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en/docs/chapter_computational_complexity/index.md
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# Chapter 2.   Complexity Analysis
<div class="center-table" markdown>
![complexity_analysis](../assets/covers/chapter_complexity_analysis.jpg){ class="cover-image" }
</div>
!!! abstract
Complexity analysis is like a space-time navigator in the vast universe of algorithms.

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en/docs/chapter_computational_complexity/space_complexity.md
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<p align="center"> Figure 2-16 &nbsp; Common Types of Space Complexity </p>
### 1. &nbsp; Constant Order $O(1)$ {data-toc-label="Constant Order"}
### 1. &nbsp; Constant Order $O(1)$ {data-toc-label="1. &nbsp; Constant Order"}
Constant order is common in constants, variables, objects that are independent of the size of input data $n$.
@@ -1124,7 +1124,7 @@ Note that memory occupied by initializing variables or calling functions in a lo
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### 2. &nbsp; Linear Order $O(n)$ {data-toc-label="Linear Order"}
### 2. &nbsp; Linear Order $O(n)$ {data-toc-label="2. &nbsp; Linear Order"}
Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to $n$:
@@ -1589,7 +1589,7 @@ As shown below, this function's recursive depth is $n$, meaning there are $n$ in
<p align="center"> Figure 2-17 &nbsp; Recursive Function Generating Linear Order Space Complexity </p>
### 3. &nbsp; Quadratic Order $O(n^2)$ {data-toc-label="Quadratic Order"}
### 3. &nbsp; Quadratic Order $O(n^2)$ {data-toc-label="3. &nbsp; Quadratic Order"}
Quadratic order is common in matrices and graphs, where the number of elements is quadratic to $n$:
@@ -2025,7 +2025,7 @@ As shown below, the recursive depth of this function is $n$, and in each recursi
<p align="center"> Figure 2-18 &nbsp; Recursive Function Generating Quadratic Order Space Complexity </p>
### 4. &nbsp; Exponential Order $O(2^n)$ {data-toc-label="Exponential Order"}
### 4. &nbsp; Exponential Order $O(2^n)$ {data-toc-label="4. &nbsp; Exponential Order"}
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:
@@ -2225,7 +2225,7 @@ Exponential order is common in binary trees. Observe the below image, a "full bi
<p align="center"> Figure 2-19 &nbsp; Full Binary Tree Generating Exponential Order Space Complexity </p>
### 5. &nbsp; Logarithmic Order $O(\log n)$ {data-toc-label="Logarithmic Order"}
### 5. &nbsp; Logarithmic Order $O(\log n)$ {data-toc-label="5. &nbsp; Logarithmic Order"}
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.

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en/docs/chapter_computational_complexity/time_complexity.md
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<p align="center"> Figure 2-9 &nbsp; Common Types of Time Complexity </p>
### 1. &nbsp; Constant Order $O(1)$ {data-toc-label="Constant Order"}
### 1. &nbsp; Constant Order $O(1)$ {data-toc-label="1. &nbsp; Constant Order"}
Constant order means the number of operations is independent of the input data size $n$. In the following function, although the number of operations `size` might be large, the time complexity remains $O(1)$ as it's unrelated to $n$:
@@ -1167,7 +1167,7 @@ Constant order means the number of operations is independent of the input data s
<div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20constant%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E5%B8%B8%E6%95%B0%E9%98%B6%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20size%20%3D%2010%0A%20%20%20%20for%20_%20in%20range%28size%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22%E8%BE%93%E5%85%A5%E6%95%B0%E6%8D%AE%E5%A4%A7%E5%B0%8F%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20constant%28n%29%0A%20%20%20%20print%28%22%E5%B8%B8%E6%95%B0%E9%98%B6%E7%9A%84%E6%93%8D%E4%BD%9C%E6%95%B0%E9%87%8F%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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### 2. &nbsp; Linear Order $O(n)$ {data-toc-label="Linear Order"}
### 2. &nbsp; Linear Order $O(n)$ {data-toc-label="2. &nbsp; Linear Order"}
Linear order indicates the number of operations grows linearly with the input data size $n$. Linear order commonly appears in single-loop structures:
@@ -1538,7 +1538,7 @@ Operations like array traversal and linked list traversal have a time complexity
It'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.
### 3. &nbsp; Quadratic Order $O(n^2)$ {data-toc-label="Quadratic Order"}
### 3. &nbsp; Quadratic Order $O(n^2)$ {data-toc-label="3. &nbsp; Quadratic Order"}
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 $O(n^2)$:
@@ -2073,7 +2073,7 @@ For instance, in bubble sort, the outer loop runs $n - 1$ times, and the inner l
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### 4. &nbsp; Exponential Order $O(2^n)$ {data-toc-label="Exponential Order"}
### 4. &nbsp; Exponential Order $O(2^n)$ {data-toc-label="4. &nbsp; Exponential Order"}
Biological "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.
@@ -2491,7 +2491,7 @@ In practice, exponential order often appears in recursive functions. For example
Exponential 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.
### 5. &nbsp; Logarithmic Order $O(\log n)$ {data-toc-label="Logarithmic Order"}
### 5. &nbsp; Logarithmic Order $O(\log n)$ {data-toc-label="5. &nbsp; Logarithmic Order"}
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 $2^n$.
@@ -2861,7 +2861,7 @@ Logarithmic order is typical in algorithms based on the divide-and-conquer strat
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 $O(\log n)$.
### 6. &nbsp; Linear-Logarithmic Order $O(n \log n)$ {data-toc-label="Linear-Logarithmic Order"}
### 6. &nbsp; Linear-Logarithmic Order $O(n \log n)$ {data-toc-label="6. &nbsp; Linear-Logarithmic Order"}
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:
@@ -3076,7 +3076,7 @@ The image below demonstrates how linear-logarithmic order is generated. Each lev
Mainstream sorting algorithms typically have a time complexity of $O(n \log n)$, such as quicksort, mergesort, and heapsort.
### 7. &nbsp; Factorial Order $O(n!)$ {data-toc-label="Factorial Order"}
### 7. &nbsp; Factorial Order $O(n!)$ {data-toc-label="7. &nbsp; Factorial Order"}
Factorial order corresponds to the mathematical problem of "full permutation." Given $n$ distinct elements, the total number of possible permutations is: