverse()\n return arr\n
linkedlist_stack.java/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n private:\n ListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n int stkSize; // \u6808\u7684\u957f\u5ea6\n\n public:\n LinkedListStack() {\n stackTop = nullptr;\n stkSize = 0;\n }\n\n ~LinkedListStack() {\n // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n freeMemoryLinkedList(stackTop);\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n int size() {\n return stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u6808 */\n void push(int num) {\n ListNode *node = new ListNode(num);\n node->next = stackTop;\n stackTop = node;\n stkSize++;\n }\n\n /* \u51fa\u6808 */\n int pop() {\n int num = top();\n ListNode *tmp = stackTop;\n stackTop = stackTop->next;\n // \u91ca\u653e\u5185\u5b58\n delete tmp;\n stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n int top() {\n if (isEmpty())\n throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n return stackTop->val;\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n vector<int> toVector() {\n ListNode *node = stackTop;\n vector<int> res(size());\n for (int i = res.size() - 1; i >= 0; i--) {\n res[i] = node->val;\n node = node->next;\n }\n return res;\n }\n};\n
linkedlist_stack.cs/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n private ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n private int stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n public LinkedListStack() {\n stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n public int size() {\n return stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u6808 */\n public void push(int num) {\n ListNode node = new ListNode(num);\n node.next = stackPeek;\n stackPeek = node;\n stkSize++;\n }\n\n /* \u51fa\u6808 */\n public int pop() {\n int num = peek();\n stackPeek = stackPeek.next;\n stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n public int peek() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return stackPeek.val;\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] toArray() {\n ListNode node = stackPeek;\n int[] res = new int[size()];\n for (int i = res.length - 1; i >= 0; i--) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_stack.go/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n ListNode? stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n int stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n public LinkedListStack() {\n stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n public int Size() {\n return stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return Size() == 0;\n }\n\n /* \u5165\u6808 */\n public void Push(int num) {\n ListNode node = new(num) {\n next = stackPeek\n };\n stackPeek = node;\n stkSize++;\n }\n\n /* \u51fa\u6808 */\n public int Pop() {\n int num = Peek();\n stackPeek = stackPeek!.next;\n stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n public int Peek() {\n if (IsEmpty())\n throw new Exception();\n return stackPeek!.val;\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] ToArray() {\n if (stackPeek == null)\n return [];\n\n ListNode? node = stackPeek;\n int[] res = new int[Size()];\n for (int i = res.Length - 1; i >= 0; i--) {\n res[i] = node!.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_stack.swift/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\n data *list.List\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\n return &linkedListStack{\n data: list.New(),\n }\n}\n\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\n s.data.PushBack(value)\n}\n\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Back()\n s.data.Remove(e)\n return e.Value\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\n if s.isEmpty() {\n return nil\n }\n e := s.data.Back()\n return e.Value\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\n return s.data.Len()\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\n return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\n return s.data\n}\n
linkedlist_stack.js/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n private var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n private var _size = 0 // \u6808\u7684\u957f\u5ea6\n\n init() {}\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n func size() -> Int {\n _size\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n size() == 0\n }\n\n /* \u5165\u6808 */\n func push(num: Int) {\n let node = ListNode(x: num)\n node.next = _peek\n _peek = node\n _size += 1\n }\n\n /* \u51fa\u6808 */\n @discardableResult\n func pop() -> Int {\n let num = peek()\n _peek = _peek?.next\n _size -= 1\n return num\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n func peek() -> Int {\n if isEmpty() {\n fatalError(\"\u6808\u4e3a\u7a7a\")\n }\n return _peek!.val\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n func toArray() -> [Int] {\n var node = _peek\n var res = Array(repeating: 0, count: _size)\n for i in sequence(first: res.count - 1, next: { $0 >= 0 + 1 ? $0 - 1 : nil }) {\n res[i] = node!.val\n node = node?.next\n }\n return res\n }\n}\n
linkedlist_stack.ts/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n #stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n #stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n constructor() {\n this.#stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n get size() {\n return this.#stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.size === 0;\n }\n\n /* \u5165\u6808 */\n push(num) {\n const node = new ListNode(num);\n node.next = this.#stackPeek;\n this.#stackPeek = node;\n this.#stkSize++;\n }\n\n /* \u51fa\u6808 */\n pop() {\n const num = this.peek();\n this.#stackPeek = this.#stackPeek.next;\n this.#stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n peek() {\n if (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n return this.#stackPeek.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n toArray() {\n let node = this.#stackPeek;\n const res = new Array(this.size);\n for (let i = res.length - 1; i >= 0; i--) {\n res[i] = node.val;\n node = node.next;\n }\n return res;\n }\n}\n
linkedlist_stack.dart/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n private stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n private stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\n\n constructor() {\n this.stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n get size(): number {\n return this.stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.size === 0;\n }\n\n /* \u5165\u6808 */\n push(num: number): void {\n const node = new ListNode(num);\n node.next = this.stackPeek;\n this.stackPeek = node;\n this.stkSize++;\n }\n\n /* \u51fa\u6808 */\n pop(): number {\n const num = this.peek();\n if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n this.stackPeek = this.stackPeek.next;\n this.stkSize--;\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n peek(): number {\n if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n return this.stackPeek.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n toArray(): number[] {\n let node = this.stackPeek;\n const res = new Array<number>(this.size);\n for (let i = res.length - 1; i >= 0; i--) {\n res[i] = node!.val;\n node = node!.next;\n }\n return res;\n }\n}\n
linkedlist_stack.rs/* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n ListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n int _stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n LinkedListStack() {\n _stackPeek = null;\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n int size() {\n return _stkSize;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _stkSize == 0;\n }\n\n /* \u5165\u6808 */\n void push(int _num) {\n final ListNode node = ListNode(_num);\n node.next = _stackPeek;\n _stackPeek = node;\n _stkSize++;\n }\n\n /* \u51fa\u6808 */\n int pop() {\n final int _num = peek();\n _stackPeek = _stackPeek!.next;\n _stkSize--;\n return _num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n int peek() {\n if (_stackPeek == null) {\n throw Exception(\"\u6808\u4e3a\u7a7a\");\n }\n return _stackPeek!.val;\n }\n\n /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\n List<int> toList() {\n ListNode? node = _stackPeek;\n List<int> list = [];\n while (node != null) {\n list.add(node.val);\n node = node.next;\n }\n list = list.reversed.toList();\n return list;\n }\n}\n
linkedlist_stack.c/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\n#[allow(dead_code)]\npub struct LinkedListStack<T> {\n stack_peek: Option<Rc<RefCell<ListNode<T>>>>, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n stk_size: usize, // \u6808\u7684\u957f\u5ea6\n}\n\nimpl<T: Copy> LinkedListStack<T> {\n pub fn new() -> Self {\n Self {\n stack_peek: None,\n stk_size: 0,\n }\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n pub fn size(&self) -> usize {\n return self.stk_size;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n pub fn is_empty(&self) -> bool {\n return self.size() == 0;\n }\n\n /* \u5165\u6808 */\n pub fn push(&mut self, num: T) {\n let node = ListNode::new(num);\n node.borrow_mut().next = self.stack_peek.take();\n self.stack_peek = Some(node);\n self.stk_size += 1;\n }\n\n /* \u51fa\u6808 */\n pub fn pop(&mut self) -> Option<T> {\n self.stack_peek.take().map(|old_head| {\n match old_head.borrow_mut().next.take() {\n Some(new_head) => {\n self.stack_peek = Some(new_head);\n }\n None => {\n self.stack_peek = None;\n }\n }\n self.stk_size -= 1;\n Rc::try_unwrap(old_head).ok().unwrap().into_inner().val\n })\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n pub fn peek(&self) -> Option<&Rc<RefCell<ListNode<T>>>> {\n self.stack_peek.as_ref()\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n pub fn to_array(&self, head: Option<&Rc<RefCell<ListNode<T>>>>) -> Vec<T> {\n if let Some(node) = head {\n let mut nums = self.to_array(node.borrow().next.as_ref());\n nums.push(node.borrow().val);\n return nums;\n }\n return Vec::new();\n }\n}\n
linkedlist_stack.zig/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n ListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n int size; // \u6808\u7684\u957f\u5ea6\n} LinkedListStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListStack *newLinkedListStack() {\n LinkedListStack *s = malloc(sizeof(LinkedListStack));\n s->top = NULL;\n s->size = 0;\n return s;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(LinkedListStack *s) {\n while (s->top) {\n ListNode *n = s->top->next;\n free(s->top);\n s->top = n;\n }\n free(s);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(LinkedListStack *s) {\n return s->size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(LinkedListStack *s) {\n return size(s) == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(LinkedListStack *s, int num) {\n ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n node->next = s->top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\n node->val = num; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\n s->top = node; // \u66f4\u65b0\u6808\u9876\n s->size++; // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(LinkedListStack *s) {\n if (s->size == 0) {\n printf(\"\u6808\u4e3a\u7a7a\\n\");\n return INT_MAX;\n }\n return s->top->val;\n}\n\n/* \u51fa\u6808 */\nint pop(LinkedListStack *s) {\n int val = peek(s);\n ListNode *tmp = s->top;\n s->top = s->top->next;\n // \u91ca\u653e\u5185\u5b58\n free(tmp);\n s->size--;\n return val;\n}\n
"},{"location":"chapter_stack_and_queue/stack/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\n return struct {\n const Self = @This();\n\n stack_top: ?*inc.ListNode(T) = null, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n stk_size: usize = 0, // \u6808\u7684\u957f\u5ea6\n mem_arena: ?std.heap.ArenaAllocator = null,\n mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n if (self.mem_arena == null) {\n self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n self.mem_allocator = self.mem_arena.?.allocator();\n }\n self.stack_top = null;\n self.stk_size = 0;\n }\n\n // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.mem_arena == null) return;\n self.mem_arena.?.deinit();\n }\n\n // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.stk_size;\n }\n\n // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.size() == 0;\n }\n\n // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n pub fn peek(self: *Self) T {\n if (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\n return self.stack_top.?.val;\n } \n\n // \u5165\u6808\n pub fn push(self: *Self, num: T) !void {\n var node = try self.mem_allocator.create(inc.ListNode(T));\n node.init(num);\n node.next = self.stack_top;\n self.stack_top = node;\n self.stk_size += 1;\n } \n\n // \u51fa\u6808\n pub fn pop(self: *Self) T {\n var num = self.peek();\n self.stack_top = self.stack_top.?.next;\n self.stk_size -= 1;\n return num;\n } \n\n // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\n pub fn toArray(self: *Self) ![]T {\n var node = self.stack_top;\n var res = try self.mem_allocator.alloc(T, self.size());\n @memset(res, @as(T, 0));\n var i: usize = 0;\n while (i < res.len) : (i += 1) {\n res[res.len - i - 1] = node.?.val;\n node = node.?.next;\n }\n return res;\n }\n };\n}\n\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u4f5c\u4e3a\u6808\u9876\u3002\u5982\u56fe 5-3 \u6240\u793a\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u5206\u522b\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u4e0e\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002
ArrayStackpush()pop()\u56fe 5-3 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c
\u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u4f1a\u6e90\u6e90\u4e0d\u65ad\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u6570\u7ec4\uff0c\u8fd9\u6837\u5c31\u65e0\u987b\u81ea\u884c\u5904\u7406\u6570\u7ec4\u6269\u5bb9\u95ee\u9898\u3002\u4ee5\u4e0b\u4e3a\u793a\u4f8b\u4ee3\u7801\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig array_stack.py
array_stack.cppclass ArrayStack:\n \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\n\n def __init__(self):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._stack: list[int] = []\n\n def size(self) -> int:\n \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n return len(self._stack)\n\n def is_empty(self) -> bool:\n \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n return self._stack == []\n\n def push(self, item: int):\n \"\"\"\u5165\u6808\"\"\"\n self._stack.append(item)\n\n def pop(self) -> int:\n \"\"\"\u51fa\u6808\"\"\"\n if self.is_empty():\n raise IndexError(\"\u6808\u4e3a\u7a7a\")\n return self._stack.pop()\n\n def peek(self) -> int:\n \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n if self.is_empty():\n raise IndexError(\"\u6808\u4e3a\u7a7a\")\n return self._stack[-1]\n\n def to_list(self) -> list[int]:\n \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n return self._stack\n
array_stack.java/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n private:\n vector<int> stack;\n\n public:\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n int size() {\n return stack.size();\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return stack.size() == 0;\n }\n\n /* \u5165\u6808 */\n void push(int num) {\n stack.push_back(num);\n }\n\n /* \u51fa\u6808 */\n int pop() {\n int num = top();\n stack.pop_back();\n return num;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n int top() {\n if (isEmpty())\n throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n return stack.back();\n }\n\n /* \u8fd4\u56de Vector */\n vector<int> toVector() {\n return stack;\n }\n};\n
array_stack.cs/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n private ArrayList<Integer> stack;\n\n public ArrayStack() {\n // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n stack = new ArrayList<>();\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n public int size() {\n return stack.size();\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n public boolean isEmpty() {\n return size() == 0;\n }\n\n /* \u5165\u6808 */\n public void push(int num) {\n stack.add(num);\n }\n\n /* \u51fa\u6808 */\n public int pop() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return stack.remove(size() - 1);\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n public int peek() {\n if (isEmpty())\n throw new IndexOutOfBoundsException();\n return stack.get(size() - 1);\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public Object[] toArray() {\n return stack.toArray();\n }\n}\n
array_stack.go/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n List<int> stack;\n public ArrayStack() {\n // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n stack = [];\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n public int Size() {\n return stack.Count;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n public bool IsEmpty() {\n return Size() == 0;\n }\n\n /* \u5165\u6808 */\n public void Push(int num) {\n stack.Add(num);\n }\n\n /* \u51fa\u6808 */\n public int Pop() {\n if (IsEmpty())\n throw new Exception();\n var val = Peek();\n stack.RemoveAt(Size() - 1);\n return val;\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n public int Peek() {\n if (IsEmpty())\n throw new Exception();\n return stack[Size() - 1];\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n public int[] ToArray() {\n return [.. stack];\n }\n}\n
array_stack.swift/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\n data []int // \u6570\u636e\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\n return &arrayStack{\n // \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\n data: make([]int, 0, 16),\n }\n}\n\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\n return len(s.data)\n}\n\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\n return s.size() == 0\n}\n\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n // \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\n s.data = append(s.data, v)\n}\n\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\n val := s.peek()\n s.data = s.data[:len(s.data)-1]\n return val\n}\n\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\n if s.isEmpty() {\n return nil\n }\n val := s.data[len(s.data)-1]\n return val\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\n return s.data\n}\n
array_stack.js/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n private var stack: [Int]\n\n init() {\n // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n stack = []\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n func size() -> Int {\n stack.count\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n func isEmpty() -> Bool {\n stack.isEmpty\n }\n\n /* \u5165\u6808 */\n func push(num: Int) {\n stack.append(num)\n }\n\n /* \u51fa\u6808 */\n @discardableResult\n func pop() -> Int {\n if isEmpty() {\n fatalError(\"\u6808\u4e3a\u7a7a\")\n }\n return stack.removeLast()\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n func peek() -> Int {\n if isEmpty() {\n fatalError(\"\u6808\u4e3a\u7a7a\")\n }\n return stack.last!\n }\n\n /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n func toArray() -> [Int] {\n stack\n }\n}\n
array_stack.ts/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n #stack;\n constructor() {\n this.#stack = [];\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n get size() {\n return this.#stack.length;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n isEmpty() {\n return this.#stack.length === 0;\n }\n\n /* \u5165\u6808 */\n push(num) {\n this.#stack.push(num);\n }\n\n /* \u51fa\u6808 */\n pop() {\n if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n return this.#stack.pop();\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n top() {\n if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n return this.#stack[this.#stack.length - 1];\n }\n\n /* \u8fd4\u56de Array */\n toArray() {\n return this.#stack;\n }\n}\n
array_stack.dart/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n private stack: number[];\n constructor() {\n this.stack = [];\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n get size(): number {\n return this.stack.length;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n isEmpty(): boolean {\n return this.stack.length === 0;\n }\n\n /* \u5165\u6808 */\n push(num: number): void {\n this.stack.push(num);\n }\n\n /* \u51fa\u6808 */\n pop(): number | undefined {\n if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n return this.stack.pop();\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n top(): number | undefined {\n if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n return this.stack[this.stack.length - 1];\n }\n\n /* \u8fd4\u56de Array */\n toArray() {\n return this.stack;\n }\n}\n
array_stack.rs/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n late List<int> _stack;\n ArrayStack() {\n _stack = [];\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n int size() {\n return _stack.length;\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n bool isEmpty() {\n return _stack.isEmpty;\n }\n\n /* \u5165\u6808 */\n void push(int _num) {\n _stack.add(_num);\n }\n\n /* \u51fa\u6808 */\n int pop() {\n if (isEmpty()) {\n throw Exception(\"\u6808\u4e3a\u7a7a\");\n }\n return _stack.removeLast();\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n int peek() {\n if (isEmpty()) {\n throw Exception(\"\u6808\u4e3a\u7a7a\");\n }\n return _stack.last;\n }\n\n /* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n List<int> toArray() => _stack;\n}\n
array_stack.c/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct ArrayStack<T> {\n stack: Vec<T>,\n}\n\nimpl<T> ArrayStack<T> {\n /* \u521d\u59cb\u5316\u6808 */\n fn new() -> ArrayStack<T> {\n ArrayStack::<T> { stack: Vec::<T>::new() }\n }\n\n /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n fn size(&self) -> usize {\n self.stack.len()\n }\n\n /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n fn is_empty(&self) -> bool {\n self.size() == 0\n }\n\n /* \u5165\u6808 */\n fn push(&mut self, num: T) {\n self.stack.push(num);\n }\n\n /* \u51fa\u6808 */\n fn pop(&mut self) -> Option<T> {\n match self.stack.pop() {\n Some(num) => Some(num),\n None => None,\n }\n }\n\n /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n fn peek(&self) -> Option<&T> {\n if self.is_empty() { panic!(\"\u6808\u4e3a\u7a7a\") };\n self.stack.last()\n }\n\n /* \u8fd4\u56de &Vec */\n fn to_array(&self) -> &Vec<T> {\n &self.stack\n }\n}\n
array_stack.zig/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n int *data;\n int size;\n} ArrayStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayStack *newArrayStack() {\n ArrayStack *stack = malloc(sizeof(ArrayStack));\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\n stack->data = malloc(sizeof(int) * MAX_SIZE);\n stack->size = 0;\n return stack;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayStack(ArrayStack *stack) {\n free(stack->data);\n free(stack);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(ArrayStack *stack) {\n return stack->size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(ArrayStack *stack) {\n return stack->size == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(ArrayStack *stack, int num) {\n if (stack->size == MAX_SIZE) {\n printf(\"\u6808\u5df2\u6ee1\\n\");\n return;\n }\n stack->data[stack->size] = num;\n stack->size++;\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(ArrayStack *stack) {\n if (stack->size == 0) {\n printf(\"\u6808\u4e3a\u7a7a\\n\");\n return INT_MAX;\n }\n return stack->data[stack->size - 1];\n}\n\n/* \u51fa\u6808 */\nint pop(ArrayStack *stack) {\n int val = peek(stack);\n stack->size--;\n return val;\n}\n
"},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3 \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":"// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\n return struct {\n const Self = @This();\n\n stack: ?std.ArrayList(T) = null, \n\n // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n pub fn init(self: *Self, allocator: std.mem.Allocator) void {\n if (self.stack == null) {\n self.stack = std.ArrayList(T).init(allocator);\n }\n }\n\n // \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\n pub fn deinit(self: *Self) void {\n if (self.stack == null) return;\n self.stack.?.deinit();\n }\n\n // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n pub fn size(self: *Self) usize {\n return self.stack.?.items.len;\n }\n\n // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n pub fn isEmpty(self: *Self) bool {\n return self.size() == 0;\n }\n\n // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n pub fn peek(self: *Self) T {\n if (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\n return self.stack.?.items[self.size() - 1];\n } \n\n // \u5165\u6808\n pub fn push(self: *Self, num: T) !void {\n try self.stack.?.append(num);\n } \n\n // \u51fa\u6808\n pub fn pop(self: *Self) T {\n var num = self.stack.?.pop();\n return num;\n } \n\n // \u8fd4\u56de ArrayList\n pub fn toList(self: *Self) std.ArrayList(T) {\n return self.stack.?;\n }\n };\n}\n\u652f\u6301\u64cd\u4f5c
\u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u3002\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u8d85\u51fa\u4e86\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002
\u65f6\u95f4\u6548\u7387
\u5728\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u548c\u51fa\u6808\u64cd\u4f5c\u90fd\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u8fdb\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f83\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u5bfc\u81f4\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d8\u4e3a \\(O(n)\\) \u3002
\u5728\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u6548\u7387\u964d\u4f4e\u7684\u95ee\u9898\u3002\u4f46\u662f\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u8282\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5bf9\u8f83\u4f4e\u3002\u4e0d\u8fc7\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u672c\u8eab\u5c31\u662f\u8282\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u9ad8\u6548\u7387\u3002
\u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\u65f6\uff0c\u4f8b\u5982
int\u6216double\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002- \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u6548\u7387\u4f1a\u964d\u4f4e\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002
- \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002
\u7a7a\u95f4\u6548\u7387
\u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u4e3a\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\uff1b\u5e76\u4e14\uff0c\u6269\u5bb9\u673a\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\u7684\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002
\u7136\u800c\uff0c\u7531\u4e8e\u94fe\u8868\u8282\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64\u94fe\u8868\u8282\u70b9\u5360\u7528\u7684\u7a7a\u95f4\u76f8\u5bf9\u8f83\u5927\u3002
\u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u8282\u7701\u5185\u5b58\uff0c\u9700\u8981\u9488\u5bf9\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002
"},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4 \u00a0 \u6808\u5178\u578b\u5e94\u7528","text":"- \u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u4f1a\u5bf9\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u540e\u9000\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u4e2a\u7f51\u9875\u3002\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002
- \u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u4e8e\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u9636\u6bb5\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u5219\u4f1a\u4e0d\u65ad\u6267\u884c\u51fa\u6808\u64cd\u4f5c\u3002
- \u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002
- \u5728\u65f6\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u8f83\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u6269\u5bb9\u8fc7\u7a0b\u4e2d\uff0c\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6808\u7684\u94fe\u8868\u5b9e\u73b0\u5177\u6709\u66f4\u4e3a\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002
- \u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u53ef\u80fd\u5bfc\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u94fe\u8868\u8282\u70b9\u6240\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u6bd4\u6570\u7ec4\u5143\u7d20\u66f4\u5927\u3002
- \u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u540c\u6837\u53ef\u4ee5\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u7684\u5bf9\u6bd4\u4e0a\uff0c\u961f\u5217\u7684\u7ed3\u8bba\u4e0e\u524d\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u4f3c\u3002
- \u53cc\u5411\u961f\u5217\u662f\u4e00\u79cd\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u961f\u5217\uff0c\u5b83\u5141\u8bb8\u5728\u4e24\u7aef\u8fdb\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u3002
\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u662f\u5426\u662f\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\uff1f
\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u529f\u80fd\u672c\u8d28\u4e0a\u662f\u201c\u6808\u201d\u7684\u4f53\u73b0\u3002\u5f53\u7528\u6237\u8bbf\u95ee\u4e00\u4e2a\u65b0\u9875\u9762\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u88ab\u6dfb\u52a0\u5230\u6808\u9876\uff1b\u5f53\u7528\u6237\u70b9\u51fb\u540e\u9000\u6309\u94ae\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u4ece\u6808\u9876\u5f39\u51fa\u3002\u4f7f\u7528\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u65b9\u4fbf\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u5728\u201c\u53cc\u5411\u961f\u5217\u201d\u7ae0\u8282\u6709\u63d0\u5230\u3002
\u5728\u51fa\u6808\u540e\uff0c\u662f\u5426\u9700\u8981\u91ca\u653e\u51fa\u6808\u8282\u70b9\u7684\u5185\u5b58\uff1f
\u5982\u679c\u540e\u7eed\u4ecd\u9700\u8981\u4f7f\u7528\u5f39\u51fa\u8282\u70b9\uff0c\u5219\u4e0d\u9700\u8981\u91ca\u653e\u5185\u5b58\u3002\u82e5\u4e4b\u540e\u4e0d\u9700\u8981\u7528\u5230\uff0c
Java\u548cPython\u7b49\u8bed\u8a00\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\uff1b\u5728C\u548cC++\u4e2d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\u3002\u53cc\u5411\u961f\u5217\u50cf\u662f\u4e24\u4e2a\u6808\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u4e48\uff1f
\u53cc\u5411\u961f\u5217\u5c31\u50cf\u662f\u6808\u548c\u961f\u5217\u7684\u7ec4\u5408\uff0c\u6216\u4e24\u4e2a\u6808\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73b0\u7684\u662f\u6808 + \u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u6808\u4e0e\u961f\u5217\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u66f4\u52a0\u7075\u6d3b\u3002
\u64a4\u9500\uff08undo\uff09\u548c\u53cd\u64a4\u9500\uff08redo\uff09\u5177\u4f53\u662f\u5982\u4f55\u5b9e\u73b0\u7684\uff1f
\u4f7f\u7528\u4e24\u4e2a\u6808\uff0c\u6808
A\u7528\u4e8e\u64a4\u9500\uff0c\u6808B\u7528\u4e8e\u53cd\u64a4\u9500\u3002- \u6bcf\u5f53\u7528\u6237\u6267\u884c\u4e00\u4e2a\u64cd\u4f5c\uff0c\u5c06\u8fd9\u4e2a\u64cd\u4f5c\u538b\u5165\u6808
A\uff0c\u5e76\u6e05\u7a7a\u6808B\u3002 - \u5f53\u7528\u6237\u6267\u884c\u201c\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808
A\u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808B\u3002 - \u5f53\u7528\u6237\u6267\u884c\u201c\u53cd\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808
B\u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808A\u3002
Abstract
\u53c2\u5929\u5927\u6811\u5145\u6ee1\u751f\u547d\u529b\uff0c\u6839\u6df1\u53f6\u8302\uff0c\u5206\u679d\u6276\u758f\u3002
\u5b83\u4e3a\u6211\u4eec\u5c55\u73b0\u4e86\u6570\u636e\u5206\u6cbb\u7684\u751f\u52a8\u5f62\u6001\u3002
"},{"location":"chapter_tree/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"- 7.1 \u00a0 \u4e8c\u53c9\u6811
- 7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386
- 7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a
- 7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811
- 7.5 \u00a0 AVL \u6811 *
- 7.6 \u00a0 \u5c0f\u7ed3
\u5728\u94fe\u8868\u8868\u793a\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u5b58\u50a8\u5355\u5143\u4e3a\u8282\u70b9
TreeNode\uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u76f8\u8fde\u63a5\u3002\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u94fe\u8868\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002\u90a3\u4e48\uff0c\u6211\u4eec\u80fd\u5426\u7528\u6570\u7ec4\u6765\u8868\u793a\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002
"},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1 \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"\u5148\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\u3002\u7ed9\u5b9a\u4e00\u68f5\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u6240\u6709\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u5b58\u50a8\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u5219\u6bcf\u4e2a\u8282\u70b9\u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u6570\u7ec4\u7d22\u5f15\u3002
\u6839\u636e\u5c42\u5e8f\u904d\u5386\u7684\u7279\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u7236\u8282\u70b9\u7d22\u5f15\u4e0e\u5b50\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u201c\u6620\u5c04\u516c\u5f0f\u201d\uff1a\u82e5\u67d0\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002\u56fe 7-12 \u5c55\u793a\u4e86\u5404\u4e2a\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\u3002
\u56fe 7-12 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a
\u6620\u5c04\u516c\u5f0f\u7684\u89d2\u8272\u76f8\u5f53\u4e8e\u94fe\u8868\u4e2d\u7684\u6307\u9488\u3002\u7ed9\u5b9a\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u8282\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u901a\u8fc7\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5b83\u7684\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3002
"},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2 \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u53c9\u6811","text":"\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u4e00\u4e2a\u7279\u4f8b\uff0c\u5728\u4e8c\u53c9\u6811\u7684\u4e2d\u95f4\u5c42\u901a\u5e38\u5b58\u5728\u8bb8\u591a \\(\\text{None}\\) \u3002\u7531\u4e8e\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b \\(\\text{None}\\) \uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u6cd5\u4ec5\u51ed\u8be5\u5e8f\u5217\u6765\u63a8\u6d4b \\(\\text{None}\\) \u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\u3002\u8fd9\u610f\u5473\u7740\u5b58\u5728\u591a\u79cd\u4e8c\u53c9\u6811\u7ed3\u6784\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002
\u5982\u56fe 7-13 \u6240\u793a\uff0c\u7ed9\u5b9a\u4e00\u68f5\u975e\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u4e0a\u8ff0\u6570\u7ec4\u8868\u793a\u65b9\u6cd5\u5df2\u7ecf\u5931\u6548\u3002
\u56fe 7-13 \u00a0 \u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027
\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u663e\u5f0f\u5730\u5199\u51fa\u6240\u6709 \\(\\text{None}\\) \u3002\u5982\u56fe 7-14 \u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig# \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\n# \u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c INT_MAX \u6807\u8bb0\u7a7a\u4f4d\nvector<int> tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nList<int?> tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 None \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n\n\u56fe 7-14 \u00a0 \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a
\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u56de\u987e\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u5b9a\u4e49\uff0c\\(\\text{None}\\) \u53ea\u51fa\u73b0\u5728\u6700\u5e95\u5c42\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u56e0\u6b64\u6240\u6709 \\(\\text{None}\\) \u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002
\u8fd9\u610f\u5473\u7740\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u6240\u6709 \\(\\text{None}\\) \uff0c\u975e\u5e38\u65b9\u4fbf\u3002\u56fe 7-15 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u4f8b\u5b50\u3002
\u56fe 7-15 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a
\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u68f5\u57fa\u4e8e\u6570\u7ec4\u8868\u793a\u7684\u4e8c\u53c9\u6811\uff0c\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u64cd\u4f5c\u3002
- \u7ed9\u5b9a\u67d0\u8282\u70b9\uff0c\u83b7\u53d6\u5b83\u7684\u503c\u3001\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3001\u7236\u8282\u70b9\u3002
- \u83b7\u53d6\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3001\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002
array_binary_tree.cppclass ArrayBinaryTree:\n \"\"\"\u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b\"\"\"\n\n def __init__(self, arr: list[int | None]):\n \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n self._tree = list(arr)\n\n def size(self):\n \"\"\"\u8282\u70b9\u6570\u91cf\"\"\"\n return len(self._tree)\n\n def val(self, i: int) -> int:\n \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c\"\"\"\n # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if i < 0 or i >= self.size():\n return None\n return self._tree[i]\n\n def left(self, i: int) -> int | None:\n \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n return 2 * i + 1\n\n def right(self, i: int) -> int | None:\n \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n return 2 * i + 2\n\n def parent(self, i: int) -> int | None:\n \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n return (i - 1) // 2\n\n def level_order(self) -> list[int]:\n \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n self.res = []\n # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for i in range(self.size()):\n if self.val(i) is not None:\n self.res.append(self.val(i))\n return self.res\n\n def dfs(self, i: int, order: str):\n \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n if self.val(i) is None:\n return\n # \u524d\u5e8f\u904d\u5386\n if order == \"pre\":\n self.res.append(self.val(i))\n self.dfs(self.left(i), order)\n # \u4e2d\u5e8f\u904d\u5386\n if order == \"in\":\n self.res.append(self.val(i))\n self.dfs(self.right(i), order)\n # \u540e\u5e8f\u904d\u5386\n if order == \"post\":\n self.res.append(self.val(i))\n\n def pre_order(self) -> list[int]:\n \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n self.res = []\n self.dfs(0, order=\"pre\")\n return self.res\n\n def in_order(self) -> list[int]:\n \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n self.res = []\n self.dfs(0, order=\"in\")\n return self.res\n\n def post_order(self) -> list[int]:\n \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n self.res = []\n self.dfs(0, order=\"post\")\n return self.res\n
array_binary_tree.java/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n public:\n /* \u6784\u9020\u65b9\u6cd5 */\n ArrayBinaryTree(vector<int> arr) {\n tree = arr;\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n int size() {\n return tree.size();\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n int val(int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= size())\n return INT_MAX;\n return tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n int left(int i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n int right(int i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n int parent(int i) {\n return (i - 1) / 2;\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n vector<int> levelOrder() {\n vector<int> res;\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size(); i++) {\n if (val(i) != INT_MAX)\n res.push_back(val(i));\n }\n return res;\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n vector<int> preOrder() {\n vector<int> res;\n dfs(0, \"pre\", res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n vector<int> inOrder() {\n vector<int> res;\n dfs(0, \"in\", res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n vector<int> postOrder() {\n vector<int> res;\n dfs(0, \"post\", res);\n return res;\n }\n\n private:\n vector<int> tree;\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n void dfs(int i, string order, vector<int> &res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (val(i) == INT_MAX)\n return;\n // \u524d\u5e8f\u904d\u5386\n if (order == \"pre\")\n res.push_back(val(i));\n dfs(left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order == \"in\")\n res.push_back(val(i));\n dfs(right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order == \"post\")\n res.push_back(val(i));\n }\n};\n
array_binary_tree.cs/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n private List<Integer> tree;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n public ArrayBinaryTree(List<Integer> arr) {\n tree = new ArrayList<>(arr);\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n public int size() {\n return tree.size();\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n public Integer val(int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= size())\n return null;\n return tree.get(i);\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n public Integer left(int i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n public Integer right(int i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n public Integer parent(int i) {\n return (i - 1) / 2;\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n public List<Integer> levelOrder() {\n List<Integer> res = new ArrayList<>();\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size(); i++) {\n if (val(i) != null)\n res.add(val(i));\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n private void dfs(Integer i, String order, List<Integer> res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (val(i) == null)\n return;\n // \u524d\u5e8f\u904d\u5386\n if (\"pre\".equals(order))\n res.add(val(i));\n dfs(left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (\"in\".equals(order))\n res.add(val(i));\n dfs(right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (\"post\".equals(order))\n res.add(val(i));\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n public List<Integer> preOrder() {\n List<Integer> res = new ArrayList<>();\n dfs(0, \"pre\", res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n public List<Integer> inOrder() {\n List<Integer> res = new ArrayList<>();\n dfs(0, \"in\", res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n public List<Integer> postOrder() {\n List<Integer> res = new ArrayList<>();\n dfs(0, \"post\", res);\n return res;\n }\n}\n
array_binary_tree.go/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(List<int?> arr) {\n List<int?> tree = new(arr);\n\n /* \u8282\u70b9\u6570\u91cf */\n public int Size() {\n return tree.Count;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n public int? Val(int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= Size())\n return null;\n return tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n public int Left(int i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n public int Right(int i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n public int Parent(int i) {\n return (i - 1) / 2;\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n public List<int> LevelOrder() {\n List<int> res = [];\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < Size(); i++) {\n if (Val(i).HasValue)\n res.Add(Val(i)!.Value);\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n void DFS(int i, string order, List<int> res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (!Val(i).HasValue)\n return;\n // \u524d\u5e8f\u904d\u5386\n if (order == \"pre\")\n res.Add(Val(i)!.Value);\n DFS(Left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order == \"in\")\n res.Add(Val(i)!.Value);\n DFS(Right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order == \"post\")\n res.Add(Val(i)!.Value);\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n public List<int> PreOrder() {\n List<int> res = [];\n DFS(0, \"pre\", res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n public List<int> InOrder() {\n List<int> res = [];\n DFS(0, \"in\", res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n public List<int> PostOrder() {\n List<int> res = [];\n DFS(0, \"post\", res);\n return res;\n }\n}\n
array_binary_tree.swift/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\ntype arrayBinaryTree struct {\n tree []any\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n return &arrayBinaryTree{\n tree: arr,\n }\n}\n\n/* \u8282\u70b9\u6570\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\n return len(abt.tree)\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if i < 0 || i >= abt.size() {\n return nil\n }\n return abt.tree[i]\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n return 2*i + 1\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n return (i - 1) / 2\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n var res []any\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for i := 0; i < abt.size(); i++ {\n if abt.val(i) != nil {\n res = append(res, abt.val(i))\n }\n }\n return res\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if abt.val(i) == nil {\n return\n }\n // \u524d\u5e8f\u904d\u5386\n if order == \"pre\" {\n *res = append(*res, abt.val(i))\n }\n abt.dfs(abt.left(i), order, res)\n // \u4e2d\u5e8f\u904d\u5386\n if order == \"in\" {\n *res = append(*res, abt.val(i))\n }\n abt.dfs(abt.right(i), order, res)\n // \u540e\u5e8f\u904d\u5386\n if order == \"post\" {\n *res = append(*res, abt.val(i))\n }\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n var res []any\n abt.dfs(0, \"pre\", &res)\n return res\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n var res []any\n abt.dfs(0, \"in\", &res)\n return res\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n var res []any\n abt.dfs(0, \"post\", &res)\n return res\n}\n
array_binary_tree.js/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n private var tree: [Int?]\n\n /* \u6784\u9020\u65b9\u6cd5 */\n init(arr: [Int?]) {\n tree = arr\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n func size() -> Int {\n tree.count\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n func val(i: Int) -> Int? {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if i < 0 || i >= size() {\n return nil\n }\n return tree[i]\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n func left(i: Int) -> Int {\n 2 * i + 1\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n func right(i: Int) -> Int {\n 2 * i + 2\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n func parent(i: Int) -> Int {\n (i - 1) / 2\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n func levelOrder() -> [Int] {\n var res: [Int] = []\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for i in stride(from: 0, to: size(), by: 1) {\n if let val = val(i: i) {\n res.append(val)\n }\n }\n return res\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n private func dfs(i: Int, order: String, res: inout [Int]) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n guard let val = val(i: i) else {\n return\n }\n // \u524d\u5e8f\u904d\u5386\n if order == \"pre\" {\n res.append(val)\n }\n dfs(i: left(i: i), order: order, res: &res)\n // \u4e2d\u5e8f\u904d\u5386\n if order == \"in\" {\n res.append(val)\n }\n dfs(i: right(i: i), order: order, res: &res)\n // \u540e\u5e8f\u904d\u5386\n if order == \"post\" {\n res.append(val)\n }\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n func preOrder() -> [Int] {\n var res: [Int] = []\n dfs(i: 0, order: \"pre\", res: &res)\n return res\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n func inOrder() -> [Int] {\n var res: [Int] = []\n dfs(i: 0, order: \"in\", res: &res)\n return res\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n func postOrder() -> [Int] {\n var res: [Int] = []\n dfs(i: 0, order: \"post\", res: &res)\n return res\n }\n}\n
