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https://github.com/krahets/hello-algo.git
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Retranslate the complete English edition from Simplified Chinese (#2010)
* Retranslate the complete English edition from Simplified Chinese * Polish three English explanations after reader-focused review
This commit is contained in:
Yudong Jin
@@ -7,7 +7,7 @@
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package chapter_computational_complexity;
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public class complexity_exercises {
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/* Iterative summation */
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/* Sum using iteration */
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static int sumIter(int n) {
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int res = 0;
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for (int i = 1; i <= n; i++) {
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@@ -16,7 +16,7 @@ public class complexity_exercises {
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return res;
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}
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/* Recursive summation */
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/* Sum using recursion */
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static int sumRecur(int n) {
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if (n == 1) {
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return 1;
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@@ -24,7 +24,7 @@ public class complexity_exercises {
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return n + sumRecur(n - 1);
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}
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/* Linear loop */
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/* Linear-time loop */
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static int linearLoop(int n) {
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int res = 0;
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for (int i = 0; i < n; i++) {
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@@ -33,7 +33,7 @@ public class complexity_exercises {
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return res;
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}
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/* Quadratic loop */
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/* Quadratic-time loop */
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static int quadraticLoop(int n) {
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int res = 0;
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for (int i = 0; i < n; i++) {
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@@ -44,7 +44,7 @@ public class complexity_exercises {
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return res;
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}
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/* Logarithmic loop */
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/* Logarithmic-time loop */
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static int logarithmicLoop(int n) {
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while (n > 1) {
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n /= 2;
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@@ -10,7 +10,7 @@ public class iteration {
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/* for loop */
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static int forLoop(int n) {
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int res = 0;
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// Sum 1, 2, ..., n-1, n
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// Sum 1, 2, ..., n-1, n in a loop
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for (int i = 1; i <= n; i++) {
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res += i;
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}
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@@ -20,11 +20,11 @@ public class iteration {
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/* while loop */
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static int whileLoop(int n) {
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int res = 0;
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int i = 1; // Initialize condition variable
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// Sum 1, 2, ..., n-1, n
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int i = 1; // Initialize the loop control variable
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// Sum 1, 2, ..., n-1, n in a loop
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while (i <= n) {
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res += i;
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i++; // Update condition variable
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i++; // Update the loop control variable
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}
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return res;
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}
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@@ -32,23 +32,23 @@ public class iteration {
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/* while loop (two updates) */
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static int whileLoopII(int n) {
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int res = 0;
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int i = 1; // Initialize condition variable
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// Sum 1, 4, 10, ...
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int i = 1; // Initialize the loop control variable
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// Sum 1, 4, 10, ... in a loop
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while (i <= n) {
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res += i;
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// Update condition variable
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// Update the loop control variable
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i++;
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i *= 2;
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}
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return res;
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}
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/* Nested for loop */
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/* Nested for loops */
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static String nestedForLoop(int n) {
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StringBuilder res = new StringBuilder();
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// Loop i = 1, 2, ..., n-1, n
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// Loop over i = 1, 2, ..., n-1, n
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for (int i = 1; i <= n; i++) {
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// Loop j = 1, 2, ..., n-1, n
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// Loop over j = 1, 2, ..., n-1, n
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for (int j = 1; j <= n; j++) {
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res.append("(" + i + ", " + j + "), ");
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}
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@@ -56,21 +56,21 @@ public class iteration {
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return res.toString();
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}
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/* Driver Code */
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/* Driver code */
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public static void main(String[] args) {
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int n = 5;
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int res;
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res = forLoop(n);
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System.out.println("\nfor loop sum result res = " + res);
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System.out.println("\nSum from the for loop: res = " + res);
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res = whileLoop(n);
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System.out.println("\nwhile loop sum result res = " + res);
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System.out.println("\nSum from the while loop: res = " + res);
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res = whileLoopII(n);
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System.out.println("\nwhile loop (two updates) sum result res = " + res);
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System.out.println("\nSum from the while loop (two updates): res = " + res);
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String resStr = nestedForLoop(n);
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System.out.println("\nDouble for loop traversal result " + resStr);
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System.out.println("\nTraversal result from the nested for loops: " + resStr);
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}
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}
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@@ -11,12 +11,12 @@ import java.util.Stack;
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public class recursion {
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/* Recursion */
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static int recur(int n) {
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// Termination condition
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// Base case
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if (n == 1)
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return 1;
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// Recurse: recursive call
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// Recurse: make a recursive call
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int res = recur(n - 1);
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// Return: return result
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// Unwind: return the result
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return n + res;
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}
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@@ -25,14 +25,14 @@ public class recursion {
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// Use an explicit stack to simulate the system call stack
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Stack<Integer> stack = new Stack<>();
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int res = 0;
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// Recurse: recursive call
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// Recurse: make a recursive call
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for (int i = n; i > 0; i--) {
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// Simulate "recurse" with "push"
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// Simulate recursive calls by pushing onto the stack
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stack.push(i);
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}
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// Return: return result
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// Unwind: return the result
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while (!stack.isEmpty()) {
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// Simulate "return" with "pop"
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// Simulate unwinding by popping from the stack
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res += stack.pop();
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}
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// res = 1+2+3+...+n
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@@ -41,39 +41,39 @@ public class recursion {
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/* Tail recursion */
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static int tailRecur(int n, int res) {
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// Termination condition
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// Base case
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if (n == 0)
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return res;
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// Tail recursive call
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// Tail-recursive call
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return tailRecur(n - 1, res + n);
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}
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/* Fibonacci sequence: recursion */
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static int fib(int n) {
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// Termination condition f(1) = 0, f(2) = 1
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// Base cases: f(1) = 0, f(2) = 1
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if (n == 1 || n == 2)
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return n - 1;
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// Recursive call f(n) = f(n-1) + f(n-2)
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// Recursive calls: f(n) = f(n-1) + f(n-2)
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int res = fib(n - 1) + fib(n - 2);
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// Return result f(n)
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// Return the result f(n)
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return res;
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}
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/* Driver Code */
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/* Driver code */
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public static void main(String[] args) {
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int n = 5;
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int res;
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res = recur(n);
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System.out.println("\nRecursive function sum result res = " + res);
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System.out.println("\nSum from recursion: res = " + res);
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res = forLoopRecur(n);
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System.out.println("\nUsing iteration to simulate recursive sum result res = " + res);
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System.out.println("\nSum from simulating recursion with iteration: res = " + res);
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res = tailRecur(n, 0);
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System.out.println("\nTail recursive function sum result res = " + res);
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System.out.println("\nSum from tail recursion: res = " + res);
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res = fib(n);
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System.out.println("\nThe " + n + "th term of the Fibonacci sequence is " + res);
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System.out.println("\nFibonacci term " + n + " is " + res);
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}
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}
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@@ -16,9 +16,9 @@ public class space_complexity {
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return 0;
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}
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/* Constant order */
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/* Constant complexity */
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static void constant(int n) {
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// Constants, variables, objects occupy O(1) space
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// Constants, variables, and objects occupy O(1) space
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final int a = 0;
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int b = 0;
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int[] nums = new int[10000];
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@@ -27,41 +27,41 @@ public class space_complexity {
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for (int i = 0; i < n; i++) {
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int c = 0;
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}
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// Functions in the loop occupy O(1) space
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// Function calls in the loop occupy O(1) space
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for (int i = 0; i < n; i++) {
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function();
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}
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}
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/* Linear order */
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/* Linear complexity */
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static void linear(int n) {
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// Array of length n uses O(n) space
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// An array of length n uses O(n) space
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int[] nums = new int[n];
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// A list of length n occupies O(n) space
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List<ListNode> nodes = new ArrayList<>();
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for (int i = 0; i < n; i++) {
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nodes.add(new ListNode(i));
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}
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// A hash table of length n occupies O(n) space
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// A hash table with n entries occupies O(n) space
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Map<Integer, String> map = new HashMap<>();
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for (int i = 0; i < n; i++) {
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map.put(i, String.valueOf(i));
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}
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}
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/* Linear order (recursive implementation) */
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/* Linear complexity (recursive implementation) */
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static void linearRecur(int n) {
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System.out.println("Recursion n = " + n);
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System.out.println("Recursive call: n = " + n);
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if (n == 1)
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return;
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linearRecur(n - 1);
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}
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/* Exponential order */
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/* Quadratic complexity */
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static void quadratic(int n) {
