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Add python codes and for the chapter of
computational complexity. Update Java codes. Update Contributors.
This commit is contained in:
parent
83629f3d2c
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daf25d5e64
17 changed files with 571 additions and 82 deletions
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@ -48,7 +48,7 @@
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## To-Dos
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## To-Dos
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- [ ] [代码翻译](https://github.com/krahets/hello-algo/issues/15)(JavaScript, TypeScript, C, C#, ... 请求大佬帮助)
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- [x] [代码翻译](https://github.com/krahets/hello-algo/issues/15)(JavaScript, TypeScript, C, C#, ... 请求大佬帮助)
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- [ ] 数据结构:散列表、堆(优先队列)、图
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- [ ] 数据结构:散列表、堆(优先队列)、图
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- [ ] 算法:搜索与回溯、选择 / 堆排序、动态规划、贪心、分治
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- [ ] 算法:搜索与回溯、选择 / 堆排序、动态规划、贪心、分治
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@ -7,6 +7,7 @@ package chapter_computational_complexity
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// twoSumBruteForce
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// twoSumBruteForce
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func twoSumBruteForce(nums []int, target int) []int {
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func twoSumBruteForce(nums []int, target int) []int {
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size := len(nums)
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size := len(nums)
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// 两层循环,时间复杂度 O(n^2)
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for i := 0; i < size-1; i++ {
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for i := 0; i < size-1; i++ {
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for j := i + 1; i < size; j++ {
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for j := i + 1; i < size; j++ {
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if nums[i]+nums[j] == target {
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if nums[i]+nums[j] == target {
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@ -19,7 +20,9 @@ func twoSumBruteForce(nums []int, target int) []int {
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// twoSumHashTable
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// twoSumHashTable
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func twoSumHashTable(nums []int, target int) []int {
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func twoSumHashTable(nums []int, target int) []int {
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// 辅助哈希表,空间复杂度 O(n)
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hashTable := map[int]int{}
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hashTable := map[int]int{}
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// 单层循环,时间复杂度 O(n)
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for idx, val := range nums {
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for idx, val := range nums {
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if preIdx, ok := hashTable[target-val]; ok {
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if preIdx, ok := hashTable[target-val]; ok {
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return []int{preIdx, idx}
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return []int{preIdx, idx}
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@ -8,9 +8,10 @@ package chapter_computational_complexity;
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import java.util.*;
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import java.util.*;
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class solution_brute_force {
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class SolutionBruteForce {
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public int[] twoSum(int[] nums, int target) {
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public int[] twoSum(int[] nums, int target) {
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int size = nums.length;
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int size = nums.length;
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// 两层循环,时间复杂度 O(n^2)
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for (int i = 0; i < size - 1; i++) {
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for (int i = 0; i < size - 1; i++) {
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for (int j = i + 1; j < size; j++) {
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for (int j = i + 1; j < size; j++) {
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if (nums[i] + nums[j] == target)
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if (nums[i] + nums[j] == target)
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@ -21,10 +22,12 @@ class solution_brute_force {
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}
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}
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}
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}
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class solution_hash_map {
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class SolutionHashMap {
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public int[] twoSum(int[] nums, int target) {
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public int[] twoSum(int[] nums, int target) {
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int size = nums.length;
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int size = nums.length;
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// 辅助哈希表,空间复杂度 O(n)
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Map<Integer, Integer> dic = new HashMap<>();
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Map<Integer, Integer> dic = new HashMap<>();
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// 单层循环,时间复杂度 O(n)
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for (int i = 0; i < size; i++) {
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for (int i = 0; i < size; i++) {
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if (dic.containsKey(target - nums[i])) {
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if (dic.containsKey(target - nums[i])) {
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return new int[] { dic.get(target - nums[i]), i };
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return new int[] { dic.get(target - nums[i]), i };
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@ -43,11 +46,11 @@ public class leetcode_two_sum {
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// ====== Driver Code ======
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// ====== Driver Code ======
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// 方法一
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// 方法一
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solution_brute_force slt1 = new solution_brute_force();
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SolutionBruteForce slt1 = new SolutionBruteForce();
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int[] res = slt1.twoSum(nums, target);
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int[] res = slt1.twoSum(nums, target);
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System.out.println(Arrays.toString(res));
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System.out.println(Arrays.toString(res));
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// 方法二
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// 方法二
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solution_hash_map slt2 = new solution_hash_map();
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SolutionHashMap slt2 = new SolutionHashMap();
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res = slt2.twoSum(nums, target);
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res = slt2.twoSum(nums, target);
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System.out.println(Arrays.toString(res));
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System.out.println(Arrays.toString(res));
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}
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}
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@ -100,7 +100,7 @@ public class space_complexity_types {
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quadratic(n);
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quadratic(n);
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quadraticRecur(n);
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quadraticRecur(n);
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// 指数阶
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// 指数阶
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TreeNode tree = buildTree(n);
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TreeNode root = buildTree(n);
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PrintUtil.printTree(tree);
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PrintUtil.printTree(root);
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}
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}
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}
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}
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@ -29,7 +29,6 @@ public class time_complexity_types {
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int count = 0;
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int count = 0;
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// 循环次数与数组长度成正比
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// 循环次数与数组长度成正比
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for (int num : nums) {
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for (int num : nums) {
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// System.out.println(num);
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count++;
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count++;
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}
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}
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return count;
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return count;
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@ -38,6 +37,7 @@ public class time_complexity_types {
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/* 平方阶 */
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/* 平方阶 */
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static int quadratic(int n) {
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static int quadratic(int n) {
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int count = 0;
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int count = 0;
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// 循环次数与数组长度成平方关系
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for (int i = 0; i < n; i++) {
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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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for (int j = 0; j < n; j++) {
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count++;
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count++;
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}
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}
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/* 平方阶(冒泡排序) */
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/* 平方阶(冒泡排序) */
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static void bubbleSort(int[] nums) {
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static int bubbleSort(int[] nums) {
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int n = nums.length;
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int count = 0; // 计数器
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for (int i = 0; i < n - 1; i++) {
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// 外循环:待排序元素数量为 n-1, n-2, ..., 1
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for (int j = 0; j < n - 1 - i; j++) {
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for (int i = nums.length - 1; i > 0; i--) {
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// 内循环:冒泡操作
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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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if (nums[j] > nums[j + 1]) {
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// 交换 nums[j] 和 nums[j + 1]
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// 交换 nums[j] 与 nums[j + 1]
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int tmp = nums[j];
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int tmp = nums[j];
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nums[j] = nums[j + 1];
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nums[j] = nums[j + 1];
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nums[j + 1] = tmp;
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nums[j + 1] = tmp;
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count += 3; // 元素交换包含 3 个单元操作
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}
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}
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}
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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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}
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/* 指数阶(循环实现) */
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/* 指数阶(循环实现) */
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count = quadratic(n);
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count = quadratic(n);
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System.out.println("平方阶的计算操作数量 = " + count);
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System.out.println("平方阶的计算操作数量 = " + 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("平方阶(冒泡排序)的计算操作数量 = " + count);
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count = exponential(n);
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count = exponential(n);
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System.out.println("指数阶(循环实现)的计算操作数量 = " + count);
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System.out.println("指数阶(循环实现)的计算操作数量 = " + count);
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@ -57,6 +57,7 @@ def find(nums, target):
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return i
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return i
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return -1
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return -1
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""" Driver Code """
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""" Driver Code """
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if __name__ == "__main__":
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if __name__ == "__main__":
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""" 初始化数组 """
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""" 初始化数组 """
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index += 1
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index += 1
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return -1
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return -1
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"""
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Driver Code
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""" Driver Code """