array_binary_tree.ts/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n #tree;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(arr) {\n this.#tree = arr;\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n size() {\n return this.#tree.length;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n val(i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= this.size()) return null;\n return this.#tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n left(i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n right(i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n parent(i) {\n return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n levelOrder() {\n let res = [];\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (let i = 0; i < this.size(); i++) {\n if (this.val(i) !== null) res.push(this.val(i));\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n #dfs(i, order, res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (this.val(i) === null) return;\n // \u524d\u5e8f\u904d\u5386\n if (order === 'pre') res.push(this.val(i));\n this.#dfs(this.left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order === 'in') res.push(this.val(i));\n this.#dfs(this.right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order === 'post') res.push(this.val(i));\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n preOrder() {\n const res = [];\n this.#dfs(0, 'pre', res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n inOrder() {\n const res = [];\n this.#dfs(0, 'in', res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n postOrder() {\n const res = [];\n this.#dfs(0, 'post', res);\n return res;\n }\n}\n
array_binary_tree.dart/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n #tree: (number | null)[];\n\n /* \u6784\u9020\u65b9\u6cd5 */\n constructor(arr: (number | null)[]) {\n this.#tree = arr;\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n size(): number {\n return this.#tree.length;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n val(i: number): number | null {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= this.size()) return null;\n return this.#tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n left(i: number): number {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n right(i: number): number {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n parent(i: number): number {\n return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n levelOrder(): number[] {\n let res = [];\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (let i = 0; i < this.size(); i++) {\n if (this.val(i) !== null) res.push(this.val(i));\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n #dfs(i: number, order: Order, res: (number | null)[]): void {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (this.val(i) === null) return;\n // \u524d\u5e8f\u904d\u5386\n if (order === 'pre') res.push(this.val(i));\n this.#dfs(this.left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order === 'in') res.push(this.val(i));\n this.#dfs(this.right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order === 'post') res.push(this.val(i));\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n preOrder(): (number | null)[] {\n const res = [];\n this.#dfs(0, 'pre', res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n inOrder(): (number | null)[] {\n const res = [];\n this.#dfs(0, 'in', res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n postOrder(): (number | null)[] {\n const res = [];\n this.#dfs(0, 'post', res);\n return res;\n }\n}\n
array_binary_tree.rs/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n late List<int?> _tree;\n\n /* \u6784\u9020\u65b9\u6cd5 */\n ArrayBinaryTree(this._tree);\n\n /* \u8282\u70b9\u6570\u91cf */\n int size() {\n return _tree.length;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n int? val(int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= size()) {\n return null;\n }\n return _tree[i];\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n int? left(int i) {\n return 2 * i + 1;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n int? right(int i) {\n return 2 * i + 2;\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n int? parent(int i) {\n return (i - 1) ~/ 2;\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n List<int> levelOrder() {\n List<int> res = [];\n for (int i = 0; i < size(); i++) {\n if (val(i) != null) {\n res.add(val(i)!);\n }\n }\n return res;\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n void dfs(int i, String order, List<int?> res) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (val(i) == null) {\n return;\n }\n // \u524d\u5e8f\u904d\u5386\n if (order == 'pre') {\n res.add(val(i));\n }\n dfs(left(i)!, order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if (order == 'in') {\n res.add(val(i));\n }\n dfs(right(i)!, order, res);\n // \u540e\u5e8f\u904d\u5386\n if (order == 'post') {\n res.add(val(i));\n }\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n List<int?> preOrder() {\n List<int?> res = [];\n dfs(0, 'pre', res);\n return res;\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n List<int?> inOrder() {\n List<int?> res = [];\n dfs(0, 'in', res);\n return res;\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n List<int?> postOrder() {\n List<int?> res = [];\n dfs(0, 'post', res);\n return res;\n }\n}\n
array_binary_tree.c/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nstruct ArrayBinaryTree {\n tree: Vec<Option<i32>>,\n}\n\nimpl ArrayBinaryTree {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new(arr: Vec<Option<i32>>) -> Self {\n Self { tree: arr }\n }\n\n /* \u8282\u70b9\u6570\u91cf */\n fn size(&self) -> i32 {\n self.tree.len() as i32\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n fn val(&self, i: i32) -> Option<i32> {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if i < 0 || i >= self.size() {\n None\n } else {\n self.tree[i as usize]\n }\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n fn left(&self, i: i32) -> i32 {\n 2 * i + 1\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n fn right(&self, i: i32) -> i32 {\n 2 * i + 2\n }\n\n /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n fn parent(&self, i: i32) -> i32 {\n (i - 1) / 2\n }\n\n /* \u5c42\u5e8f\u904d\u5386 */\n fn level_order(&self) -> Vec<i32> {\n let mut res = vec![];\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for i in 0..self.size() {\n if let Some(val) = self.val(i) {\n res.push(val)\n }\n }\n res\n }\n\n /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n fn dfs(&self, i: i32, order: &str, res: &mut Vec<i32>) {\n if self.val(i).is_none() {\n return;\n }\n let val = self.val(i).unwrap();\n // \u524d\u5e8f\u904d\u5386\n if order == \"pre\" {\n res.push(val);\n }\n self.dfs(self.left(i), order, res);\n // \u4e2d\u5e8f\u904d\u5386\n if order == \"in\" {\n res.push(val);\n }\n self.dfs(self.right(i), order, res);\n // \u540e\u5e8f\u904d\u5386\n if order == \"post\" {\n res.push(val);\n }\n }\n\n /* \u524d\u5e8f\u904d\u5386 */\n fn pre_order(&self) -> Vec<i32> {\n let mut res = vec![];\n self.dfs(0, \"pre\", &mut res);\n res\n }\n\n /* \u4e2d\u5e8f\u904d\u5386 */\n fn in_order(&self) -> Vec<i32> {\n let mut res = vec![];\n self.dfs(0, \"in\", &mut res);\n res\n }\n\n /* \u540e\u5e8f\u904d\u5386 */\n fn post_order(&self) -> Vec<i32> {\n let mut res = vec![];\n self.dfs(0, \"post\", &mut res);\n res\n }\n}\n
array_binary_tree.zig/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7ed3\u6784\u4f53 */\ntypedef struct {\n int *tree;\n int size;\n} ArrayBinaryTree;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n abt->tree = malloc(sizeof(int) * arrSize);\n memcpy(abt->tree, arr, sizeof(int) * arrSize);\n abt->size = arrSize;\n return abt;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n free(abt->tree);\n free(abt);\n}\n\n/* \u8282\u70b9\u6570\u91cf */\nint size(ArrayBinaryTree *abt) {\n return abt->size;\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n if (i < 0 || i >= size(abt))\n return INT_MAX;\n return abt->tree[i];\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n int *res = (int *)malloc(sizeof(int) * size(abt));\n int index = 0;\n // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n for (int i = 0; i < size(abt); i++) {\n if (val(abt, i) != INT_MAX)\n res[index++] = val(abt, i);\n }\n *returnSize = index;\n return res;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n if (val(abt, i) == INT_MAX)\n return;\n // \u524d\u5e8f\u904d\u5386\n if (strcmp(order, \"pre\") == 0)\n res[(*index)++] = val(abt, i);\n dfs(abt, left(i), order, res, index);\n // \u4e2d\u5e8f\u904d\u5386\n if (strcmp(order, \"in\") == 0)\n res[(*index)++] = val(abt, i);\n dfs(abt, right(i), order, res, index);\n // \u540e\u5e8f\u904d\u5386\n if (strcmp(order, \"post\") == 0)\n res[(*index)++] = val(abt, i);\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n int *res = (int *)malloc(sizeof(int) * size(abt));\n int index = 0;\n dfs(abt, 0, \"pre\", res, &index);\n *returnSize = index;\n return res;\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n int *res = (int *)malloc(sizeof(int) * size(abt));\n int index = 0;\n dfs(abt, 0, \"in\", res, &index);\n *returnSize = index;\n return res;\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n int *res = (int *)malloc(sizeof(int) * size(abt));\n int index = 0;\n dfs(abt, 0, \"post\", res, &index);\n *returnSize = index;\n return res;\n}\n
"},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"[class]{ArrayBinaryTree}-[func]{}\n\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\u4e3b\u8981\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002
- \u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u5bf9\u7f13\u5b58\u53cb\u597d\uff0c\u8bbf\u95ee\u4e0e\u904d\u5386\u901f\u5ea6\u8f83\u5feb\u3002
- \u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u6bd4\u8f83\u8282\u7701\u7a7a\u95f4\u3002
- \u5141\u8bb8\u968f\u673a\u8bbf\u95ee\u8282\u70b9\u3002
\u7136\u800c\uff0c\u6570\u7ec4\u8868\u793a\u4e5f\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002
- \u6570\u7ec4\u5b58\u50a8\u9700\u8981\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5b58\u50a8\u6570\u636e\u91cf\u8fc7\u5927\u7684\u6811\u3002
- \u589e\u5220\u8282\u70b9\u9700\u8981\u901a\u8fc7\u6570\u7ec4\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u5b9e\u73b0\uff0c\u6548\u7387\u8f83\u4f4e\u3002
- \u5f53\u4e8c\u53c9\u6811\u4e2d\u5b58\u5728\u5927\u91cf \\(\\text{None}\\) \u65f6\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u7684\u8282\u70b9\u6570\u636e\u6bd4\u91cd\u8f83\u4f4e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u8f83\u4f4e\u3002
\u5728\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\uff0c\u5728\u591a\u6b21\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u9000\u5316\u4e3a\u94fe\u8868\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c06\u4ece \\(O(\\log n)\\) \u6076\u5316\u4e3a \\(O(n)\\) \u3002
\u5982\u56fe 7-24 \u6240\u793a\uff0c\u7ecf\u8fc7\u4e24\u6b21\u5220\u9664\u8282\u70b9\u64cd\u4f5c\uff0c\u8fd9\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\u4fbf\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002
\u56fe 7-24 \u00a0 AVL \u6811\u5728\u5220\u9664\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316
\u518d\u4f8b\u5982\uff0c\u5728\u56fe 7-25 \u6240\u793a\u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u8282\u70b9\u540e\uff0c\u6811\u5c06\u4e25\u91cd\u5411\u5de6\u503e\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u6076\u5316\u3002
\u56fe 7-25 \u00a0 AVL \u6811\u5728\u63d2\u5165\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316
1962 \u5e74 G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u8bba\u6587 \"An algorithm for the organization of information\" \u4e2d\u63d0\u51fa\u4e86\u300cAVL \u6811\u300d\u3002\u8bba\u6587\u4e2d\u8be6\u7ec6\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u6301\u7eed\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4e0d\u4f1a\u9000\u5316\uff0c\u4ece\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u9700\u8981\u9891\u7e41\u8fdb\u884c\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u573a\u666f\u4e2d\uff0cAVL \u6811\u80fd\u59cb\u7ec8\u4fdd\u6301\u9ad8\u6548\u7684\u6570\u636e\u64cd\u4f5c\u6027\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002
"},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1 \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"AVL \u6811\u65e2\u662f\u4e8c\u53c9\u641c\u7d22\u6811\u4e5f\u662f\u5e73\u8861\u4e8c\u53c9\u6811\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u7c7b\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u4e5f\u88ab\u79f0\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 balanced binary search tree\u300d\u3002
"},{"location":"chapter_tree/avl_tree/#1","title":"1. \u00a0 \u8282\u70b9\u9ad8\u5ea6","text":"\u7531\u4e8e AVL \u6811\u7684\u76f8\u5173\u64cd\u4f5c\u9700\u8981\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e3a\u8282\u70b9\u7c7b\u6dfb\u52a0
PythonC++JavaC#GoSwiftJSTSDartRustCZigheight\u53d8\u91cf\uff1aclass TreeNode:\n \"\"\"AVL \u6811\u8282\u70b9\u7c7b\"\"\"\n def __init__(self, val: int):\n self.val: int = val # \u8282\u70b9\u503c\n self.height: int = 0 # \u8282\u70b9\u9ad8\u5ea6\n self.left: TreeNode | None = None # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n self.right: TreeNode | None = None # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n/* AVL \u6811\u8282\u70b9\u7c7b */\nstruct TreeNode {\n int val{}; // \u8282\u70b9\u503c\n int height = 0; // \u8282\u70b9\u9ad8\u5ea6\n TreeNode *left{}; // \u5de6\u5b50\u8282\u70b9\n TreeNode *right{}; // \u53f3\u5b50\u8282\u70b9\n TreeNode() = default;\n explicit TreeNode(int x) : val(x){}\n};\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n public int val; // \u8282\u70b9\u503c\n public int height; // \u8282\u70b9\u9ad8\u5ea6\n public TreeNode left; // \u5de6\u5b50\u8282\u70b9\n public TreeNode right; // \u53f3\u5b50\u8282\u70b9\n public TreeNode(int x) { val = x; }\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n public int? val = x; // \u8282\u70b9\u503c\n public int height; // \u8282\u70b9\u9ad8\u5ea6\n public TreeNode? left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n Val int // \u8282\u70b9\u503c\n Height int // \u8282\u70b9\u9ad8\u5ea6\n Left *TreeNode // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n Right *TreeNode // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n var val: Int // \u8282\u70b9\u503c\n var height: Int // \u8282\u70b9\u9ad8\u5ea6\n var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\n var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\n\n init(x: Int) {\n val = x\n height = 0\n }\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n val; // \u8282\u70b9\u503c\n height; //\u8282\u70b9\u9ad8\u5ea6\n left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n constructor(val, left, right, height) {\n this.val = val === undefined ? 0 : val;\n this.height = height === undefined ? 0 : height;\n this.left = left === undefined ? null : left;\n this.right = right === undefined ? null : right;\n }\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n val: number; // \u8282\u70b9\u503c\n height: number; // \u8282\u70b9\u9ad8\u5ea6\n left: TreeNode | null; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n right: TreeNode | null; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n this.val = val === undefined ? 0 : val;\n this.height = height === undefined ? 0 : height; \n this.left = left === undefined ? null : left; \n this.right = right === undefined ? null : right; \n }\n}\n/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n int val; // \u8282\u70b9\u503c\n int height; // \u8282\u70b9\u9ad8\u5ea6\n TreeNode? left; // \u5de6\u5b50\u8282\u70b9\n TreeNode? right; // \u53f3\u5b50\u8282\u70b9\n TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n val: i32, // \u8282\u70b9\u503c\n height: i32, // \u8282\u70b9\u9ad8\u5ea6\n left: Option<Rc<RefCell<TreeNode>>>, // \u5de6\u5b50\u8282\u70b9\n right: Option<Rc<RefCell<TreeNode>>>, // \u53f3\u5b50\u8282\u70b9\n}\n\nimpl TreeNode {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new(val: i32) -> Rc<RefCell<Self>> {\n Rc::new(RefCell::new(Self {\n val,\n height: 0,\n left: None,\n right: None\n }))\n }\n}\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nTreeNode struct TreeNode {\n int val;\n int height;\n struct TreeNode *left;\n struct TreeNode *right;\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n TreeNode *node;\n\n node = (TreeNode *)malloc(sizeof(TreeNode));\n node->val = val;\n node->height = 0;\n node->left = NULL;\n node->right = NULL;\n return node;\n}\n\n\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u662f\u6307\u4ece\u8be5\u8282\u70b9\u5230\u5176\u6700\u8fdc\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u6240\u7ecf\u8fc7\u7684\u201c\u8fb9\u201d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\u7684\u662f\uff0c\u53f6\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(0\\) \uff0c\u800c\u7a7a\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(-1\\) \u3002\u6211\u4eec\u5c06\u521b\u5efa\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u548c\u66f4\u65b0\u8282\u70b9\u7684\u9ad8\u5ea6\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef height(self, node: TreeNode | None) -> int:\n \"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n # \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n if node is not None:\n return node.height\n return -1\n\ndef update_height(self, node: TreeNode | None):\n \"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n # \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height = max([self.height(node.left), self.height(node.right)]) + 1\n
avl_tree.java/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node == nullptr ? -1 : node->height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node->height = max(height(node->left), height(node->right)) + 1;\n}\n
avl_tree.cs/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n
avl_tree.go/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n
avl_tree.swift/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n if node != nil {\n return node.Height\n }\n return -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n lh := t.height(node.Left)\n rh := t.height(node.Right)\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n if lh > rh {\n node.Height = lh + 1\n } else {\n node.Height = rh + 1\n }\n}\n
avl_tree.js/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -> Int {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n node == nil ? -1 : node!.height\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n
avl_tree.ts/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height =\n Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
avl_tree.dart/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.height =\n Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
avl_tree.rs/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node!.height = max(height(node.left), height(node.right)) + 1;\n}\n
avl_tree.c/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -> i32 {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n match node {\n Some(node) => node.borrow().height,\n None => -1,\n }\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\n if let Some(node) = node {\n let left = node.borrow().left.clone();\n let right = node.borrow().right.clone();\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n }\n}\n
avl_tree.zig/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n if (node != NULL) {\n return node->height;\n }\n return -1;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n int lh = height(node->left);\n int rh = height(node->right);\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n if (lh > rh) {\n node->height = lh + 1;\n } else {\n node->height = rh + 1;\n }\n}\n
"},{"location":"chapter_tree/avl_tree/#2","title":"2. \u00a0 \u8282\u70b9\u5e73\u8861\u56e0\u5b50","text":"// \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n _ = self;\n // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n return if (node == null) -1 else node.?.height;\n}\n\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n\u8282\u70b9\u7684\u300c\u5e73\u8861\u56e0\u5b50 balance factor\u300d\u5b9a\u4e49\u4e3a\u8282\u70b9\u5de6\u5b50\u6811\u7684\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\uff0c\u540c\u65f6\u89c4\u5b9a\u7a7a\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a \\(0\\) \u3002\u6211\u4eec\u540c\u6837\u5c06\u83b7\u53d6\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef balance_factor(self, node: TreeNode | None) -> int:\n \"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n # \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if node is None:\n return 0\n # \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return self.height(node.left) - self.height(node.right)\n
avl_tree.java/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == nullptr)\n return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node->left) - height(node->right);\n}\n
avl_tree.cs/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == null)\n return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node.left) - height(node.right);\n}\n
avl_tree.go/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return Height(node.left) - Height(node.right);\n}\n
avl_tree.swift/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if node == nil {\n return 0\n }\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return t.height(node.Left) - t.height(node.Right)\n}\n
avl_tree.js/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -> Int {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n guard let node = node else { return 0 }\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node: node.left) - height(node: node.right)\n}\n
avl_tree.ts/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node === null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return this.height(node.left) - this.height(node.right);\n}\n
avl_tree.dart/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node === null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return this.height(node.left) - this.height(node.right);\n}\n
avl_tree.rs/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node.left) - height(node.right);\n}\n
avl_tree.c/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -> i32 {\n match node {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n None => 0,\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n Some(node) => {\n Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n }\n }\n}\n
avl_tree.zig/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == NULL) {\n return 0;\n }\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return height(node->left) - height(node->right);\n}\n// \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n if (node == null) return 0;\n // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n return self.height(node.?.left) - self.height(node.?.right);\n}\nNote
\u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002
"},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2 \u00a0 AVL \u6811\u65cb\u8f6c","text":"AVL \u6811\u7684\u7279\u70b9\u5728\u4e8e\u201c\u65cb\u8f6c\u201d\u64cd\u4f5c\uff0c\u5b83\u80fd\u591f\u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7684\u6027\u8d28\uff0c\u4e5f\u80fd\u4f7f\u6811\u91cd\u65b0\u53d8\u4e3a\u201c\u5e73\u8861\u4e8c\u53c9\u6811\u201d\u3002