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// Matrix uses O(n^2) space
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// The matrix uses O(n^2) space
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int[][] numMatrix = new int[n][n];
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// 2D list uses O(n^2) space
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// The two-dimensional list occupies O(n^2) space
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List<List<Integer>> numList = new ArrayList<>();
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for (int i = 0; i < n; i++) {
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List<Integer> tmp = new ArrayList<>();
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@@ -72,17 +72,17 @@ public class space_complexity {
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}
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}
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/* Quadratic order (recursive implementation) */
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/* Quadratic complexity (recursive implementation) */
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static int quadraticRecur(int n) {
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if (n <= 0)
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return 0;
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// Array nums has length n, n-1, ..., 2, 1
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// The lengths of array nums are n, n-1, ..., 2, 1
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int[] nums = new int[n];
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System.out.println("In recursion n = " + n + ", nums length = " + nums.length);
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System.out.println("Recursive call: n = " + n + " with nums length = " + nums.length);
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return quadraticRecur(n - 1);
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}
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/* Driver Code */
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/* Exponential complexity (building a perfect binary tree) */
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static TreeNode buildTree(int n) {
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if (n == 0)
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return null;
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@@ -92,18 +92,18 @@ public class space_complexity {
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return root;
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}
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/* Driver Code */
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/* Driver code */
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public static void main(String[] args) {
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int n = 5;
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// Constant order
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// Constant complexity
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constant(n);
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// Linear order
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// Linear complexity
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linear(n);
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linearRecur(n);
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// Exponential order
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// Quadratic complexity
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quadratic(n);
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quadraticRecur(n);
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// Exponential order
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// Exponential complexity
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TreeNode root = buildTree(n);
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PrintUtil.printTree(root);
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}
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@@ -7,7 +7,7 @@
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package chapter_computational_complexity;
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public class time_complexity {
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/* Constant order */
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/* Constant complexity */
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static int constant(int n) {
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int count = 0;
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int size = 100000;
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@@ -16,7 +16,7 @@ public class time_complexity {
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return count;
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}
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/* Linear order */
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/* Linear complexity */
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static int linear(int n) {
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int count = 0;
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for (int i = 0; i < n; i++)
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@@ -24,20 +24,20 @@ public class time_complexity {
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return count;
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}
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/* Linear order (traversing array) */
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/* Linear complexity (array traversal) */
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static int arrayTraversal(int[] nums) {
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int count = 0;
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// Number of iterations is proportional to the array length
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// The number of iterations is proportional to the array length
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for (int num : nums) {
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count++;
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}
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return count;
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}
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/* Exponential order */
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/* Quadratic complexity */
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static int quadratic(int n) {
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int count = 0;
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// Number of iterations is quadratically related to the data size n
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// The number of iterations is proportional to the square of the input size n
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for (int i = 0; i < n; i++) {
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for (int j = 0; j < n; j++) {
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count++;
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@@ -46,29 +46,29 @@ public class time_complexity {
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return count;
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}
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/* Quadratic order (bubble sort) */
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/* Quadratic complexity (bubble sort) */
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static int bubbleSort(int[] nums) {
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int count = 0; // Counter
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// Outer loop: unsorted range is [0, i]
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// Outer loop: the unsorted range is [0, i]
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for (int i = nums.length - 1; i > 0; i--) {
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// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range
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// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps
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for (int j = 0; j < i; j++) {
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if (nums[j] > nums[j + 1]) {
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// Swap nums[j] and nums[j + 1]
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int tmp = nums[j];
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nums[j] = nums[j + 1];
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nums[j + 1] = tmp;
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count += 3; // Element swap includes 3 unit operations
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count += 3; // Swapping two elements takes 3 elementary operations
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}
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}
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}
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return count;
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}
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/* Exponential order (loop implementation) */
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/* Exponential complexity (iterative implementation) */
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static int exponential(int n) {
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int count = 0, base = 1;
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// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)
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// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)
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for (int i = 0; i < n; i++) {