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"""
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if __name__ == "__main__":
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if __name__ == "__main__":
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""" 初始化链表 """
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""" 初始化链表 """
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# 初始化各个结点
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# 初始化各个结点
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@ -8,9 +8,8 @@ import sys, os.path as osp
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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from include import *
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from include import *
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"""
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Driver Code
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""" Driver Code """
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"""
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if __name__ == "__main__":
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if __name__ == "__main__":
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""" 初始化列表 """
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""" 初始化列表 """
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list = [1, 3, 2, 5, 4]
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list = [1, 3, 2, 5, 4]
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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from include import *
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from include import *
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""" 列表类简易实现 """
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""" 列表类简易实现 """
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class MyList:
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class MyList:
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""" 构造函数 """
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""" 构造函数 """
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return self._nums[:self._size]
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return self._nums[:self._size]
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"""
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""" Driver Code """
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Driver Code
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"""
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if __name__ == "__main__":
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if __name__ == "__main__":
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""" 初始化列表 """
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""" 初始化列表 """
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list = MyList()
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list = MyList()
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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from include import *
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from include import *
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class SolutionBruteForce:
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def twoSum(self, nums: List[int], target: int) -> List[int]:
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for i in range(len(nums) - 1):
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for j in range(i + 1, len(nums)):
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if nums[i] + nums[j] == target:
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return i, j
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return []
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class SolutionHashMap:
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def twoSum(self, nums: List[int], target: int) -> List[int]:
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dic = {}
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for i in range(len(nums)):
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if target - nums[i] in dic:
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return dic[target - nums[i]], i
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dic[nums[i]] = i
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return []
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""" Driver Code """
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if __name__ == '__main__':
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# ======= Test Case =======
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nums = [ 2,7,11,15 ];
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target = 9;
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# ====== Driver Code ======
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# 方法一
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res = SolutionBruteForce().twoSum(nums, target);
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print(res)
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# 方法二
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res = SolutionHashMap().twoSum(nums, target);
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print(res)
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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from include import *
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from include import *
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""" 函数 """
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def function():
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# do something
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return 0
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""" 常数阶 """
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def constant(n):
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# 常量、变量、对象占用 O(1) 空间
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a = 0
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nums = [0] * 10000
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node = ListNode(0)
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# 循环中的变量占用 O(1) 空间
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for _ in range(n):
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c = 0
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# 循环中的函数占用 O(1) 空间
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for _ in range(n):
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function()
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""" 线性阶 """
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def linear(n):
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# 长度为 n 的列表占用 O(n) 空间
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nums = [0] * n
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# 长度为 n 的哈希表占用 O(n) 空间
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mapp = {}
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for i in range(n):
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mapp[i] = str(i)
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""" 线性阶(递归实现) """
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def linearRecur(n):
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print("递归 n = ", n)
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if n == 1: return
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linearRecur(n - 1)
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""" 平方阶 """
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def quadratic(n):
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# 二维列表占用 O(n^2) 空间
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num_matrix = [[0] * n for _ in range(n)]
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""" 平方阶(递归实现) """
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def quadratic_recur(n):
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if n <= 0: return 0