\u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7edd\u5bf9\u503c \\(> 1\\) \u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5931\u8861\u8282\u70b9\u201d\u3002\u6839\u636e\u8282\u70b9\u5931\u8861\u60c5\u51b5\u7684\u4e0d\u540c\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u56db\u79cd\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u4e0b\u9762\u8be6\u7ec6\u4ecb\u7ecd\u8fd9\u4e9b\u65cb\u8f6c\u64cd\u4f5c\u3002
"},{"location":"chapter_tree/avl_tree/#1_1","title":"1. \u00a0 \u53f3\u65cb","text":"\u5982\u56fe 7-26 \u6240\u793a\uff0c\u8282\u70b9\u4e0b\u65b9\u4e3a\u5e73\u8861\u56e0\u5b50\u3002\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u8282\u70b9\u662f\u201c\u8282\u70b9 3\u201d\u3002\u6211\u4eec\u5173\u6ce8\u4ee5\u8be5\u5931\u8861\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\uff0c\u5c06\u8be5\u8282\u70b9\u8bb0\u4e3a
<1><2><3><4>node\uff0c\u5176\u5de6\u5b50\u8282\u70b9\u8bb0\u4e3achild\uff0c\u6267\u884c\u201c\u53f3\u65cb\u201d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u5b50\u6811\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u3002\u56fe 7-26 \u00a0 \u53f3\u65cb\u64cd\u4f5c\u6b65\u9aa4
\u5982\u56fe 7-27 \u6240\u793a\uff0c\u5f53\u8282\u70b9
child\u6709\u53f3\u5b50\u8282\u70b9\uff08\u8bb0\u4e3agrandChild\uff09\u65f6\uff0c\u9700\u8981\u5728\u53f3\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06grandChild\u4f5c\u4e3anode\u7684\u5de6\u5b50\u8282\u70b9\u3002\u56fe 7-27 \u00a0 \u6709 grandChild \u7684\u53f3\u65cb\u64cd\u4f5c
\u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u4e0a\u9700\u8981\u901a\u8fc7\u4fee\u6539\u8282\u70b9\u6307\u9488\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef right_rotate(self, node: TreeNode | None) -> TreeNode | None:\n \"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\n child = node.left\n grand_child = child.right\n # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node\n node.left = grand_child\n # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.update_height(node)\n self.update_height(child)\n # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n
avl_tree.java/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n TreeNode *child = node->left;\n TreeNode *grandChild = child->right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child->right = node;\n node->left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.cs/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\n TreeNode child = node.left;\n TreeNode grandChild = child.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.go/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\n TreeNode? child = node?.left;\n TreeNode? grandChild = child?.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n UpdateHeight(node);\n UpdateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.swift/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n child := node.Left\n grandChild := child.Right\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.Right = node\n node.Left = grandChild\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n t.updateHeight(node)\n t.updateHeight(child)\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n}\n
avl_tree.js/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -> TreeNode? {\n let child = node?.left\n let grandChild = child?.right\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child?.right = node\n node?.left = grandChild\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node: node)\n updateHeight(node: child)\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n}\n
avl_tree.ts/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\n const child = node.left;\n const grandChild = child.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n this.#updateHeight(node);\n this.#updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.dart/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\n const child = node.left;\n const grandChild = child.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n this.updateHeight(node);\n this.updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.rs/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\n TreeNode? child = node!.left;\n TreeNode? grandChild = child!.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.right = node;\n node.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.c/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {\n match node {\n Some(node) => {\n let child = node.borrow().left.clone().unwrap();\n let grand_child = child.borrow().right.clone();\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.borrow_mut().right = Some(node.clone());\n node.borrow_mut().left = grand_child;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n Self::update_height(Some(node));\n Self::update_height(Some(child.clone()));\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n Some(child)\n }\n None => None,\n }\n}\n
avl_tree.zig/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n TreeNode *child, *grandChild;\n child = node->left;\n grandChild = child->right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child->right = node;\n node->left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
"},{"location":"chapter_tree/avl_tree/#2_1","title":"2. \u00a0 \u5de6\u65cb","text":"// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n var child = node.?.left;\n var grandChild = child.?.right;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n child.?.right = node;\n node.?.left = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.updateHeight(node);\n self.updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n\u76f8\u5e94\u5730\uff0c\u5982\u679c\u8003\u8651\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u5219\u9700\u8981\u6267\u884c\u56fe 7-28 \u6240\u793a\u7684\u201c\u5de6\u65cb\u201d\u64cd\u4f5c\u3002
\u56fe 7-28 \u00a0 \u5de6\u65cb\u64cd\u4f5c
\u540c\u7406\uff0c\u5982\u56fe 7-29 \u6240\u793a\uff0c\u5f53\u8282\u70b9
child\u6709\u5de6\u5b50\u8282\u70b9\uff08\u8bb0\u4e3agrandChild\uff09\u65f6\uff0c\u9700\u8981\u5728\u5de6\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06grandChild\u4f5c\u4e3anode\u7684\u53f3\u5b50\u8282\u70b9\u3002\u56fe 7-29 \u00a0 \u6709 grandChild \u7684\u5de6\u65cb\u64cd\u4f5c
\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u903b\u8f91\u4e0a\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u5b83\u4eec\u5206\u522b\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u57fa\u4e8e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u53f3\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\u4e2d\u7684\u6240\u6709\u7684
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.pyleft\u66ff\u6362\u4e3aright\uff0c\u5c06\u6240\u6709\u7684right\u66ff\u6362\u4e3aleft\uff0c\u5373\u53ef\u5f97\u5230\u5de6\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a
avl_tree.cppdef left_rotate(self, node: TreeNode | None) -> TreeNode | None:\n \"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\n child = node.right\n grand_child = child.left\n # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node\n node.right = grand_child\n # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.update_height(node)\n self.update_height(child)\n # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n
avl_tree.java/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n TreeNode *child = node->right;\n TreeNode *grandChild = child->left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child->left = node;\n node->right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.cs/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\n TreeNode child = node.right;\n TreeNode grandChild = child.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.go/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\n TreeNode? child = node?.right;\n TreeNode? grandChild = child?.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n UpdateHeight(node);\n UpdateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.swift/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n child := node.Right\n grandChild := child.Left\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.Left = node\n node.Right = grandChild\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n t.updateHeight(node)\n t.updateHeight(child)\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n}\n
avl_tree.js/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -> TreeNode? {\n let child = node?.right\n let grandChild = child?.left\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child?.left = node\n node?.right = grandChild\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node: node)\n updateHeight(node: child)\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child\n}\n
avl_tree.ts/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\n const child = node.right;\n const grandChild = child.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n this.#updateHeight(node);\n this.#updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.dart/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\n const child = node.right;\n const grandChild = child.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n this.updateHeight(node);\n this.updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.rs/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\n TreeNode? child = node!.right;\n TreeNode? grandChild = child!.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.left = node;\n node.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
avl_tree.c/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {\n match node {\n Some(node) => {\n let child = node.borrow().right.clone().unwrap();\n let grand_child = child.borrow().left.clone();\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.borrow_mut().left = Some(node.clone());\n node.borrow_mut().right = grand_child;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n Self::update_height(Some(node));\n Self::update_height(Some(child.clone()));\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n Some(child)\n }\n None => None,\n }\n}\n
avl_tree.zig/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n TreeNode *child, *grandChild;\n child = node->right;\n grandChild = child->left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child->left = node;\n node->right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n
"},{"location":"chapter_tree/avl_tree/#3","title":"3. \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb","text":"// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n var child = node.?.right;\n var grandChild = child.?.left;\n // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n child.?.left = node;\n node.?.right = grandChild;\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.updateHeight(node);\n self.updateHeight(child);\n // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n return child;\n}\n\u5bf9\u4e8e\u56fe 7-30 \u4e2d\u7684\u5931\u8861\u8282\u70b9 3 \uff0c\u4ec5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\u3002\u6b64\u65f6\u9700\u8981\u5148\u5bf9
child\u6267\u884c\u201c\u5de6\u65cb\u201d\uff0c\u518d\u5bf9node\u6267\u884c\u201c\u53f3\u65cb\u201d\u3002\u56fe 7-30 \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb
"},{"location":"chapter_tree/avl_tree/#4","title":"4. \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb","text":"\u5982\u56fe 7-31 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\u60c5\u51b5\uff0c\u9700\u8981\u5148\u5bf9
child\u6267\u884c\u201c\u53f3\u65cb\u201d\uff0c\u518d\u5bf9node\u6267\u884c\u201c\u5de6\u65cb\u201d\u3002\u56fe 7-31 \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb
"},{"location":"chapter_tree/avl_tree/#5","title":"5. \u00a0 \u65cb\u8f6c\u7684\u9009\u62e9","text":"\u56fe 7-32 \u5c55\u793a\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0\u6848\u4f8b\u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u8981\u91c7\u7528\u53f3\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5de6\u65cb\u7684\u64cd\u4f5c\u3002
\u56fe 7-32 \u00a0 AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5
\u5982\u4e0b\u8868\u6240\u793a\uff0c\u6211\u4eec\u901a\u8fc7\u5224\u65ad\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f83\u9ad8\u4e00\u4fa7\u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8d1f\u53f7\uff0c\u6765\u786e\u5b9a\u5931\u8861\u8282\u70b9\u5c5e\u4e8e\u56fe 7-32 \u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002
\u8868 7-3 \u00a0 \u56db\u79cd\u65cb\u8f6c\u60c5\u51b5\u7684\u9009\u62e9\u6761\u4ef6
\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(> 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(> 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(<0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(< -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(< -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(>0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb\u4e3a\u4e86\u4fbf\u4e8e\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u6709\u4e86\u8fd9\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5c31\u80fd\u5bf9\u5404\u79cd\u5931\u8861\u60c5\u51b5\u8fdb\u884c\u65cb\u8f6c\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef rotate(self, node: TreeNode | None) -> TreeNode | None:\n \"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n # \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n balance_factor = self.balance_factor(node)\n # \u5de6\u504f\u6811\n if balance_factor > 1:\n if self.balance_factor(node.left) >= 0:\n # \u53f3\u65cb\n return self.right_rotate(node)\n else:\n # \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = self.left_rotate(node.left)\n return self.right_rotate(node)\n # \u53f3\u504f\u6811\n elif balance_factor < -1:\n if self.balance_factor(node.right) <= 0:\n # \u5de6\u65cb\n return self.left_rotate(node)\n else:\n # \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = self.right_rotate(node.right)\n return self.left_rotate(node)\n # \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n
avl_tree.java/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int _balanceFactor = balanceFactor(node);\n // \u5de6\u504f\u6811\n if (_balanceFactor > 1) {\n if (balanceFactor(node->left) >= 0) {\n // \u53f3\u65cb\n return rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node->left = leftRotate(node->left);\n return rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (_balanceFactor < -1) {\n if (balanceFactor(node->right) <= 0) {\n // \u5de6\u65cb\n return leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node->right = rightRotate(node->right);\n return leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.cs/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int balanceFactor = balanceFactor(node);\n // \u5de6\u504f\u6811\n if (balanceFactor > 1) {\n if (balanceFactor(node.left) >= 0) {\n // \u53f3\u65cb\n return rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = leftRotate(node.left);\n return rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balanceFactor < -1) {\n if (balanceFactor(node.right) <= 0) {\n // \u5de6\u65cb\n return leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = rightRotate(node.right);\n return leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.go/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int balanceFactorInt = BalanceFactor(node);\n // \u5de6\u504f\u6811\n if (balanceFactorInt > 1) {\n if (BalanceFactor(node?.left) >= 0) {\n // \u53f3\u65cb\n return RightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node!.left = LeftRotate(node!.left);\n return RightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balanceFactorInt < -1) {\n if (BalanceFactor(node?.right) <= 0) {\n // \u5de6\u65cb\n return LeftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node!.right = RightRotate(node!.right);\n return LeftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.swift/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n // Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\n bf := t.balanceFactor(node)\n // \u5de6\u504f\u6811\n if bf > 1 {\n if t.balanceFactor(node.Left) >= 0 {\n // \u53f3\u65cb\n return t.rightRotate(node)\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.Left = t.leftRotate(node.Left)\n return t.rightRotate(node)\n }\n }\n // \u53f3\u504f\u6811\n if bf < -1 {\n if t.balanceFactor(node.Right) <= 0 {\n // \u5de6\u65cb\n return t.leftRotate(node)\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.Right = t.rightRotate(node.Right)\n return t.leftRotate(node)\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n}\n
avl_tree.js/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -> TreeNode? {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n let balanceFactor = balanceFactor(node: node)\n // \u5de6\u504f\u6811\n if balanceFactor > 1 {\n if self.balanceFactor(node: node?.left) >= 0 {\n // \u53f3\u65cb\n return rightRotate(node: node)\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node?.left = leftRotate(node: node?.left)\n return rightRotate(node: node)\n }\n }\n // \u53f3\u504f\u6811\n if balanceFactor < -1 {\n if self.balanceFactor(node: node?.right) <= 0 {\n // \u5de6\u65cb\n return leftRotate(node: node)\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node?.right = rightRotate(node: node?.right)\n return leftRotate(node: node)\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n}\n
avl_tree.ts/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n const balanceFactor = this.balanceFactor(node);\n // \u5de6\u504f\u6811\n if (balanceFactor > 1) {\n if (this.balanceFactor(node.left) >= 0) {\n // \u53f3\u65cb\n return this.#rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = this.#leftRotate(node.left);\n return this.#rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balanceFactor < -1) {\n if (this.balanceFactor(node.right) <= 0) {\n // \u5de6\u65cb\n return this.#leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = this.#rightRotate(node.right);\n return this.#leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.dart/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n const balanceFactor = this.balanceFactor(node);\n // \u5de6\u504f\u6811\n if (balanceFactor > 1) {\n if (this.balanceFactor(node.left) >= 0) {\n // \u53f3\u65cb\n return this.rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = this.leftRotate(node.left);\n return this.rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balanceFactor < -1) {\n if (this.balanceFactor(node.right) <= 0) {\n // \u5de6\u65cb\n return this.leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = this.rightRotate(node.right);\n return this.leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.rs/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int factor = balanceFactor(node);\n // \u5de6\u504f\u6811\n if (factor > 1) {\n if (balanceFactor(node!.left) >= 0) {\n // \u53f3\u65cb\n return rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.left = leftRotate(node.left);\n return rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (factor < -1) {\n if (balanceFactor(node!.right) <= 0) {\n // \u5de6\u65cb\n return leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.right = rightRotate(node.right);\n return leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
avl_tree.c/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n let balance_factor = Self::balance_factor(node.clone());\n // \u5de6\u504f\u6811\n if balance_factor > 1 {\n let node = node.unwrap();\n if Self::balance_factor(node.borrow().left.clone()) >= 0 {\n // \u53f3\u65cb\n Self::right_rotate(Some(node))\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n let left = node.borrow().left.clone();\n node.borrow_mut().left = Self::left_rotate(left);\n Self::right_rotate(Some(node))\n }\n }\n // \u53f3\u504f\u6811\n else if balance_factor < -1 {\n let node = node.unwrap();\n if Self::balance_factor(node.borrow().right.clone()) <= 0 {\n // \u5de6\u65cb\n Self::left_rotate(Some(node))\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n let right = node.borrow().right.clone();\n node.borrow_mut().right = Self::right_rotate(right);\n Self::left_rotate(Some(node))\n }\n } else {\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n node\n }\n}\n
avl_tree.zig/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n int bf = balanceFactor(node);\n // \u5de6\u504f\u6811\n if (bf > 1) {\n if (balanceFactor(node->left) >= 0) {\n // \u53f3\u65cb\n return rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node->left = leftRotate(node->left);\n return rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (bf < -1) {\n if (balanceFactor(node->right) <= 0) {\n // \u5de6\u65cb\n return leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node->right = rightRotate(node->right);\n return leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\n
"},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3 \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#1_2","title":"1. \u00a0 \u63d2\u5165\u8282\u70b9","text":"// \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n var balance_factor = self.balanceFactor(node);\n // \u5de6\u504f\u6811\n if (balance_factor > 1) {\n if (self.balanceFactor(node.?.left) >= 0) {\n // \u53f3\u65cb\n return self.rightRotate(node);\n } else {\n // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n node.?.left = self.leftRotate(node.?.left);\n return self.rightRotate(node);\n }\n }\n // \u53f3\u504f\u6811\n if (balance_factor < -1) {\n if (self.balanceFactor(node.?.right) <= 0) {\n // \u5de6\u65cb\n return self.leftRotate(node);\n } else {\n // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n node.?.right = self.rightRotate(node.?.right);\n return self.leftRotate(node);\n }\n }\n // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n}\nAVL \u6811\u7684\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u5728\u4e3b\u4f53\u4e0a\u7c7b\u4f3c\u3002\u552f\u4e00\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5728 AVL \u6811\u4e2d\u63d2\u5165\u8282\u70b9\u540e\uff0c\u4ece\u8be5\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u53ef\u80fd\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u5931\u8861\u8282\u70b9\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8fd9\u4e2a\u8282\u70b9\u5f00\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef insert(self, val):\n \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -> TreeNode:\n \"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n if node is None:\n return TreeNode(val)\n # 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\n if val < node.val:\n node.left = self.insert_helper(node.left, val)\n elif val > node.val:\n node.right = self.insert_helper(node.right, val)\n else:\n # \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.update_height(node)\n # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n return self.rotate(node)\n