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for (int j = 0; j < base; j++) {
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count++;
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@@ -79,14 +79,14 @@ public class time_complexity {
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return count;
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}
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/* Exponential order (recursive implementation) */
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/* Exponential complexity (recursive implementation) */
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static int expRecur(int n) {
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if (n == 1)
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return 1;
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return expRecur(n - 1) + expRecur(n - 1) + 1;
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}
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/* Logarithmic order (loop implementation) */
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/* Logarithmic complexity (iterative implementation) */
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static int logarithmic(int n) {
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int count = 0;
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while (n > 1) {
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@@ -96,14 +96,14 @@ public class time_complexity {
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return count;
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}
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/* Logarithmic order (recursive implementation) */
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/* Logarithmic complexity (recursive implementation) */
|
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static int logRecur(int n) {
|
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if (n <= 1)
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return 0;
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return logRecur(n / 2) + 1;
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}
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/* Linearithmic order */
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/* Linearithmic complexity */
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static int linearLogRecur(int n) {
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if (n <= 1)
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return 1;
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@@ -114,54 +114,54 @@ public class time_complexity {
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return count;
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}
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/* Factorial order (recursive implementation) */
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/* Factorial complexity (recursive implementation) */
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static int factorialRecur(int n) {
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if (n == 0)
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return 1;
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int count = 0;
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// Split from 1 into n
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// Split one problem into n subproblems
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for (int i = 0; i < n; i++) {
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count += factorialRecur(n - 1);
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}
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return count;
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}
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/* Driver Code */
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/* Driver code */
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public static void main(String[] args) {
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// You can modify n to run and observe the trend of the number of operations for various complexities
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// Try different values of n to see how the operation count grows for each complexity
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int n = 8;
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System.out.println("Input data size n = " + n);
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System.out.println("Input size n = " + n);
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int count = constant(n);
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System.out.println("Constant order operation count = " + count);
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System.out.println("Operation count for constant complexity = " + count);
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count = linear(n);
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System.out.println("Linear order operation count = " + count);
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System.out.println("Operation count for linear complexity = " + count);
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count = arrayTraversal(new int[n]);
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System.out.println("Linear order (array traversal) operation count = " + count);
|
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System.out.println("Operation count for linear complexity (array traversal) = " + count);
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count = quadratic(n);
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System.out.println("Quadratic order operation count = " + count);
|
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System.out.println("Operation count for quadratic complexity = " + count);
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int[] nums = new int[n];
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for (int i = 0; i < n; i++)
|
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nums[i] = n - i; // [n,n-1,...,2,1]
|
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count = bubbleSort(nums);
|
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System.out.println("Quadratic order (bubble sort) operation count = " + count);
|
||||
System.out.println("Operation count for quadratic complexity (bubble sort) = " + count);
|
||||
|
||||
count = exponential(n);
|
||||
System.out.println("Exponential order (loop implementation) operation count = " + count);
|
||||
System.out.println("Operation count for exponential complexity (iterative implementation) = " + count);
|
||||
count = expRecur(n);
|
||||
System.out.println("Exponential order (recursive implementation) operation count = " + count);
|
||||
System.out.println("Operation count for exponential complexity (recursive implementation) = " + count);
|
||||
|
||||
count = logarithmic(n);
|
||||
System.out.println("Logarithmic order (loop implementation) operation count = " + count);
|
||||
System.out.println("Operation count for logarithmic complexity (iterative implementation) = " + count);
|
||||
count = logRecur(n);
|
||||
System.out.println("Logarithmic order (recursive implementation) operation count = " + count);
|
||||
System.out.println("Operation count for logarithmic complexity (recursive implementation) = " + count);
|
||||
|
||||
count = linearLogRecur(n);
|
||||
System.out.println("Linearithmic order (recursive implementation) operation count = " + count);
|
||||
System.out.println("Operation count for linearithmic complexity (recursive implementation) = " + count);
|
||||
|
||||
count = factorialRecur(n);
|
||||
System.out.println("Factorial order (recursive implementation) operation count = " + count);
|
||||
System.out.println("Operation count for factorial complexity (recursive implementation) = " + count);
|
||||
}
|
||||
}
|
||||
|
||||
@@ -9,14 +9,14 @@ package chapter_computational_complexity;
|
||||
import java.util.*;
|
||||
|
||||
public class worst_best_time_complexity {
|
||||
/* Generate an array with elements { 1, 2, ..., n }, order shuffled */
|
||||
/* Generate an array containing { 1, 2, ..., n } in shuffled order */
|
||||
static int[] randomNumbers(int n) {
|
||||
Integer[] nums = new Integer[n];
|
||||
// Generate array nums = { 1, 2, 3, ..., n }
|
||||
// Generate the array nums = { 1, 2, 3, ..., n }
|
||||
for (int i = 0; i < n; i++) {
|
||||
nums[i] = i + 1;
|
||||
}
|
||||
// Randomly shuffle array elements
|
||||
// Randomly shuffle the array elements
|
||||
Collections.shuffle(Arrays.asList(nums));
|
||||
// Integer[] -> int[]
|
||||
int[] res = new int[n];
|
||||
@@ -26,25 +26,25 @@ public class worst_best_time_complexity {
|
||||
return res;
|
||||
}
|
||||
|
||||
/* Find the index of number 1 in array nums */
|
||||
/* Find the index of the number 1 in array nums */
|
||||
static int findOne(int[] nums) {
|
||||
for (int i = 0; i < nums.length; i++) {
|
||||
// When element 1 is at the head of the array, best time complexity O(1) is achieved
|
||||
// When element 1 is at the tail of the array, worst time complexity O(n) is achieved
|
||||
// When 1 is at the start of the array, the best-case time complexity is O(1)
|
||||
// When 1 is at the end of the array, the worst-case time complexity is O(n)
|
||||
if (nums[i] == 1)
|
||||
return i;
|
||||
}
|
||||
return -1;
|
||||
}
|
||||
|
||||
/* Driver Code */
|
||||
/* Driver code */
|
||||
public static void main(String[] args) {
|
||||
for (int i = 0; i < 10; i++) {
|
||||
int n = 100;
|
||||
int[] nums = randomNumbers(n);
|
||||
int index = findOne(nums);
|
||||
System.out.println("\nArray [ 1, 2, ..., n ] after shuffling = " + Arrays.toString(nums));
|
||||
System.out.println("Index of number 1 is " + index);
|
||||
System.out.println("The index of the number 1 is " + index);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
Reference in New Issue
Block a user