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nums = [0] * n
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print("递归 n = {} 中的 nums 长度 = {}".format(n, len(nums)))
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return quadratic_recur(n - 1)
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""" 指数阶(建立满二叉树) """
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def build_tree(n):
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if n == 0: return None
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root = TreeNode(0)
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root.left = build_tree(n - 1)
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root.right = build_tree(n - 1)
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return root
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""" Driver Code """
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if __name__ == "__main__":
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n = 5
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# 常数阶
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constant(n)
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# 线性阶
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linear(n)
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linearRecur(n)
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# 平方阶
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quadratic(n)
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quadratic_recur(n)
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# 指数阶
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root = build_tree(n)
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print_tree(root)
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@ -8,3 +8,133 @@ import sys, os.path as osp
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
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from include import *
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from include import *
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""" 常数阶 """
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def constant(n):
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count = 0
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size = 100000
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for _ in range(size):
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count += 1
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return count
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""" 线性阶 """
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def linear(n):
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count = 0
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for _ in range(n):
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count += 1
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return count
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""" 线性阶(遍历数组)"""
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def array_traversal(nums):
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count = 0
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# 循环次数与数组长度成正比
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for num in nums:
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count += 1
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return count
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||||||
|
""" 平方阶 """
|
||||||
|
def quadratic(n):
|
||||||
|
count = 0
|
||||||
|
# 循环次数与数组长度成平方关系
|
||||||
|
for i in range(n):
|
||||||
|
for j in range(n):
|
||||||
|
count += 1
|
||||||
|
return count
|
||||||
|
|
||||||
|
""" 平方阶(冒泡排序)"""
|
||||||
|
def bubble_sort(nums):
|
||||||
|
count = 0 # 计数器
|
||||||
|
# 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
||||||
|
for i in range(len(nums) - 1, 0, -1):
|
||||||
|
# 内循环:冒泡操作
|
||||||
|
for j in range(i):
|
||||||
|
if nums[j] > nums[j + 1]:
|
||||||
|
# 交换 nums[j] 与 nums[j + 1]
|
||||||
|
tmp = nums[j]
|
||||||
|
nums[j] = nums[j + 1]
|
||||||
|
nums[j + 1] = tmp
|
||||||
|
count += 3 # 元素交换包含 3 个单元操作
|
||||||
|
return count
|
||||||
|
|
||||||
|
""" 指数阶(循环实现)"""
|
||||||
|
def exponential(n):
|
||||||
|
count, base = 0, 1
|
||||||
|
# cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
|
||||||
|
for _ in range(n):
|
||||||
|
for _ in range(base):
|
||||||
|
count += 1
|
||||||
|
base *= 2
|
||||||
|
# count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
|
||||||
|
return count
|
||||||
|
|
||||||
|
""" 指数阶(递归实现)"""
|
||||||
|
def exp_recur(n):
|
||||||
|
if n == 1: return 1
|
||||||
|
return exp_recur(n - 1) + exp_recur(n - 1) + 1
|
||||||
|
|
||||||
|
""" 对数阶(循环实现)"""
|
||||||
|
def logarithmic(n):
|
||||||
|
count = 0
|
||||||
|
while n > 1:
|
||||||
|
n = n / 2
|
||||||
|
count += 1
|
||||||
|
return count
|
||||||
|
|
||||||
|
""" 对数阶(递归实现)"""
|
||||||
|
def log_recur(n):
|
||||||
|
if n <= 1: return 0
|
||||||
|
return log_recur(n / 2) + 1
|
||||||
|
|
||||||
|
""" 线性对数阶 """
|
||||||
|
def linear_log_recur(n):
|
||||||
|
if n <= 1: return 1
|
||||||
|
count = linear_log_recur(n // 2) + \
|
||||||
|
linear_log_recur(n // 2)
|
||||||
|
for _ in range(n):
|
||||||
|
count += 1
|
||||||
|
return count
|
||||||
|
|
||||||
|
""" 阶乘阶(递归实现)"""
|
||||||
|
def factorial_recur(n):
|
||||||
|
if n == 0: return 1
|
||||||
|
count = 0
|
||||||
|
# 从 1 个分裂出 n 个
|
||||||
|
for _ in range(n):
|
||||||
|
count += factorial_recur(n - 1)
|
||||||
|
return count
|
||||||
|
|
||||||
|
|
||||||
|
""" Driver Code """
|
||||||
|
if __name__ == "__main__":
|
||||||
|
# 可以修改 n 运行,体会一下各种复杂度的操作数量变化趋势
|
||||||
|
n = 8
|
||||||
|
print("输入数据大小 n =", n)
|
||||||
|
|
||||||
|
count = constant(n)
|
||||||
|
print("常数阶的计算操作数量 =", count)
|
||||||
|
|
||||||
|
count = linear(n)
|
||||||
|
print("线性阶的计算操作数量 =", count)
|
||||||
|
count = array_traversal([0] * n)
|
||||||
|
print("线性阶(遍历数组)的计算操作数量 =", count)
|
||||||
|
|
||||||
|
count = quadratic(n)
|
||||||
|
print("平方阶的计算操作数量 =", count)
|
||||||
|
nums = [i for i in range(n, 0, -1)] # [n,n-1,...,2,1]
|
||||||
|
count = bubble_sort(nums)
|
||||||
|
print("平方阶(冒泡排序)的计算操作数量 =", count)
|
||||||
|
|
||||||
|
count = exponential(n)
|
||||||
|
print("指数阶(循环实现)的计算操作数量 =", count)
|
||||||
|
count = exp_recur(n)
|
||||||
|
print("指数阶(递归实现)的计算操作数量 =", count)
|
||||||
|
|
||||||
|
count = logarithmic(n)
|
||||||
|
print("对数阶(循环实现)的计算操作数量 =", count)
|
||||||
|
count = log_recur(n)
|
||||||
|
print("对数阶(递归实现)的计算操作数量 =", count)
|
||||||
|
|
||||||
|
count = linear_log_recur(n)
|
||||||
|
print("线性对数阶(递归实现)的计算操作数量 =", count)
|
||||||
|
|
||||||
|
count = factorial_recur(n)
|
||||||
|
print("阶乘阶(递归实现)的计算操作数量 =", count)
|
||||||
|
|
|
@ -8,3 +8,27 @@ import sys, os.path as osp
|
||||||
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
|
sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
|
||||||
from include import *
|
from include import *
|
||||||
|
|
||||||
|
""" 生成一个数组,元素为: 1, 2, ..., n ,顺序被打乱 """
|
||||||
|
def random_numbers(n):
|
||||||
|
# 生成数组 nums =: 1, 2, 3, ..., n
|
||||||
|
nums = [i for i in range(1, n + 1)]
|
||||||
|
# 随机打乱数组元素
|
||||||
|
random.shuffle(nums)
|
||||||
|
return nums
|
||||||
|
|
||||||
|
""" 查找数组 nums 中数字 1 所在索引 """
|
||||||
|
def find_one(nums):
|
||||||
|
for i in range(len(nums)):
|
||||||
|
if nums[i] == 1:
|
||||||
|
return i
|
||||||
|
return -1
|
||||||
|
|
||||||
|
|
||||||
|
""" Driver Code """
|
||||||
|
if __name__ == "__main__":
|
||||||
|
for i in range(10):
|
||||||
|
n = 100
|
||||||
|
nums = random_numbers(n)
|
||||||
|
index = find_one(nums)
|
||||||
|
print("\n数组 [ 1, 2, ..., n ] 被打乱后 =", nums)
|
||||||
|
print("数字 1 的索引为", index)
|
||||||
|
|
|
@ -62,7 +62,23 @@ comments: true
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title=""
|
```python title=""
|
||||||
|
""" 类 """
|
||||||
|
class Node:
|
||||||
|
def __init__(self, x):
|
||||||
|
self.val = x # 结点值
|
||||||
|
self.next = None # 指向下一结点的指针(引用)
|
||||||
|
|
||||||
|
""" 函数(或称方法) """
|
||||||
|
def function():