avl_tree.java/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n if (node == nullptr)\n return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node->val)\n node->left = insertHelper(node->left, val);\n else if (val > node->val)\n node->right = insertHelper(node->right, val);\n else\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.cs/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\n if (node == null)\n return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val)\n node.left = insertHelper(node.left, val);\n else if (val > node.val)\n node.right = insertHelper(node.right, val);\n else\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.go/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int val) {\n root = InsertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n if (node == null) return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val)\n node.left = InsertHelper(node.left, val);\n else if (val > node.val)\n node.right = InsertHelper(node.right, val);\n else\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n UpdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = Rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.swift/* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\n t.root = t.insertHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n if node == nil {\n return NewTreeNode(val)\n }\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if val < node.Val.(int) {\n node.Left = t.insertHelper(node.Left, val)\n } else if val > node.Val.(int) {\n node.Right = t.insertHelper(node.Right, val)\n } else {\n // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n return node\n }\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n t.updateHeight(node)\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = t.rotate(node)\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node\n}\n
avl_tree.js/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(val: Int) {\n root = insertHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -> TreeNode? {\n var node = node\n if node == nil {\n return TreeNode(x: val)\n }\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if val < node!.val {\n node?.left = insertHelper(node: node?.left, val: val)\n } else if val > node!.val {\n node?.right = insertHelper(node: node?.right, val: val)\n } else {\n return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n }\n updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node: node)\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node\n}\n
avl_tree.ts/* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\n this.root = this.#insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\n if (node === null) return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val) node.left = this.#insertHelper(node.left, val);\n else if (val > node.val)\n node.right = this.#insertHelper(node.right, val);\n else return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = this.#rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.dart/* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\n this.root = this.insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n if (node === null) return new TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val) {\n node.left = this.insertHelper(node.left, val);\n } else if (val > node.val) {\n node.right = this.insertHelper(node.right, val);\n } else {\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n }\n this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = this.rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.rs/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\n if (node == null) return TreeNode(val);\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node.val)\n node.left = insertHelper(node.left, val);\n else if (val > node.val)\n node.right = insertHelper(node.right, val);\n else\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.c/* \u63d2\u5165\u8282\u70b9 */\nfn insert(&mut self, val: i32) {\n self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -> OptionTreeNodeRc {\n match node {\n Some(mut node) => {\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n match {\n let node_val = node.borrow().val;\n node_val\n }\n .cmp(&val)\n {\n Ordering::Greater => {\n let left = node.borrow().left.clone();\n node.borrow_mut().left = Self::insert_helper(left, val);\n }\n Ordering::Less => {\n let right = node.borrow().right.clone();\n node.borrow_mut().right = Self::insert_helper(right, val);\n }\n Ordering::Equal => {\n return Some(node); // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n }\n }\n Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = Self::rotate(Some(node)).unwrap();\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n Some(node)\n }\n None => Some(TreeNode::new(val)),\n }\n}\n
avl_tree.zig/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(AVLTree *tree, int val) {\n tree->root = insertHelper(tree->root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n if (node == NULL) {\n return newTreeNode(val);\n }\n /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n if (val < node->val) {\n node->left = insertHelper(node->left, val);\n } else if (val > node->val) {\n node->right = insertHelper(node->right, val);\n } else {\n // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n return node;\n }\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
"},{"location":"chapter_tree/avl_tree/#2_2","title":"2. \u00a0 \u5220\u9664\u8282\u70b9","text":"// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\n self.root = (try self.insertHelper(self.root, val)).?;\n}\n\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\n var node = node_;\n if (node == null) {\n var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\n tmp_node.init(val);\n return tmp_node;\n }\n // 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\n if (val < node.?.val) {\n node.?.left = try self.insertHelper(node.?.left, val);\n } else if (val > node.?.val) {\n node.?.right = try self.insertHelper(node.?.right, val);\n } else {\n return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n }\n self.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n node = self.rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n\u7c7b\u4f3c\u5730\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5220\u9664\u8282\u70b9\u65b9\u6cd5\u7684\u57fa\u7840\u4e0a\uff0c\u9700\u8981\u4ece\u5e95\u81f3\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py
avl_tree.cppdef remove(self, val: int):\n \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -> TreeNode | None:\n \"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n if node is None:\n return None\n # 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\n if val < node.val:\n node.left = self.remove_helper(node.left, val)\n elif val > node.val:\n node.right = self.remove_helper(node.right, val)\n else:\n if node.left is None or node.right is None:\n child = node.left or node.right\n # \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if child is None:\n return None\n # \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else:\n node = child\n else:\n # \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n temp = node.right\n while temp.left is not None:\n temp = temp.left\n node.right = self.remove_helper(node.right, temp.val)\n node.val = temp.val\n # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n self.update_height(node)\n # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n return self.rotate(node)\n
avl_tree.java/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n if (node == nullptr)\n return nullptr;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node->val)\n node->left = removeHelper(node->left, val);\n else if (val > node->val)\n node->right = removeHelper(node->right, val);\n else {\n if (node->left == nullptr || node->right == nullptr) {\n TreeNode *child = node->left != nullptr ? node->left : node->right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == nullptr) {\n delete node;\n return nullptr;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else {\n delete node;\n node = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode *temp = node->right;\n while (temp->left != nullptr) {\n temp = temp->left;\n }\n int tempVal = temp->val;\n node->right = removeHelper(node->right, temp->val);\n node->val = tempVal;\n }\n }\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.cs/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\n if (node == null)\n return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val)\n node.left = removeHelper(node.left, val);\n else if (val > node.val)\n node.right = removeHelper(node.right, val);\n else {\n if (node.left == null || node.right == null) {\n TreeNode child = node.left != null ? node.left : node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == null)\n return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else\n node = child;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode temp = node.right;\n while (temp.left != null) {\n temp = temp.left;\n }\n node.right = removeHelper(node.right, temp.val);\n node.val = temp.val;\n }\n }\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.go/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int val) {\n root = RemoveHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n if (node == null) return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val)\n node.left = RemoveHelper(node.left, val);\n else if (val > node.val)\n node.right = RemoveHelper(node.right, val);\n else {\n if (node.left == null || node.right == null) {\n TreeNode? child = node.left ?? node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == null)\n return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else\n node = child;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode? temp = node.right;\n while (temp.left != null) {\n temp = temp.left;\n }\n node.right = RemoveHelper(node.right, temp.val!.Value);\n node.val = temp.val;\n }\n }\n UpdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = Rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.swift/* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\n t.root = t.removeHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n if node == nil {\n return nil\n }\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if val < node.Val.(int) {\n node.Left = t.removeHelper(node.Left, val)\n } else if val > node.Val.(int) {\n node.Right = t.removeHelper(node.Right, val)\n } else {\n if node.Left == nil || node.Right == nil {\n child := node.Left\n if node.Right != nil {\n child = node.Right\n }\n if child == nil {\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n return nil\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n node = child\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n temp := node.Right\n for temp.Left != nil {\n temp = temp.Left\n }\n node.Right = t.removeHelper(node.Right, temp.Val.(int))\n node.Val = temp.Val\n }\n }\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n t.updateHeight(node)\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = t.rotate(node)\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node\n}\n
avl_tree.js/* \u5220\u9664\u8282\u70b9 */\nfunc remove(val: Int) {\n root = removeHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -> TreeNode? {\n var node = node\n if node == nil {\n return nil\n }\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if val < node!.val {\n node?.left = removeHelper(node: node?.left, val: val)\n } else if val > node!.val {\n node?.right = removeHelper(node: node?.right, val: val)\n } else {\n if node?.left == nil || node?.right == nil {\n let child = node?.left != nil ? node?.left : node?.right\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if child == nil {\n return nil\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else {\n node = child\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n var temp = node?.right\n while temp?.left != nil {\n temp = temp?.left\n }\n node?.right = removeHelper(node: node?.right, val: temp!.val)\n node?.val = temp!.val\n }\n }\n updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node: node)\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node\n}\n
avl_tree.ts/* \u5220\u9664\u8282\u70b9 */\nremove(val) {\n this.root = this.#removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\n if (node === null) return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val) node.left = this.#removeHelper(node.left, val);\n else if (val > node.val)\n node.right = this.#removeHelper(node.right, val);\n else {\n if (node.left === null || node.right === null) {\n const child = node.left !== null ? node.left : node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child === null) return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else node = child;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n let temp = node.right;\n while (temp.left !== null) {\n temp = temp.left;\n }\n node.right = this.#removeHelper(node.right, temp.val);\n node.val = temp.val;\n }\n }\n this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = this.#rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.dart/* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\n this.root = this.removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n if (node === null) return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val) {\n node.left = this.removeHelper(node.left, val);\n } else if (val > node.val) {\n node.right = this.removeHelper(node.right, val);\n } else {\n if (node.left === null || node.right === null) {\n const child = node.left !== null ? node.left : node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child === null) {\n return null;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n node = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n let temp = node.right;\n while (temp.left !== null) {\n temp = temp.left;\n }\n node.right = this.removeHelper(node.right, temp.val);\n node.val = temp.val;\n }\n }\n this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = this.rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.rs/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\n if (node == null) return null;\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node.val)\n node.left = removeHelper(node.left, val);\n else if (val > node.val)\n node.right = removeHelper(node.right, val);\n else {\n if (node.left == null || node.right == null) {\n TreeNode? child = node.left ?? node.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == null)\n return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n else\n node = child;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode? temp = node.right;\n while (temp!.left != null) {\n temp = temp.left;\n }\n node.right = removeHelper(node.right, temp.val);\n node.val = temp.val;\n }\n }\n updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
avl_tree.c/* \u5220\u9664\u8282\u70b9 */\nfn remove(&self, val: i32) {\n Self::remove_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -> OptionTreeNodeRc {\n match node {\n Some(mut node) => {\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if val < node.borrow().val {\n let left = node.borrow().left.clone();\n node.borrow_mut().left = Self::remove_helper(left, val);\n } else if val > node.borrow().val {\n let right = node.borrow().right.clone();\n node.borrow_mut().right = Self::remove_helper(right, val);\n } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n let child = if node.borrow().left.is_some() {\n node.borrow().left.clone()\n } else {\n node.borrow().right.clone()\n };\n match child {\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n None => {\n return None;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n Some(child) => node = child,\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n let mut temp = node.borrow().right.clone().unwrap();\n loop {\n let temp_left = temp.borrow().left.clone();\n if temp_left.is_none() {\n break;\n }\n temp = temp_left.unwrap();\n }\n let right = node.borrow().right.clone();\n node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n node.borrow_mut().val = temp.borrow().val;\n }\n Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = Self::rotate(Some(node)).unwrap();\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n Some(node)\n }\n None => None,\n }\n}\n
avl_tree.zig/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(AVLTree *tree, int val) {\n TreeNode *root = removeHelper(tree->root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n TreeNode *child, *grandChild;\n if (node == NULL) {\n return NULL;\n }\n /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n if (val < node->val) {\n node->left = removeHelper(node->left, val);\n } else if (val > node->val) {\n node->right = removeHelper(node->right, val);\n } else {\n if (node->left == NULL || node->right == NULL) {\n child = node->left;\n if (node->right != NULL) {\n child = node->right;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == NULL) {\n return NULL;\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n node = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n TreeNode *temp = node->right;\n while (temp->left != NULL) {\n temp = temp->left;\n }\n int tempVal = temp->val;\n node->right = removeHelper(node->right, temp->val);\n node->val = tempVal;\n }\n }\n // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n updateHeight(node);\n /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n node = rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\n
"},{"location":"chapter_tree/avl_tree/#3_1","title":"3. \u00a0 \u67e5\u627e\u8282\u70b9","text":"// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\n self.root = self.removeHelper(self.root, val).?;\n}\n\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\n var node = node_;\n if (node == null) return null;\n // 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\n if (val < node.?.val) {\n node.?.left = self.removeHelper(node.?.left, val);\n } else if (val > node.?.val) {\n node.?.right = self.removeHelper(node.?.right, val);\n } else {\n if (node.?.left == null or node.?.right == null) {\n var child = if (node.?.left != null) node.?.left else node.?.right;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n if (child == null) {\n return null;\n // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n } else {\n node = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n var temp = node.?.right;\n while (temp.?.left != null) {\n temp = temp.?.left;\n }\n node.?.right = self.removeHelper(node.?.right, temp.?.val);\n node.?.val = temp.?.val;\n }\n }\n self.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n node = self.rotate(node);\n // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n return node;\n}\nAVL \u6811\u7684\u8282\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002
"},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4 \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"- \u7ec4\u7ec7\u548c\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u7684\u573a\u666f\u3002
- \u7528\u4e8e\u6784\u5efa\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\u3002
- \u7ea2\u9ed1\u6811\u5728\u8bb8\u591a\u5e94\u7528\u4e2d\u6bd4 AVL \u6811\u66f4\u53d7\u6b22\u8fce\u3002\u8fd9\u662f\u56e0\u4e3a\u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u76f8\u5bf9\u5bbd\u677e\uff0c\u5728\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u5176\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u7684\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002
\u5982\u56fe 7-16 \u6240\u793a\uff0c\u300c\u4e8c\u53c9\u641c\u7d22\u6811 binary search tree\u300d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u3002
- \u5bf9\u4e8e\u6839\u8282\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c \\(<\\) \u6839\u8282\u70b9\u7684\u503c \\(<\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c\u3002
- \u4efb\u610f\u8282\u70b9\u7684\u5de6\u3001\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u540c\u6837\u6ee1\u8db3\u6761\u4ef6
1.\u3002
\u56fe 7-16 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811
"},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":"\u6211\u4eec\u5c06\u4e8c\u53c9\u641c\u7d22\u6811\u5c01\u88c5\u4e3a\u4e00\u4e2a\u7c7b
"},{"location":"chapter_tree/binary_search_tree/#1","title":"1. \u00a0 \u67e5\u627e\u8282\u70b9","text":"BinarySearchTree\uff0c\u5e76\u58f0\u660e\u4e00\u4e2a\u6210\u5458\u53d8\u91cfroot\uff0c\u6307\u5411\u6811\u7684\u6839\u8282\u70b9\u3002\u7ed9\u5b9a\u76ee\u6807\u8282\u70b9\u503c
num\uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u5982\u56fe 7-17 \u6240\u793a\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u8282\u70b9cur\uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9root\u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u8282\u70b9\u503ccur.val\u548cnum\u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb\u3002- \u82e5
cur.val < num\uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728cur\u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884ccur = cur.right\u3002 - \u82e5
cur.val > num\uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728cur\u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884ccur = cur.left\u3002 - \u82e5
cur.val = num\uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u8282\u70b9\u3002
\u56fe 7-17 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u8282\u70b9\u793a\u4f8b
\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py
binary_search_tree.cppdef search(self, num: int) -> TreeNode | None:\n \"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\n cur = self._root\n # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while cur is not None:\n # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur.val < num:\n cur = cur.right\n # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n elif cur.val > num:\n cur = cur.left\n # \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else:\n break\n return cur\n
binary_search_tree.java/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\n TreeNode *cur = root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != nullptr) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur->val < num)\n cur = cur->right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur->val > num)\n cur = cur->left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else\n break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.cs/* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\n TreeNode cur = root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > num)\n cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else\n break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.go/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? Search(int num) {\n TreeNode? cur = root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur =\n cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > num)\n cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else\n break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.swift/* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n node := bst.root\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n for node != nil {\n if node.Val.(int) < num {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n node = node.Right\n } else if node.Val.(int) > num {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n node = node.Left\n } else {\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n break\n }\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return node\n}\n
binary_search_tree.js/* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -> TreeNode? {\n var cur = root\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while cur != nil {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur!.val < num {\n cur = cur?.right\n }\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if cur!.val > num {\n cur = cur?.left\n }\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else {\n break\n }\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur\n}\n
binary_search_tree.ts/* \u67e5\u627e\u8282\u70b9 */\nsearch(num) {\n let cur = this.root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > num) cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.dart/* \u67e5\u627e\u8282\u70b9 */\nsearch(num: number): TreeNode | null {\n let cur = this.root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > num) cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.rs/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int _num) {\n TreeNode? cur = _root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < _num)\n cur = cur.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else if (cur.val > _num)\n cur = cur.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n else\n break;\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
binary_search_tree.c/* \u67e5\u627e\u8282\u70b9 */\npub fn search(&self, num: i32) -> OptionTreeNodeRc {\n let mut cur = self.root.clone();\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while let Some(node) = cur.clone() {\n match num.cmp(&node.borrow().val) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n Ordering::Greater => cur = node.borrow().right.clone(),\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n Ordering::Less => cur = node.borrow().left.clone(),\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n Ordering::Equal => break,\n }\n }\n\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n cur\n}\n
binary_search_tree.zig/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(BinarySearchTree *bst, int num) {\n TreeNode *cur = bst->root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != NULL) {\n if (cur->val < num) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n cur = cur->right;\n } else if (cur->val > num) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n cur = cur->left;\n } else {\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n break;\n }\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n
"},{"location":"chapter_tree/binary_search_tree/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"// \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\n var cur = self.root;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.?.val < num) {\n cur = cur.?.right;\n // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n } else if (cur.?.val > num) {\n cur = cur.?.left;\n // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n } else {\n break;\n }\n }\n // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n return cur;\n}\n\u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20
num\uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 < \u6839\u8282\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-18 \u6240\u793a\u3002- \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u76f8\u4f3c\uff0c\u4ece\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u8282\u70b9\u503c\u548c
num\u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\uff08\u904d\u5386\u81f3 \\(\\text{None}\\) \uff09\u65f6\u8df3\u51fa\u5faa\u73af\u3002 - \u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u8282\u70b9\uff1a\u521d\u59cb\u5316\u8282\u70b9
num\uff0c\u5c06\u8be5\u8282\u70b9\u7f6e\u4e8e \\(\\text{None}\\) \u7684\u4f4d\u7f6e\u3002
\u56fe 7-18 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u8282\u70b9
\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002
- \u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u8282\u70b9\uff0c\u5426\u5219\u5c06\u8fdd\u53cd\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u8282\u70b9\u5728\u6811\u4e2d\u5df2\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002
- \u4e3a\u4e86\u5b9e\u73b0\u63d2\u5165\u8282\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u8282\u70b9
pre\u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u8282\u70b9\u3002\u8fd9\u6837\u5728\u904d\u5386\u81f3 \\(\\text{None}\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u8282\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u3002
binary_search_tree.cppdef insert(self, num: int):\n \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n # \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if self._root is None:\n self._root = TreeNode(num)\n return\n # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n cur, pre = self._root, None\n while cur is not None:\n # \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if cur.val == num:\n return\n pre = cur\n # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur.val < num:\n cur = cur.right\n # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else:\n cur = cur.left\n # \u63d2\u5165\u8282\u70b9\n node = TreeNode(num)\n if pre.val < num:\n pre.right = node\n else:\n pre.left = node\n
binary_search_tree.java/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (root == nullptr) {\n root = new TreeNode(num);\n return;\n }\n TreeNode *cur = root, *pre = nullptr;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != nullptr) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur->val == num)\n return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur->val < num)\n cur = cur->right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur->left;\n }\n // \u63d2\u5165\u8282\u70b9\n TreeNode *node = new TreeNode(num);\n if (pre->val < num)\n pre->right = node;\n else\n pre->left = node;\n}\n
binary_search_tree.cs/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (root == null) {\n root = new TreeNode(num);\n return;\n }\n TreeNode cur = root, pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val == num)\n return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u63d2\u5165\u8282\u70b9\n TreeNode node = new TreeNode(num);\n if (pre.val < num)\n pre.right = node;\n else\n pre.left = node;\n}\n
binary_search_tree.go/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (root == null) {\n root = new TreeNode(num);\n return;\n }\n TreeNode? cur = root, pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val == num)\n return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n\n // \u63d2\u5165\u8282\u70b9\n TreeNode node = new(num);\n if (pre != null) {\n if (pre.val < num)\n pre.right = node;\n else\n pre.left = node;\n }\n}\n
binary_search_tree.swift/* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\n cur := bst.root\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if cur == nil {\n bst.root = NewTreeNode(num)\n return\n }\n // \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n var pre *TreeNode = nil\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n for cur != nil {\n if cur.Val == num {\n return\n }\n pre = cur\n if cur.Val.(int) < num {\n cur = cur.Right\n } else {\n cur = cur.Left\n }\n }\n // \u63d2\u5165\u8282\u70b9\n node := NewTreeNode(num)\n if pre.Val.(int) < num {\n pre.Right = node\n } else {\n pre.Left = node\n }\n}\n
binary_search_tree.js/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if root == nil {\n root = TreeNode(x: num)\n return\n }\n var cur = root\n var pre: TreeNode?\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while cur != nil {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if cur!.val == num {\n return\n }\n pre = cur\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur!.val < num {\n cur = cur?.right\n }\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else {\n cur = cur?.left\n }\n }\n // \u63d2\u5165\u8282\u70b9\n let node = TreeNode(x: num)\n if pre!.val < num {\n pre?.right = node\n } else {\n pre?.left = node\n }\n}\n
binary_search_tree.ts/* \u63d2\u5165\u8282\u70b9 */\ninsert(num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (this.root === null) {\n this.root = new TreeNode(num);\n return;\n }\n let cur = this.root,\n pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val === num) return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else cur = cur.left;\n }\n // \u63d2\u5165\u8282\u70b9\n const node = new TreeNode(num);\n if (pre.val < num) pre.right = node;\n else pre.left = node;\n}\n
binary_search_tree.dart/* \u63d2\u5165\u8282\u70b9 */\ninsert(num: number): void {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (this.root === null) {\n this.root = new TreeNode(num);\n return;\n }\n let cur: TreeNode | null = this.root,\n pre: TreeNode | null = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val === num) return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else cur = cur.left;\n }\n // \u63d2\u5165\u8282\u70b9\n const node = new TreeNode(num);\n if (pre!.val < num) pre!.right = node;\n else pre!.left = node;\n}\n
binary_search_tree.rs/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int _num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (_root == null) {\n _root = TreeNode(_num);\n return;\n }\n TreeNode? cur = _root;\n TreeNode? pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.val == _num) return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < _num)\n cur = cur.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u63d2\u5165\u8282\u70b9\n TreeNode? node = TreeNode(_num);\n if (pre!.val < _num)\n pre.right = node;\n else\n pre.left = node;\n}\n
binary_search_tree.c/* \u63d2\u5165\u8282\u70b9 */\npub fn insert(&mut self, num: i32) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if self.root.is_none() {\n self.root = Some(TreeNode::new(num));\n return;\n }\n let mut cur = self.root.clone();\n let mut pre = None;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while let Some(node) = cur.clone() {\n match num.cmp(&node.borrow().val) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n Ordering::Equal => return,\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n Ordering::Greater => {\n pre = cur.clone();\n cur = node.borrow().right.clone();\n }\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n Ordering::Less => {\n pre = cur.clone();\n cur = node.borrow().left.clone();\n }\n }\n }\n // \u63d2\u5165\u8282\u70b9\n let pre = pre.unwrap();\n let node = Some(TreeNode::new(num));\n if num > pre.borrow().val {\n pre.borrow_mut().right = node;\n } else {\n pre.borrow_mut().left = node;\n }\n}\n
binary_search_tree.zig/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(BinarySearchTree *bst, int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (bst->root == NULL) {\n bst->root = newTreeNode(num);\n return;\n }\n TreeNode *cur = bst->root, *pre = NULL;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != NULL) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur->val == num) {\n return;\n }\n pre = cur;\n if (cur->val < num) {\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n cur = cur->right;\n } else {\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n cur = cur->left;\n }\n }\n // \u63d2\u5165\u8282\u70b9\n TreeNode *node = newTreeNode(num);\n if (pre->val < num) {\n pre->right = node;\n } else {\n pre->left = node;\n }\n}\n// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n if (self.root == null) {\n self.root = try self.mem_allocator.create(inc.TreeNode(T));\n return;\n }\n var cur = self.root;\n var pre: ?*inc.TreeNode(T) = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur.?.val == num) return;\n pre = cur;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.?.val < num) {\n cur = cur.?.right;\n // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n } else {\n cur = cur.?.left;\n }\n }\n // \u63d2\u5165\u8282\u70b9\n var node = try self.mem_allocator.create(inc.TreeNode(T));\n node.init(num);\n if (pre.?.val < num) {\n pre.?.right = node;\n } else {\n pre.?.left = node;\n }\n}\n\u4e0e\u67e5\u627e\u8282\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u8282\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002
"},{"location":"chapter_tree/binary_search_tree/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"\u5148\u5728\u4e8c\u53c9\u6811\u4e2d\u67e5\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u518d\u5c06\u5176\u5220\u9664\u3002
\u4e0e\u63d2\u5165\u8282\u70b9\u7c7b\u4f3c\uff0c\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 < \u6839\u8282\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u4ecd\u7136\u6ee1\u8db3\u3002
\u56e0\u6b64\uff0c\u6211\u4eec\u6839\u636e\u76ee\u6807\u8282\u70b9\u7684\u5b50\u8282\u70b9\u6570\u91cf\uff0c\u5206 0\u30011 \u548c 2 \u4e09\u79cd\u60c5\u51b5\uff0c\u6267\u884c\u5bf9\u5e94\u7684\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u3002
\u5982\u56fe 7-19 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \u65f6\uff0c\u8868\u793a\u8be5\u8282\u70b9\u662f\u53f6\u8282\u70b9\uff0c\u53ef\u4ee5\u76f4\u63a5\u5220\u9664\u3002
\u56fe 7-19 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 0 \uff09
\u5982\u56fe 7-20 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u8282\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u8282\u70b9\u5373\u53ef\u3002
\u56fe 7-20 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 1 \uff09
\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(2\\) \u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u5220\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u4e2a\u8282\u70b9\u66ff\u6362\u8be5\u8282\u70b9\u3002\u7531\u4e8e\u8981\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 \\(<\\) \u6839\u8282\u70b9 \\(<\\) \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u4e2a\u8282\u70b9\u53ef\u4ee5\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\u6216\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u3002
\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\uff08\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\uff0c\u5219\u5220\u9664\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-21 \u6240\u793a\u3002
- \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\u5728\u201c\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff0c\u8bb0\u4e3a
tmp\u3002 - \u7528
tmp\u7684\u503c\u8986\u76d6\u5f85\u5220\u9664\u8282\u70b9\u7684\u503c\uff0c\u5e76\u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u8282\u70b9tmp\u3002
\u56fe 7-21 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 2 \uff09
\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u540c\u6837\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py
binary_search_tree.cppdef remove(self, num: int):\n \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n # \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if self._root is None:\n return\n # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n cur, pre = self._root, None\n while cur is not None:\n # \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if cur.val == num:\n break\n pre = cur\n # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur.val < num:\n cur = cur.right\n # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else:\n cur = cur.left\n # \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if cur is None:\n return\n\n # \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if cur.left is None or cur.right is None:\n # \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n child = cur.left or cur.right\n # \u5220\u9664\u8282\u70b9 cur\n if cur != self._root:\n if pre.left == cur:\n pre.left = child\n else:\n pre.right = child\n else:\n # \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n self._root = child\n # \u5b50\u8282\u70b9\u6570\u91cf = 2\n else:\n # \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n tmp: TreeNode = cur.right\n while tmp.left is not None:\n tmp = tmp.left\n # \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n self.remove(tmp.val)\n # \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val\n
binary_search_tree.java/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (root == nullptr)\n return;\n TreeNode *cur = root, *pre = nullptr;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != nullptr) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur->val == num)\n break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur->val < num)\n cur = cur->right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur->left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == nullptr)\n return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur->left == nullptr || cur->right == nullptr) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n TreeNode *child = cur->left != nullptr ? cur->left : cur->right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur != root) {\n if (pre->left == cur)\n pre->left = child;\n else\n pre->right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n root = child;\n }\n // \u91ca\u653e\u5185\u5b58\n delete cur;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode *tmp = cur->right;\n while (tmp->left != nullptr) {\n tmp = tmp->left;\n }\n int tmpVal = tmp->val;\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n remove(tmp->val);\n // \u7528 tmp \u8986\u76d6 cur\n cur->val = tmpVal;\n }\n}\n
binary_search_tree.cs/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (root == null)\n return;\n TreeNode cur = root, pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val == num)\n break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == null)\n return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left == null || cur.right == null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n TreeNode child = cur.left != null ? cur.left : cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur != root) {\n if (pre.left == cur)\n pre.left = child;\n else\n pre.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode tmp = cur.right;\n while (tmp.left != null) {\n tmp = tmp.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n remove(tmp.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val;\n }\n}\n
binary_search_tree.go/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (root == null)\n return;\n TreeNode? cur = root, pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val == num)\n break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num)\n cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == null)\n return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left == null || cur.right == null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n TreeNode? child = cur.left ?? cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur != root) {\n if (pre!.left == cur)\n pre.left = child;\n else\n pre.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode? tmp = cur.right;\n while (tmp.left != null) {\n tmp = tmp.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n Remove(tmp.val!.Value);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val;\n }\n}\n
binary_search_tree.swift/* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\n cur := bst.root\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if cur == nil {\n return\n }\n // \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n var pre *TreeNode = nil\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n for cur != nil {\n if cur.Val == num {\n break\n }\n pre = cur\n if cur.Val.(int) < num {\n // \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\n cur = cur.Right\n } else {\n // \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\n cur = cur.Left\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if cur == nil {\n return\n }\n // \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\n if cur.Left == nil || cur.Right == nil {\n var child *TreeNode = nil\n // \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\n if cur.Left != nil {\n child = cur.Left\n } else {\n child = cur.Right\n }\n // \u5220\u9664\u8282\u70b9 cur\n if cur != bst.root {\n if pre.Left == cur {\n pre.Left = child\n } else {\n pre.Right = child\n }\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n bst.root = child\n }\n // \u5b50\u8282\u70b9\u6570\u4e3a 2\n } else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n tmp := cur.Right\n for tmp.Left != nil {\n tmp = tmp.Left\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n bst.remove(tmp.Val.(int))\n // \u7528 tmp \u8986\u76d6 cur\n cur.Val = tmp.Val\n }\n}\n
binary_search_tree.js/* \u5220\u9664\u8282\u70b9 */\nfunc remove(num: Int) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if root == nil {\n return\n }\n var cur = root\n var pre: TreeNode?\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while cur != nil {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if cur!.val == num {\n break\n }\n pre = cur\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if cur!.val < num {\n cur = cur?.right\n }\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else {\n cur = cur?.left\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if cur == nil {\n return\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if cur?.left == nil || cur?.right == nil {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n let child = cur?.left != nil ? cur?.left : cur?.right\n // \u5220\u9664\u8282\u70b9 cur\n if cur !== root {\n if pre?.left === cur {\n pre?.left = child\n } else {\n pre?.right = child\n }\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n root = child\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n var tmp = cur?.right\n while tmp?.left != nil {\n tmp = tmp?.left\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n remove(num: tmp!.val)\n // \u7528 tmp \u8986\u76d6 cur\n cur?.val = tmp!.val\n }\n}\n
binary_search_tree.ts/* \u5220\u9664\u8282\u70b9 */\nremove(num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (this.root === null) return;\n let cur = this.root,\n pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val === num) break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur === null) return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left === null || cur.right === null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n const child = cur.left !== null ? cur.left : cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur !== this.root) {\n if (pre.left === cur) pre.left = child;\n else pre.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n this.root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n let tmp = cur.right;\n while (tmp.left !== null) {\n tmp = tmp.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n this.remove(tmp.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val;\n }\n}\n
binary_search_tree.dart/* \u5220\u9664\u8282\u70b9 */\nremove(num: number): void {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (this.root === null) return;\n let cur: TreeNode | null = this.root,\n pre: TreeNode | null = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur !== null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val === num) break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < num) cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur === null) return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left === null || cur.right === null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n const child: TreeNode | null =\n cur.left !== null ? cur.left : cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur !== this.root) {\n if (pre!.left === cur) pre!.left = child;\n else pre!.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n this.root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n let tmp: TreeNode | null = cur.right;\n while (tmp!.left !== null) {\n tmp = tmp!.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n this.remove(tmp!.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp!.val;\n }\n}\n
binary_search_tree.rs/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int _num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (_root == null) return;\n TreeNode? cur = _root;\n TreeNode? pre = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.val == _num) break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.val < _num)\n cur = cur.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n else\n cur = cur.left;\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n if (cur == null) return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.left == null || cur.right == null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n TreeNode? child = cur.left ?? cur.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (cur != _root) {\n if (pre!.left == cur)\n pre.left = child;\n else\n pre.right = child;\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n _root = child;\n }\n } else {\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode? tmp = cur.right;\n while (tmp!.left != null) {\n tmp = tmp.left;\n }\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n remove(tmp.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.val = tmp.val;\n }\n}\n