|
||||||
|
# do something...
|
||||||
|
return 0
|
||||||
|
|
||||||
|
def algorithm(n): # 输入数据
|
||||||
|
a = 0 # 暂存数据(常量)
|
||||||
|
b = 0 # 暂存数据(变量)
|
||||||
|
node = Node(0) # 暂存数据(对象)
|
||||||
|
c = function() # 栈帧空间(调用函数)
|
||||||
|
return a + b + c # 输出数据
|
||||||
```
|
```
|
||||||
|
|
||||||
## 推算方法
|
## 推算方法
|
||||||
|
@ -94,7 +110,11 @@ comments: true
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title=""
|
```python title=""
|
||||||
|
def algorithm(n):
|
||||||
|
a = 0 # O(1)
|
||||||
|
b = [0] * 10000 # O(1)
|
||||||
|
if n > 10:
|
||||||
|
nums = [0] * n # O(n)
|
||||||
```
|
```
|
||||||
|
|
||||||
**在递归函数中,需要注意统计栈帧空间。** 例如函数 `loop()`,在循环中调用了 $n$ 次 `function()` ,每轮中的 `function()` 都返回并释放了栈帧空间,因此空间复杂度仍为 $O(1)$ 。而递归函数 `recur()` 在运行中会同时存在 $n$ 个未返回的 `recur()` ,从而使用 $O(n)$ 的栈帧空间。
|
**在递归函数中,需要注意统计栈帧空间。** 例如函数 `loop()`,在循环中调用了 $n$ 次 `function()` ,每轮中的 `function()` 都返回并释放了栈帧空间,因此空间复杂度仍为 $O(1)$ 。而递归函数 `recur()` 在运行中会同时存在 $n$ 个未返回的 `recur()` ,从而使用 $O(n)$ 的栈帧空间。
|
||||||
|
@ -106,13 +126,13 @@ comments: true
|
||||||
// do something
|
// do something
|
||||||
return 0;
|
return 0;
|
||||||
}
|
}
|
||||||
/* 循环 */
|
/* 循环 O(1) */
|
||||||
void loop(int n) {
|
void loop(int n) {
|
||||||
for (int i = 0; i < n; i++) {
|
for (int i = 0; i < n; i++) {
|
||||||
function();
|
function();
|
||||||
}
|
}
|
||||||
}
|
}
|
||||||
/* 递归 */
|
/* 递归 O(n) */
|
||||||
void recur(int n) {
|
void recur(int n) {
|
||||||
if (n == 1) return;
|
if (n == 1) return;
|
||||||
return recur(n - 1);
|
return recur(n - 1);
|
||||||
|
@ -128,7 +148,19 @@ comments: true
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title=""
|
```python title=""
|
||||||
|
def function():
|
||||||
|
# do something
|
||||||
|
return 0
|
||||||
|
|
||||||
|
""" 循环 O(1) """
|
||||||
|
def loop(n):
|
||||||
|
for _ in range(n):
|
||||||
|
function()
|
||||||
|
|
||||||
|
""" 递归 O(n) """
|
||||||
|
def recur(n):
|
||||||
|
if n == 1: return
|
||||||
|
return recur(n - 1)
|
||||||
```
|
```
|
||||||
|
|
||||||
## 常见类型
|
## 常见类型
|
||||||
|
@ -186,7 +218,18 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="space_complexity_types.py"
|
```python title="space_complexity_types.py"
|
||||||
|
""" 常数阶 """
|
||||||
|
def constant(n):
|
||||||
|
# 常量、变量、对象占用 O(1) 空间
|
||||||
|
a = 0
|
||||||
|
nums = [0] * 10000
|
||||||
|
node = ListNode(0)
|
||||||
|
# 循环中的变量占用 O(1) 空间
|
||||||
|
for _ in range(n):
|
||||||
|
c = 0
|
||||||
|
# 循环中的函数占用 O(1) 空间
|
||||||
|
for _ in range(n):
|
||||||
|
function()
|
||||||
```
|
```
|
||||||
|
|
||||||
### 线性阶 $O(n)$
|
### 线性阶 $O(n)$
|
||||||
|
@ -222,7 +265,14 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="space_complexity_types.py"
|
```python title="space_complexity_types.py"
|
||||||
|
""" 线性阶 """
|
||||||
|
def linear(n):
|
||||||
|
# 长度为 n 的列表占用 O(n) 空间
|
||||||
|
nums = [0] * n
|
||||||
|
# 长度为 n 的哈希表占用 O(n) 空间
|
||||||
|
mapp = {}
|
||||||
|
for i in range(n):
|
||||||
|
mapp[i] = str(i)
|
||||||
```
|
```
|
||||||