binary_search_tree.c/* \u5220\u9664\u8282\u70b9 */\npub fn remove(&mut self, num: i32) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if self.root.is_none() { \n return; \n }\n let mut cur = self.root.clone();\n let mut pre = None;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while let Some(node) = cur.clone() {\n match num.cmp(&node.borrow().val) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n Ordering::Equal => break,\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n Ordering::Greater => {\n pre = cur.clone();\n cur = node.borrow().right.clone();\n }\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n Ordering::Less => {\n pre = cur.clone();\n cur = node.borrow().left.clone();\n }\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if cur.is_none() {\n return;\n }\n let cur = cur.unwrap();\n let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n match (left_child.clone(), right_child.clone()) {\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n (None, None) | (Some(_), None) | (None, Some(_)) => {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n let child = left_child.or(right_child);\n let pre = pre.unwrap();\n // \u5220\u9664\u8282\u70b9 cur\n if !Rc::ptr_eq(&cur, self.root.as_ref().unwrap()) {\n let left = pre.borrow().left.clone();\n if left.is_some() && Rc::ptr_eq(&left.as_ref().unwrap(), &cur) {\n pre.borrow_mut().left = child;\n } else {\n pre.borrow_mut().right = child;\n }\n } else {\n // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n self.root = child;\n }\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n (Some(_), Some(_)) => {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n let mut tmp = cur.borrow().right.clone();\n while let Some(node) = tmp.clone() {\n if node.borrow().left.is_some() {\n tmp = node.borrow().left.clone();\n } else {\n break;\n }\n }\n let tmpval = tmp.unwrap().borrow().val;\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n self.remove(tmpval);\n // \u7528 tmp \u8986\u76d6 cur\n cur.borrow_mut().val = tmpval;\n }\n }\n}\n
binary_search_tree.zig/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(BinarySearchTree *bst, int num) {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (bst->root == NULL)\n return;\n TreeNode *cur = bst->root, *pre = NULL;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != NULL) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur->val == num)\n break;\n pre = cur;\n if (cur->val < num) {\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\n cur = cur->right;\n } else {\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\n cur = cur->left;\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == NULL)\n return;\n // \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\n if (cur->left == NULL || cur->right == NULL) {\n /* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n TreeNode *child = cur->left != NULL ? cur->left : cur->right;\n // \u5220\u9664\u8282\u70b9 cur\n if (pre->left == cur) {\n pre->left = child;\n } else {\n pre->right = child;\n }\n } else {\n /* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n TreeNode *tmp = cur->right;\n while (tmp->left != NULL) {\n tmp = tmp->left;\n }\n int tmpVal = tmp->val;\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n removeItem(bst, tmp->val);\n // \u7528 tmp \u8986\u76d6 cur\n cur->val = tmpVal;\n }\n}\n
"},{"location":"chapter_tree/binary_search_tree/#4","title":"4. \u00a0 \u4e2d\u5e8f\u904d\u5386\u6709\u5e8f","text":"// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n if (self.root == null) return;\n var cur = self.root;\n var pre: ?*inc.TreeNode(T) = null;\n // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n while (cur != null) {\n // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n if (cur.?.val == num) break;\n pre = cur;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n if (cur.?.val < num) {\n cur = cur.?.right;\n // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n } else {\n cur = cur.?.left;\n }\n }\n // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n if (cur == null) return;\n // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n if (cur.?.left == null or cur.?.right == null) {\n // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n var child = if (cur.?.left != null) cur.?.left else cur.?.right;\n // \u5220\u9664\u8282\u70b9 cur\n if (pre.?.left == cur) {\n pre.?.left = child;\n } else {\n pre.?.right = child;\n }\n // \u5b50\u8282\u70b9\u6570\u91cf = 2\n } else {\n // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n var tmp = cur.?.right;\n while (tmp.?.left != null) {\n tmp = tmp.?.left;\n }\n var tmp_val = tmp.?.val;\n // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n self.remove(tmp.?.val);\n // \u7528 tmp \u8986\u76d6 cur\n cur.?.val = tmp_val;\n }\n}\n\u5982\u56fe 7-22 \u6240\u793a\uff0c\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u987a\u5e8f\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u6ee1\u8db3\u201c\u5de6\u5b50\u8282\u70b9 \\(<\\) \u6839\u8282\u70b9 \\(<\\) \u53f3\u5b50\u8282\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002
\u8fd9\u610f\u5473\u7740\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u8282\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002
\u5229\u7528\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u65e0\u987b\u8fdb\u884c\u989d\u5916\u7684\u6392\u5e8f\u64cd\u4f5c\uff0c\u975e\u5e38\u9ad8\u6548\u3002
\u56fe 7-22 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217
"},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528\u6570\u7ec4\u6216\u4e8c\u53c9\u641c\u7d22\u6811\u5b58\u50a8\u3002\u89c2\u5bdf\u8868 7-2 \uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5177\u6709\u7a33\u5b9a\u4e14\u9ad8\u6548\u7684\u6027\u80fd\u3002\u53ea\u6709\u5728\u9ad8\u9891\u6dfb\u52a0\u3001\u4f4e\u9891\u67e5\u627e\u5220\u9664\u6570\u636e\u7684\u573a\u666f\u4e0b\uff0c\u6570\u7ec4\u6bd4\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002
\u8868 7-2 \u00a0 \u6570\u7ec4\u4e0e\u641c\u7d22\u6811\u7684\u6548\u7387\u5bf9\u6bd4
\u65e0\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\)\u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u8282\u70b9\u3002
\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u4e0d\u65ad\u5730\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u56fe 7-23 \u6240\u793a\u7684\u94fe\u8868\uff0c\u8fd9\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4f1a\u9000\u5316\u4e3a \\(O(n)\\) \u3002
\u56fe 7-23 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316
"},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"- \u7528\u4f5c\u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002
- \u4f5c\u4e3a\u67d0\u4e9b\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
- \u7528\u4e8e\u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72b6\u6001\u3002
\u300c\u4e8c\u53c9\u6811 binary tree\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u5355\u5143\u662f\u8282\u70b9\uff0c\u6bcf\u4e2a\u8282\u70b9\u5305\u542b\u503c\u3001\u5de6\u5b50\u8282\u70b9\u5f15\u7528\u548c\u53f3\u5b50\u8282\u70b9\u5f15\u7528\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZigclass TreeNode:\n \"\"\"\u4e8c\u53c9\u6811\u8282\u70b9\u7c7b\"\"\"\n def __init__(self, val: int):\n self.val: int = val # \u8282\u70b9\u503c\n self.left: TreeNode | None = None # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n self.right: TreeNode | None = None # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n int val; // \u8282\u70b9\u503c\n TreeNode *left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n int val; // \u8282\u70b9\u503c\n TreeNode left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n TreeNode right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n TreeNode(int x) { val = x; }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n public int? val = x; // \u8282\u70b9\u503c\n public TreeNode? left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n Val int\n Left *TreeNode\n Right *TreeNode\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\n return &TreeNode{\n Left: nil, // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n Right: nil, // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n Val: v, // \u8282\u70b9\u503c\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n var val: Int // \u8282\u70b9\u503c\n var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n\n init(x: Int) {\n val = x\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n val; // \u8282\u70b9\u503c\n left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n constructor(val, left, right) {\n this.val = val === undefined ? 0 : val;\n this.left = left === undefined ? null : left;\n this.right = right === undefined ? null : right;\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n val: number;\n left: TreeNode | null;\n right: TreeNode | null;\n\n constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n this.left = left === undefined ? null : left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n this.right = right === undefined ? null : right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n int val; // \u8282\u70b9\u503c\n TreeNode? left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n TreeNode(this.val, [this.left, this.right]);\n}\nuse std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n val: i32, // \u8282\u70b9\u503c\n left: Option<Rc<RefCell<TreeNode>>>, // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n right: Option<Rc<RefCell<TreeNode>>>, // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n\nimpl TreeNode {\n /* \u6784\u9020\u65b9\u6cd5 */\n fn new(val: i32) -> Rc<RefCell<Self>> {\n Rc::new(RefCell::new(Self {\n val,\n left: None,\n right: None\n }))\n }\n}\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct TreeNode {\n int val; // \u8282\u70b9\u503c\n int height; // \u8282\u70b9\u9ad8\u5ea6\n struct TreeNode *left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n struct TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n TreeNode *node;\n\n node = (TreeNode *)malloc(sizeof(TreeNode));\n node->val = val;\n node->height = 0;\n node->left = NULL;\n node->right = NULL;\n return node;\n}\n\n\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u5206\u522b\u6307\u5411\u300c\u5de6\u5b50\u8282\u70b9 left-child node\u300d\u548c\u300c\u53f3\u5b50\u8282\u70b9 right-child node\u300d\uff0c\u8be5\u8282\u70b9\u88ab\u79f0\u4e3a\u8fd9\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u300c\u7236\u8282\u70b9 parent node\u300d\u3002\u5f53\u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u65f6\uff0c\u6211\u4eec\u5c06\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u300c\u5de6\u5b50\u6811 left subtree\u300d\uff0c\u540c\u7406\u53ef\u5f97\u300c\u53f3\u5b50\u6811 right subtree\u300d\u3002
\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u9664\u53f6\u8282\u70b9\u5916\uff0c\u5176\u4ed6\u6240\u6709\u8282\u70b9\u90fd\u5305\u542b\u5b50\u8282\u70b9\u548c\u975e\u7a7a\u5b50\u6811\u3002\u5982\u56fe 7-1 \u6240\u793a\uff0c\u5982\u679c\u5c06\u201c\u8282\u70b9 2\u201d\u89c6\u4e3a\u7236\u8282\u70b9\uff0c\u5219\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u5206\u522b\u662f\u201c\u8282\u70b9 4\u201d\u548c\u201c\u8282\u70b9 5\u201d\uff0c\u5de6\u5b50\u6811\u662f\u201c\u8282\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\uff0c\u53f3\u5b50\u6811\u662f\u201c\u8282\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\u3002
\u56fe 7-1 \u00a0 \u7236\u8282\u70b9\u3001\u5b50\u8282\u70b9\u3001\u5b50\u6811
"},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1 \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"\u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed\u5982\u56fe 7-2 \u6240\u793a\u3002
- \u300c\u6839\u8282\u70b9 root node\u300d\uff1a\u4f4d\u4e8e\u4e8c\u53c9\u6811\u9876\u5c42\u7684\u8282\u70b9\uff0c\u6ca1\u6709\u7236\u8282\u70b9\u3002
- \u300c\u53f6\u8282\u70b9 leaf node\u300d\uff1a\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u5747\u6307\u5411 \\(\\text{None}\\) \u3002
- \u300c\u8fb9 edge\u300d\uff1a\u8fde\u63a5\u4e24\u4e2a\u8282\u70b9\u7684\u7ebf\u6bb5\uff0c\u5373\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u3002
- \u8282\u70b9\u6240\u5728\u7684\u300c\u5c42 level\u300d\uff1a\u4ece\u9876\u81f3\u5e95\u9012\u589e\uff0c\u6839\u8282\u70b9\u6240\u5728\u5c42\u4e3a 1 \u3002
- \u8282\u70b9\u7684\u300c\u5ea6 degree\u300d\uff1a\u8282\u70b9\u7684\u5b50\u8282\u70b9\u7684\u6570\u91cf\u3002\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u53d6\u503c\u8303\u56f4\u662f 0\u30011\u30012 \u3002
- \u4e8c\u53c9\u6811\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002
- \u8282\u70b9\u7684\u300c\u6df1\u5ea6 depth\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002
- \u8282\u70b9\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u8ddd\u79bb\u8be5\u8282\u70b9\u6700\u8fdc\u7684\u53f6\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002
\u56fe 7-2 \u00a0 \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed
Tip
\u8bf7\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u201c\u9ad8\u5ea6\u201d\u548c\u201c\u6df1\u5ea6\u201d\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u53ef\u80fd\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8282\u70b9\u7684\u6570\u91cf\u201d\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002
"},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2 \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_tree/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e8c\u53c9\u6811","text":"\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u8282\u70b9\uff0c\u7136\u540e\u6784\u5efa\u5f15\u7528\uff08\u6307\u9488\uff09\u3002
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree.py
binary_tree.cpp# \u521d\u59cb\u5316\u4e8c\u53c9\u6811\n# \u521d\u59cb\u5316\u8282\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
binary_tree.java/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
binary_tree.cs// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.go/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.swift/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n
binary_tree.js// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
binary_tree.ts/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n n2 = new TreeNode(2),\n n3 = new TreeNode(3),\n n4 = new TreeNode(4),\n n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.dart/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n n2 = new TreeNode(2),\n n3 = new TreeNode(3),\n n4 = new TreeNode(4),\n n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.rs/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
binary_tree.c// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n
binary_tree.zig/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
"},{"location":"chapter_tree/binary_tree/#2","title":"2. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9","text":"\n\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u6307\u9488\u6765\u5b9e\u73b0\u3002\u56fe 7-3 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u793a\u4f8b\u3002
\u56fe 7-3 \u00a0 \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree.py
binary_tree.cpp# \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\np = TreeNode(0)\n# \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n
binary_tree.java/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1->left = n2;\n
binary_tree.csTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.go/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.swift/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.Left = n2\n
binary_tree.jslet P = TreeNode(x: 0)\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n
binary_tree.ts/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.dart/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.rs/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
binary_tree.clet p = TreeNode::new(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// \u5220\u9664\u8282\u70b9 p\nn1.borrow_mut().left = Some(n2);\n
binary_tree.zig/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1->left = n2;\n\nNote
\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u8282\u70b9\u53ef\u80fd\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u800c\u5220\u9664\u8282\u70b9\u901a\u5e38\u610f\u5473\u7740\u5220\u9664\u8be5\u8282\u70b9\u53ca\u5176\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5b9e\u73b0\u6709\u5b9e\u9645\u610f\u4e49\u7684\u64cd\u4f5c\u3002
"},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3 \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#1_1","title":"1. \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"\u5982\u56fe 7-4 \u6240\u793a\uff0c\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811 perfect binary tree\u300d\u6240\u6709\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u53f6\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u7684\u5ea6\u90fd\u4e3a \\(2\\) \uff1b\u82e5\u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \uff0c\u5219\u8282\u70b9\u603b\u6570\u4e3a \\(2^{h+1} - 1\\) \uff0c\u5448\u73b0\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u73b0\u8c61\u3002
Tip
\u8bf7\u6ce8\u610f\uff0c\u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\u3002
\u56fe 7-4 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811
"},{"location":"chapter_tree/binary_tree/#2_1","title":"2. \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811","text":"\u5982\u56fe 7-5 \u6240\u793a\uff0c\u300c\u5b8c\u5168\u4e8c\u53c9\u6811 complete binary tree\u300d\u53ea\u6709\u6700\u5e95\u5c42\u7684\u8282\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u8282\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002
\u56fe 7-5 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811
"},{"location":"chapter_tree/binary_tree/#3","title":"3. \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"\u5982\u56fe 7-6 \u6240\u793a\uff0c\u300c\u5b8c\u6ee1\u4e8c\u53c9\u6811 full binary tree\u300d\u9664\u4e86\u53f6\u8282\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\u3002
\u56fe 7-6 \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811
"},{"location":"chapter_tree/binary_tree/#4","title":"4. \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811","text":"\u5982\u56fe 7-7 \u6240\u793a\uff0c\u300c\u5e73\u8861\u4e8c\u53c9\u6811 balanced binary tree\u300d\u4e2d\u4efb\u610f\u8282\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc7 1 \u3002
\u56fe 7-7 \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811
"},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4 \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"\u56fe 7-8 \u5c55\u793a\u4e86\u4e8c\u53c9\u6811\u7684\u7406\u60f3\u7ed3\u6784\u4e0e\u9000\u5316\u7ed3\u6784\u3002\u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff1b\u800c\u5f53\u6240\u6709\u8282\u70b9\u90fd\u504f\u5411\u4e00\u4fa7\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u201c\u94fe\u8868\u201d\u3002
- \u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u60c5\u51b5\uff0c\u53ef\u4ee5\u5145\u5206\u53d1\u6325\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\u3002
- \u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002
\u56fe 7-8 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784
\u5982\u8868 7-1 \u6240\u793a\uff0c\u5728\u6700\u4f73\u7ed3\u6784\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf\u3001\u8282\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u503c\u6216\u6781\u5c0f\u503c\u3002
\u8868 7-1 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784
\u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u8282\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u8282\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \u7684\u6811\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"\u4ece\u7269\u7406\u7ed3\u6784\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u5176\u904d\u5386\u65b9\u5f0f\u662f\u901a\u8fc7\u6307\u9488\u9010\u4e2a\u8bbf\u95ee\u8282\u70b9\u3002\u7136\u800c\uff0c\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u4f7f\u5f97\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u501f\u52a9\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002
\u4e8c\u53c9\u6811\u5e38\u89c1\u7684\u904d\u5386\u65b9\u5f0f\u5305\u62ec\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u7b49\u3002
"},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1 \u00a0 \u5c42\u5e8f\u904d\u5386","text":"\u5982\u56fe 7-9 \u6240\u793a\uff0c\u300c\u5c42\u5e8f\u904d\u5386 level-order traversal\u300d\u4ece\u9876\u90e8\u5230\u5e95\u90e8\u9010\u5c42\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u4e00\u5c42\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u8282\u70b9\u3002
\u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u5c5e\u4e8e\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal, BFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\u3002
\u56fe 7-9 \u00a0 \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386
"},{"location":"chapter_tree/binary_tree_traversal/#1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u901a\u5e38\u501f\u52a9\u201c\u961f\u5217\u201d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u9075\u5faa\u201c\u5148\u8fdb\u5148\u51fa\u201d\u7684\u89c4\u5219\uff0c\u800c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5219\u9075\u5faa\u201c\u9010\u5c42\u63a8\u8fdb\u201d\u7684\u89c4\u5219\uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree_bfs.py
binary_tree_bfs.cppdef level_order(root: TreeNode | None) -> list[int]:\n \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n # \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n queue: deque[TreeNode] = deque()\n queue.append(root)\n # \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n res = []\n while queue:\n node: TreeNode = queue.popleft() # \u961f\u5217\u51fa\u961f\n res.append(node.val) # \u4fdd\u5b58\u8282\u70b9\u503c\n if node.left is not None:\n queue.append(node.left) # \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if node.right is not None:\n queue.append(node.right) # \u53f3\u5b50\u8282\u70b9\u5165\u961f\n return res\n
binary_tree_bfs.java/* \u5c42\u5e8f\u904d\u5386 */\nvector<int> levelOrder(TreeNode *root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n queue<TreeNode *> queue;\n queue.push(root);\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n vector<int> vec;\n while (!queue.empty()) {\n TreeNode *node = queue.front();\n queue.pop(); // \u961f\u5217\u51fa\u961f\n vec.push_back(node->val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node->left != nullptr)\n queue.push(node->left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node->right != nullptr)\n queue.push(node->right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return vec;\n}\n
binary_tree_bfs.cs/* \u5c42\u5e8f\u904d\u5386 */\nList<Integer> levelOrder(TreeNode root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n Queue<TreeNode> queue = new LinkedList<>();\n queue.add(root);\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n List<Integer> list = new ArrayList<>();\n while (!queue.isEmpty()) {\n TreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\n list.add(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left != null)\n queue.offer(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node.right != null)\n queue.offer(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return list;\n}\n
binary_tree_bfs.go/* \u5c42\u5e8f\u904d\u5386 */\nList<int> LevelOrder(TreeNode root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n Queue<TreeNode> queue = new();\n queue.Enqueue(root);\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n List<int> list = [];\n while (queue.Count != 0) {\n TreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\n list.Add(node.val!.Value); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left != null)\n queue.Enqueue(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node.right != null)\n queue.Enqueue(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return list;\n}\n
binary_tree_bfs.swift/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []any {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n queue := list.New()\n queue.PushBack(root)\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n nums := make([]any, 0)\n for queue.Len() > 0 {\n // \u961f\u5217\u51fa\u961f\n node := queue.Remove(queue.Front()).(*TreeNode)\n // \u4fdd\u5b58\u8282\u70b9\u503c\n nums = append(nums, node.Val)\n if node.Left != nil {\n // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n queue.PushBack(node.Left)\n }\n if node.Right != nil {\n // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n queue.PushBack(node.Right)\n }\n }\n return nums\n}\n
binary_tree_bfs.js/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -> [Int] {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n var queue: [TreeNode] = [root]\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n var list: [Int] = []\n while !queue.isEmpty {\n let node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\n list.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\n if let left = node.left {\n queue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n }\n if let right = node.right {\n queue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n }\n return list\n}\n
binary_tree_bfs.ts/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n const queue = [root];\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n const list = [];\n while (queue.length) {\n let node = queue.shift(); // \u961f\u5217\u51fa\u961f\n list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return list;\n}\n
binary_tree_bfs.dart/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n const queue = [root];\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n const list: number[] = [];\n while (queue.length) {\n let node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\n list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left) {\n queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n }\n if (node.right) {\n queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n }\n return list;\n}\n
binary_tree_bfs.rs/* \u5c42\u5e8f\u904d\u5386 */\nList<int> levelOrder(TreeNode? root) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n Queue<TreeNode?> queue = Queue();\n queue.add(root);\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n List<int> res = [];\n while (queue.isNotEmpty) {\n TreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\n res.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n if (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n }\n return res;\n}\n
binary_tree_bfs.c/* \u5c42\u5e8f\u904d\u5386 */\nfn level_order(root: &Rc<RefCell<TreeNode>>) -> Vec<i32> {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n let mut que = VecDeque::new();\n que.push_back(Rc::clone(&root));\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n let mut vec = Vec::new();\n\n while let Some(node) = que.pop_front() { // \u961f\u5217\u51fa\u961f\n vec.push(node.borrow().val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if let Some(left) = node.borrow().left.as_ref() {\n que.push_back(Rc::clone(left)); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n }\n if let Some(right) = node.borrow().right.as_ref() {\n que.push_back(Rc::clone(right)); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n };\n }\n vec\n}\n
binary_tree_bfs.zig/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n /* \u8f85\u52a9\u961f\u5217 */\n int front, rear;\n int index, *arr;\n TreeNode *node;\n TreeNode **queue;\n\n /* \u8f85\u52a9\u961f\u5217 */\n queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n // \u961f\u5217\u6307\u9488\n front = 0, rear = 0;\n // \u52a0\u5165\u6839\u8282\u70b9\n queue[rear++] = root;\n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n /* \u8f85\u52a9\u6570\u7ec4 */\n arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n // \u6570\u7ec4\u6307\u9488\n index = 0;\n while (front < rear) {\n // \u961f\u5217\u51fa\u961f\n node = queue[front++];\n // \u4fdd\u5b58\u8282\u70b9\u503c\n arr[index++] = node->val;\n if (node->left != NULL) {\n // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n queue[rear++] = node->left;\n }\n if (node->right != NULL) {\n // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n queue[rear++] = node->right;\n }\n }\n // \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n *size = index;\n arr = realloc(arr, sizeof(int) * (*size));\n\n // \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\n free(queue);\n return arr;\n}\n
"},{"location":"chapter_tree/binary_tree_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"// \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n const L = std.TailQueue(*inc.TreeNode(T));\n var queue = L{};\n var root_node = try mem_allocator.create(L.Node);\n root_node.data = root;\n queue.append(root_node); \n // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n var list = std.ArrayList(T).init(std.heap.page_allocator);\n while (queue.len > 0) {\n var queue_node = queue.popFirst().?; // \u961f\u5217\u51fa\u961f\n var node = queue_node.data;\n try list.append(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n if (node.left != null) {\n var tmp_node = try mem_allocator.create(L.Node);\n tmp_node.data = node.left.?;\n queue.append(tmp_node); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n }\n if (node.right != null) {\n var tmp_node = try mem_allocator.create(L.Node);\n tmp_node.data = node.right.?;\n queue.append(tmp_node); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n } \n }\n return list;\n}\n- \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002
- \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6ee1\u4e8c\u53c9\u6811\u65f6\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\((n + 1) / 2\\) \u4e2a\u8282\u70b9\uff0c\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002
\u76f8\u5e94\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal, DFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\u3002
\u56fe 7-10 \u5c55\u793a\u4e86\u5bf9\u4e8c\u53c9\u6811\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5de5\u4f5c\u539f\u7406\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u68f5\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u6bcf\u4e2a\u8282\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u3002
\u56fe 7-10 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386
"},{"location":"chapter_tree/binary_tree_traversal/#1_1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff1a
PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree_dfs.py
binary_tree_dfs.cppdef pre_order(root: TreeNode | None):\n \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n if root is None:\n return\n # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n res.append(root.val)\n pre_order(root=root.left)\n pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n if root is None:\n return\n # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n in_order(root=root.left)\n res.append(root.val)\n in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n if root is None:\n return\n # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n post_order(root=root.left)\n post_order(root=root.right)\n res.append(root.val)\n
binary_tree_dfs.java/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\n if (root == nullptr)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n vec.push_back(root->val);\n preOrder(root->left);\n preOrder(root->right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\n if (root == nullptr)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root->left);\n vec.push_back(root->val);\n inOrder(root->right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\n if (root == nullptr)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root->left);\n postOrder(root->right);\n vec.push_back(root->val);\n}\n
binary_tree_dfs.cs/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\n if (root == null)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.add(root.val);\n preOrder(root.left);\n preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\n if (root == null)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root.left);\n list.add(root.val);\n inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\n if (root == null)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root.left);\n postOrder(root.right);\n list.add(root.val);\n}\n
binary_tree_dfs.go/* \u524d\u5e8f\u904d\u5386 */\nvoid PreOrder(TreeNode? root) {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.Add(root.val!.Value);\n PreOrder(root.left);\n PreOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid InOrder(TreeNode? root) {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n InOrder(root.left);\n list.Add(root.val!.Value);\n InOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid PostOrder(TreeNode? root) {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n PostOrder(root.left);\n PostOrder(root.right);\n list.Add(root.val!.Value);\n}\n
binary_tree_dfs.swift/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\n if node == nil {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n nums = append(nums, node.Val)\n preOrder(node.Left)\n preOrder(node.Right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\n if node == nil {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(node.Left)\n nums = append(nums, node.Val)\n inOrder(node.Right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\n if node == nil {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(node.Left)\n postOrder(node.Right)\n nums = append(nums, node.Val)\n}\n
binary_tree_dfs.js/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.append(root.val)\n preOrder(root: root.left)\n preOrder(root: root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root: root.left)\n list.append(root.val)\n inOrder(root: root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\n guard let root = root else {\n return\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root: root.left)\n postOrder(root: root.right)\n list.append(root.val)\n}\n
binary_tree_dfs.ts/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\n if (root === null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.push(root.val);\n preOrder(root.left);\n preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\n if (root === null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root.left);\n list.push(root.val);\n inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\n if (root === null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root.left);\n postOrder(root.right);\n list.push(root.val);\n}\n
binary_tree_dfs.dart/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\n if (root === null) {\n return;\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.push(root.val);\n preOrder(root.left);\n preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\n if (root === null) {\n return;\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root.left);\n list.push(root.val);\n inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\n if (root === null) {\n return;\n }\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root.left);\n postOrder(root.right);\n list.push(root.val);\n}\n
binary_tree_dfs.rs/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\n if (node == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n list.add(node.val);\n preOrder(node.left);\n preOrder(node.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\n if (node == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(node.left);\n list.add(node.val);\n inOrder(node.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\n if (node == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(node.left);\n postOrder(node.right);\n list.add(node.val);\n}\n
binary_tree_dfs.c/* \u524d\u5e8f\u904d\u5386 */\nfn pre_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {\n let mut result = vec![];\n\n if let Some(node) = root {\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n result.push(node.borrow().val);\n result.append(&mut pre_order(node.borrow().left.as_ref()));\n result.append(&mut pre_order(node.borrow().right.as_ref()));\n }\n result\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfn in_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {\n let mut result = vec![];\n\n if let Some(node) = root {\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n result.append(&mut in_order(node.borrow().left.as_ref()));\n result.push(node.borrow().val);\n result.append(&mut in_order(node.borrow().right.as_ref()));\n }\n result\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfn post_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {\n let mut result = vec![];\n\n if let Some(node) = root {\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n result.append(&mut post_order(node.borrow().left.as_ref()));\n result.append(&mut post_order(node.borrow().right.as_ref()));\n result.push(node.borrow().val);\n }\n result\n}\n
binary_tree_dfs.zig/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\n if (root == NULL)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n arr[(*size)++] = root->val;\n preOrder(root->left, size);\n preOrder(root->right, size);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\n if (root == NULL)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n inOrder(root->left, size);\n arr[(*size)++] = root->val;\n inOrder(root->right, size);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\n if (root == NULL)\n return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n postOrder(root->left, size);\n postOrder(root->right, size);\n arr[(*size)++] = root->val;\n}\n// \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\n try list.append(root.?.val);\n try preOrder(T, root.?.left);\n try preOrder(T, root.?.right);\n}\n\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\n try inOrder(T, root.?.left);\n try list.append(root.?.val);\n try inOrder(T, root.?.right);\n}\n\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n if (root == null) return;\n // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\n try postOrder(T, root.?.left);\n try postOrder(T, root.?.right);\n try list.append(root.?.val);\n}\nTip
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\u56fe 7-11 \u00a0 \u524d\u5e8f\u904d\u5386\u7684\u9012\u5f52\u8fc7\u7a0b
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- \u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u5176\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \u3002\u5f53\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
- AVL \u6811\uff0c\u4e5f\u79f0\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5b83\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\u786e\u4fdd\u5728\u4e0d\u65ad\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\u540e\u6811\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002
- AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u5411\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002
\u5bf9\u4e8e\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u53c9\u6811\uff0c\u6811\u7684\u9ad8\u5ea6\u548c\u6839\u8282\u70b9\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u5417\uff1f
\u662f\u7684\uff0c\u56e0\u4e3a\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\u3002
\u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\uff0c\u8fd9\u91cc\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u4e48\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d44\u6e90\u7684\u5b50\u8282\u70b9\u7684\u8d44\u6e90\u91ca\u653e\u5417\uff1f
\u62ff\u4e8c\u53c9\u641c\u7d22\u6811\u6765\u4e3e\u4f8b\uff0c\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u8981\u5206\u4e09\u79cd\u60c5\u51b5\u5904\u7406\uff0c\u5176\u4e2d\u6bcf\u79cd\u60c5\u51b5\u90fd\u9700\u8981\u8fdb\u884c\u591a\u4e2a\u6b65\u9aa4\u7684\u8282\u70b9\u64cd\u4f5c\u3002
\u4e3a\u4ec0\u4e48 DFS \u904d\u5386\u4e8c\u53c9\u6811\u6709\u524d\u3001\u4e2d\u3001\u540e\u4e09\u79cd\u987a\u5e8f\uff0c\u5206\u522b\u6709\u4ec0\u4e48\u7528\u5462\uff1f
\u4e0e\u987a\u5e8f\u548c\u9006\u5e8f\u904d\u5386\u6570\u7ec4\u7c7b\u4f3c\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u662f\u4e09\u79cd\u4e8c\u53c9\u6811\u904d\u5386\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5b83\u4eec\u5f97\u5230\u4e00\u4e2a\u7279\u5b9a\u987a\u5e8f\u7684\u904d\u5386\u7ed3\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\uff0c\u7531\u4e8e\u8282\u70b9\u5927\u5c0f\u6ee1\u8db3
\u5de6\u5b50\u8282\u70b9\u503c < \u6839\u8282\u70b9\u503c < \u53f3\u5b50\u8282\u70b9\u503c\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u8981\u6309\u7167\u5de6 $\\rightarrow$ \u6839 $\\rightarrow$ \u53f3\u7684\u4f18\u5148\u7ea7\u904d\u5386\u6811\uff0c\u5c31\u53ef\u4ee5\u83b7\u5f97\u6709\u5e8f\u7684\u8282\u70b9\u5e8f\u5217\u3002\u53f3\u65cb\u64cd\u4f5c\u662f\u5904\u7406\u5931\u8861\u8282\u70b9
node\u3001child\u3001grand_child\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u90a3node\u7684\u7236\u8282\u70b9\u548cnode\u539f\u6765\u7684\u8fde\u63a5\u4e0d\u9700\u8981\u7ef4\u62a4\u5417\uff1f\u53f3\u65cb\u64cd\u4f5c\u540e\u5c82\u4e0d\u662f\u65ad\u6389\u4e86\uff1f\u6211\u4eec\u9700\u8981\u4ece\u9012\u5f52\u7684\u89c6\u89d2\u6765\u770b\u8fd9\u4e2a\u95ee\u9898\u3002\u53f3\u65cb\u64cd\u4f5c
right_rotate(root)\u4f20\u5165\u7684\u662f\u5b50\u6811\u7684\u6839\u8282\u70b9\uff0c\u6700\u7ec8return child\u8fd4\u56de\u65cb\u8f6c\u4e4b\u540e\u7684\u5b50\u6811\u7684\u6839\u8282\u70b9\u3002\u5b50\u6811\u7684\u6839\u8282\u70b9\u548c\u5176\u7236\u8282\u70b9\u7684\u8fde\u63a5\u662f\u5728\u8be5\u51fd\u6570\u8fd4\u56de\u540e\u5b8c\u6210\u7684\uff0c\u4e0d\u5c5e\u4e8e\u53f3\u65cb\u64cd\u4f5c\u7684\u7ef4\u62a4\u8303\u56f4\u3002\u5728 C++ \u4e2d\uff0c\u51fd\u6570\u88ab\u5212\u5206\u5230
private\u548cpublic\u4e2d\uff0c\u8fd9\u65b9\u9762\u6709\u4ec0\u4e48\u8003\u91cf\u5417\uff1f\u4e3a\u4ec0\u4e48\u8981\u5c06height()\u51fd\u6570\u548cupdateHeight()\u51fd\u6570\u5206\u522b\u653e\u5728public\u548cprivate\u4e2d\u5462\uff1f\u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u8303\u56f4\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u7c7b\u5185\u90e8\u4f7f\u7528\uff0c\u90a3\u4e48\u5c31\u8bbe\u8ba1\u4e3a
private\u3002\u4f8b\u5982\uff0c\u7528\u6237\u5355\u72ec\u8c03\u7528updateHeight()\u662f\u6ca1\u6709\u610f\u4e49\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800cheight()\u662f\u8bbf\u95ee\u8282\u70b9\u9ad8\u5ea6\uff0c\u7c7b\u4f3c\u4e8evector.size()\uff0c\u56e0\u6b64\u8bbe\u7f6e\u6210public\u4ee5\u4fbf\u4f7f\u7528\u3002\u5982\u4f55\u4ece\u4e00\u7ec4\u8f93\u5165\u6570\u636e\u6784\u5efa\u4e00\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\uff1f\u6839\u8282\u70b9\u7684\u9009\u62e9\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f
\u662f\u7684\uff0c\u6784\u5efa\u6811\u7684\u65b9\u6cd5\u5df2\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4ee3\u7801\u4e2d\u7684
build_tree()\u65b9\u6cd5\u4e2d\u7ed9\u51fa\u3002\u81f3\u4e8e\u6839\u8282\u70b9\u7684\u9009\u62e9\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u8f93\u5165\u6570\u636e\u6392\u5e8f\uff0c\u7136\u540e\u5c06\u4e2d\u70b9\u5143\u7d20\u4f5c\u4e3a\u6839\u8282\u70b9\uff0c\u518d\u9012\u5f52\u5730\u6784\u5efa\u5de6\u53f3\u5b50\u6811\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8bc1\u6811\u7684\u5e73\u8861\u6027\u3002\u5728 Java \u4e2d\uff0c\u5b57\u7b26\u4e32\u5bf9\u6bd4\u662f\u5426\u4e00\u5b9a\u8981\u7528
equals()\u65b9\u6cd5\uff1f\u5728 Java \u4e2d\uff0c\u5bf9\u4e8e\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c
==\u7528\u4e8e\u5bf9\u6bd4\u4e24\u4e2a\u53d8\u91cf\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002\u5bf9\u4e8e\u5f15\u7528\u7c7b\u578b\uff0c\u4e24\u79cd\u7b26\u53f7\u7684\u5de5\u4f5c\u539f\u7406\u662f\u4e0d\u540c\u7684\u3002==\uff1a\u7528\u6765\u6bd4\u8f83\u4e24\u4e2a\u53d8\u91cf\u662f\u5426\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5373\u5b83\u4eec\u5728\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u662f\u5426\u76f8\u540c\u3002equals()\uff1a\u7528\u6765\u5bf9\u6bd4\u4e24\u4e2a\u5bf9\u8c61\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002
\u56e0\u6b64\uff0c\u5982\u679c\u8981\u5bf9\u6bd4\u503c\uff0c\u6211\u4eec\u5e94\u8be5\u4f7f\u7528
equals()\u3002\u7136\u800c\uff0c\u901a\u8fc7String a = \"hi\"; String b = \"hi\";\u521d\u59cb\u5316\u7684\u5b57\u7b26\u4e32\u90fd\u5b58\u50a8\u5728\u5b57\u7b26\u4e32\u5e38\u91cf\u6c60\u4e2d\uff0c\u5b83\u4eec\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u56e0\u6b64\u4e5f\u53ef\u4ee5\u7528a == b\u6765\u6bd4\u8f83\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5185\u5bb9\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u7684\u8282\u70b9\u6570\u91cf\u662f \\(2^h\\) \u5417\uff1f
\u662f\u7684\uff0c\u4f8b\u5982\u9ad8\u5ea6 \\(h = 2\\) \u7684\u6ee1\u4e8c\u53c9\u6811\uff0c\u5176\u8282\u70b9\u603b\u6570 \\(n = 7\\) \uff0c\u5219\u5e95\u5c42\u8282\u70b9\u6570\u91cf \\(4 = 2^h = (n + 1) / 2\\) \u3002
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- 0.1.1 Target Readers
+ 0.1.1 Target Audience
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Figure 1-3 Change making process
-In the aforementioned steps, at each stage, we make the optimal choice (utilizing the highest denomination possible), ultimately deriving at a feasible change-making approach. From the perspective of data structures and algorithms, this approach is essentially a "greedy" algorithm.
-From preparing a dish to traversing interstellar realms, virtually every problem-solving endeavor relies on algorithms. The emergence of computers enables us to store data structures in memory and write code to call CPUs and GPUs to execute algorithms. Consequently, we can transfer real-life predicaments to computers, efficiently addressing a myriad of complex issues.
+In the above steps, we make the best choice at each step (using the largest denomination possible), ultimately resulting in a feasible change-making plan. From the perspective of data structures and algorithms, this method is essentially a "Greedy" algorithm.
+From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers, solving various complex issues more efficiently.
Tip
If concepts such as data structures, algorithms, arrays, and binary search still seem somewhat obsecure, I encourage you to continue reading. This book will gently guide you into the realm of understanding data structures and algorithms.
diff --git a/en/chapter_introduction/what_is_dsa/index.html b/en/chapter_introduction/what_is_dsa/index.html index d79516d45..9edfc3a42 100644 --- a/en/chapter_introduction/what_is_dsa/index.html +++ b/en/chapter_introduction/what_is_dsa/index.html @@ -912,7 +912,7 @@1.2.3 Relationship Between Data Structures and Algorithms¶
-As shown in the diagram below, data structures and algorithms are highly related and closely integrated, specifically in the following three aspects:
+As shown in the Figure 1-4 , data structures and algorithms are highly related and closely integrated, specifically in the following three aspects:
- + Skip to content @@ -410,9 +410,9 @@