|
|
||||||
以下递归函数会同时存在 $n$ 个未返回的 `algorithm()` 函数,使用 $O(n)$ 大小的栈帧空间。
|
以下递归函数会同时存在 $n$ 个未返回的 `algorithm()` 函数,使用 $O(n)$ 大小的栈帧空间。
|
||||||
|
@ -247,7 +297,11 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="space_complexity_types.py"
|
```python title="space_complexity_types.py"
|
||||||
|
""" 线性阶(递归实现) """
|
||||||
|
def linearRecur(n):
|
||||||
|
print("递归 n = ", n)
|
||||||
|
if n == 1: return
|
||||||
|
linearRecur(n - 1)
|
||||||
```
|
```
|
||||||
|
|
||||||
![space_complexity_recursive_linear](space_complexity.assets/space_complexity_recursive_linear.png)
|
![space_complexity_recursive_linear](space_complexity.assets/space_complexity_recursive_linear.png)
|
||||||
|
@ -286,7 +340,10 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="space_complexity_types.py"
|
```python title="space_complexity_types.py"
|
||||||
|
""" 平方阶 """
|
||||||
|
def quadratic(n):
|
||||||
|
# 二维列表占用 O(n^2) 空间
|
||||||
|
num_matrix = [[0] * n for _ in range(n)]
|
||||||
```
|
```
|
||||||
|
|
||||||
在以下递归函数中,同时存在 $n$ 个未返回的 `algorihtm()` ,并且每个函数中都初始化了一个数组,长度分别为 $n, n-1, n-2, ..., 2, 1$ ,平均长度为 $\frac{n}{2}$ ,因此总体使用 $O(n^2)$ 空间。
|
在以下递归函数中,同时存在 $n$ 个未返回的 `algorihtm()` ,并且每个函数中都初始化了一个数组,长度分别为 $n, n-1, n-2, ..., 2, 1$ ,平均长度为 $\frac{n}{2}$ ,因此总体使用 $O(n^2)$ 空间。
|
||||||
|
@ -297,8 +354,8 @@ $$
|
||||||
/* 平方阶(递归实现) */
|
/* 平方阶(递归实现) */
|
||||||
int quadraticRecur(int n) {
|
int quadraticRecur(int n) {
|
||||||
if (n <= 0) return 0;
|
if (n <= 0) return 0;
|
||||||
|
// 数组 nums 长度为 n, n-1, ..., 2, 1
|
||||||
int[] nums = new int[n];
|
int[] nums = new int[n];
|
||||||
System.out.println("递归 n = " + n + " 中的 nums 长度 = " + nums.length);
|
|
||||||
return quadraticRecur(n - 1);
|
return quadraticRecur(n - 1);
|
||||||
}
|
}
|
||||||
```
|
```
|
||||||
|
@ -312,7 +369,12 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="space_complexity_types.py"
|
```python title="space_complexity_types.py"
|
||||||
|
""" 平方阶(递归实现) """
|
||||||
|
def quadratic_recur(n):
|
||||||
|
if n <= 0: return 0
|
||||||
|
# 数组 nums 长度为 n, n-1, ..., 2, 1
|
||||||
|
nums = [0] * n
|
||||||
|
return quadratic_recur(n - 1)
|
||||||
```
|
```
|
||||||
|
|
||||||
![space_complexity_recursive_quadratic](space_complexity.assets/space_complexity_recursive_quadratic.png)
|
![space_complexity_recursive_quadratic](space_complexity.assets/space_complexity_recursive_quadratic.png)
|
||||||
|
@ -345,7 +407,13 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="space_complexity_types.py"
|
```python title="space_complexity_types.py"
|
||||||
|
""" 指数阶(建立满二叉树) """
|
||||||
|
def build_tree(n):
|
||||||
|
if n == 0: return None
|
||||||
|
root = TreeNode(0)
|
||||||
|
root.left = build_tree(n - 1)
|
||||||
|
root.right = build_tree(n - 1)
|
||||||
|
return root
|
||||||
```
|
```
|
||||||
|
|
||||||
![space_complexity_exponential](space_complexity.assets/space_complexity_exponential.png)
|
![space_complexity_exponential](space_complexity.assets/space_complexity_exponential.png)
|
||||||
|
|
|
@ -17,16 +17,18 @@
|
||||||
=== "Java"
|
=== "Java"
|
||||||
|
|
||||||
```java title="" title="leetcode_two_sum.java"
|
```java title="" title="leetcode_two_sum.java"
|
||||||
public int[] twoSum(int[] nums, int target) {
|
class SolutionBruteForce {
|
||||||
int size = nums.length;
|
public int[] twoSum(int[] nums, int target) {
|
||||||
// 外层 * 内层循环,时间复杂度为 O(n)
|
int size = nums.length;
|
||||||
for (int i = 0; i < size - 1; i++) {
|
// 两层循环,时间复杂度 O(n^2)
|
||||||
for (int j = i + 1; j < size; j++) {
|
for (int i = 0; i < size - 1; i++) {
|
||||||
if (nums[i] + nums[j] == target)
|
for (int j = i + 1; j < size; j++) {
|
||||||
return new int[] { i, j };
|
if (nums[i] + nums[j] == target)
|
||||||
|
return new int[] { i, j };
|
||||||
|
}
|
||||||
}
|
}
|
||||||
|
return new int[0];
|
||||||
}
|
}
|
||||||
return new int[0];
|
|
||||||
}
|
}
|
||||||
```
|
```
|
||||||
|
|
||||||
|
@ -39,14 +41,22 @@
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="leetcode_two_sum.py"
|
```python title="leetcode_two_sum.py"
|
||||||
|
class SolutionBruteForce:
|
||||||
|
def twoSum(self, nums: List[int], target: int) -> List[int]:
|
||||||
|
# 两层循环,时间复杂度 O(n^2)
|
||||||
|
for i in range(len(nums) - 1):
|
||||||
|
for j in range(i + 1, len(nums)):
|
||||||
|
if nums[i] + nums[j] == target:
|
||||||
|
return i, j
|
||||||
|
return []
|
||||||
```
|
```
|
||||||
|
|
||||||
=== "Go"
|
=== "Go"
|
||||||
|
|
||||||
```go title="leetcode_two_sum.go"
|
```go title="leetcode_two_sum.go"
|
||||||
func twoSum(nums []int, target int) []int {
|
func twoSumBruteForce(nums []int, target int) []int {
|
||||||
size := len(nums)
|
size := len(nums)
|
||||||
|
// 两层循环,时间复杂度 O(n^2)
|
||||||
for i := 0; i < size-1; i++ {
|
for i := 0; i < size-1; i++ {
|
||||||
for j := i + 1; i < size; j++ {
|
for j := i + 1; i < size; j++ {
|
||||||
if nums[i]+nums[j] == target {
|
if nums[i]+nums[j] == target {
|
||||||
|
@ -58,8 +68,6 @@
|
||||||
}
|
}
|
||||||
```
|
```
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
### 方法二:辅助哈希表
|
### 方法二:辅助哈希表
|
||||||
|
|
||||||
时间复杂度 $O(N)$ ,空间复杂度 $O(N)$ ,属于「空间换时间」。
|
时间复杂度 $O(N)$ ,空间复杂度 $O(N)$ ,属于「空间换时间」。
|
||||||
|
@ -69,18 +77,20 @@
|
||||||
=== "Java"
|
=== "Java"
|
||||||
|
|
||||||
```java title="" title="leetcode_two_sum.java"
|
```java title="" title="leetcode_two_sum.java"
|
||||||
public int[] twoSum(int[] nums, int target) {
|
class SolutionHashMap {
|
||||||
int size = nums.length;
|
public int[] twoSum(int[] nums, int target) {
|
||||||
// 辅助哈希表,空间复杂度 O(n)
|
int size = nums.length;
|
||||||
Map<Integer, Integer> dic = new HashMap<>();
|
// 辅助哈希表,空间复杂度 O(n)
|
||||||
// 单层循环,时间复杂度 O(n)
|
Map<Integer, Integer> dic = new HashMap<>();
|
||||||
for (int i = 0; i < size; i++) {
|
// 单层循环,时间复杂度 O(n)
|
||||||
if (dic.containsKey(target - nums[i])) {
|
for (int i = 0; i < size; i++) {
|
||||||
return new int[] { dic.get(target - nums[i]), i };
|
if (dic.containsKey(target - nums[i])) {
|
||||||
|
return new int[] { dic.get(target - nums[i]), i };
|
||||||
|
}
|
||||||
|
dic.put(nums[i], i);
|
||||||
}
|
}
|
||||||
dic.put(nums[i], i);
|
return new int[0];
|
||||||
}
|
}
|
||||||
return new int[0];
|
|
||||||
}
|
}
|
||||||
```
|
```
|
||||||
|
|
||||||
|
@ -93,14 +103,25 @@
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="leetcode_two_sum.py"
|
```python title="leetcode_two_sum.py"
|
||||||
|
class SolutionHashMap:
|
||||||
|
def twoSum(self, nums: List[int], target: int) -> List[int]:
|
||||||
|
# 辅助哈希表,空间复杂度 O(n)
|
||||||
|
dic = {}
|
||||||
|
# 单层循环,时间复杂度 O(n)
|
||||||
|
for i in range(len(nums)):
|
||||||
|
if target - nums[i] in dic:
|
||||||
|
return dic[target - nums[i]], i
|
||||||
|
dic[nums[i]] = i
|
||||||
|
return []
|
||||||
```
|
```
|
||||||
|
|
||||||
=== "Go"
|
=== "Go"
|
||||||
|
|
||||||
```go title="leetcode_two_sum.go"
|
```go title="leetcode_two_sum.go"
|
||||||
func twoSumHashTable(nums []int, target int) []int {
|
func twoSumHashTable(nums []int, target int) []int {
|
||||||
|
// 辅助哈希表,空间复杂度 O(n)
|
||||||
hashTable := map[int]int{}
|
hashTable := map[int]int{}
|
||||||
|
// 单层循环,时间复杂度 O(n)
|
||||||
for idx, val := range nums {
|
for idx, val := range nums {
|
||||||
if preIdx, ok := hashTable[target-val]; ok {
|
if preIdx, ok := hashTable[target-val]; ok {
|
||||||
return []int{preIdx, idx}
|
return []int{preIdx, idx}
|
||||||
|
|
|
@ -42,7 +42,14 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title=""
|
```python title=""
|
||||||
|
# 在某运行平台下
|
||||||
|
def algorithm(n):
|
||||||
|
a = 2 # 1 ns
|
||||||
|
a = a + 1 # 1 ns
|
||||||
|
a = a * 2 # 10 ns
|
||||||
|
# 循环 n 次
|
||||||
|
for _ in range(n): # 1 ns
|
||||||
|
print(0) # 5 ns
|
||||||
```
|
```
|
||||||
|
|
||||||
但实际上, **统计算法的运行时间既不合理也不现实。** 首先,我们不希望预估时间和运行平台绑定,毕竟算法需要跑在各式各样的平台之上。其次,我们很难获知每一种操作的运行时间,这为预估过程带来了极大的难度。
|
但实际上, **统计算法的运行时间既不合理也不现实。** 首先,我们不希望预估时间和运行平台绑定,毕竟算法需要跑在各式各样的平台之上。其次,我们很难获知每一种操作的运行时间,这为预估过程带来了极大的难度。
|
||||||
|
@ -87,7 +94,17 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title=""
|
```python title=""
|
||||||
|
# 算法 A 时间复杂度:常数阶
|
||||||
|
def algorithm_A(n):
|
||||||
|
print(0)
|
||||||
|
# 算法 B 时间复杂度:线性阶
|
||||||
|
def algorithm_B(n):
|
||||||
|
for _ in range(n):
|
||||||
|
print(0)
|
||||||
|
# 算法 C 时间复杂度:常数阶
|
||||||
|
def algorithm_C(n):
|
||||||
|
for _ in range(1000000):
|
||||||
|
print(0)
|
||||||
```
|
```
|
||||||
|
|
||||||
![time_complexity_first_example](time_complexity.assets/time_complexity_first_example.png)
|
![time_complexity_first_example](time_complexity.assets/time_complexity_first_example.png)
|
||||||
|
@ -105,9 +122,11 @@ $$
|
||||||
## 函数渐进上界
|
## 函数渐进上界
|
||||||
|
|
||||||
设算法「计算操作数量」为 $T(n)$ ,其是一个关于输入数据大小 $n$ 的函数。例如,以下算法的操作数量为
|
设算法「计算操作数量」为 $T(n)$ ,其是一个关于输入数据大小 $n$ 的函数。例如,以下算法的操作数量为
|
||||||
|
|
||||||
$$
|
$$
|
||||||
T(n) = 3 + 2n
|
T(n) = 3 + 2n
|
||||||
$$
|
$$
|
||||||
|
|
||||||
=== "Java"
|
=== "Java"
|
||||||
|
|
||||||
```java title=""
|
```java title=""
|
||||||
|
@ -131,14 +150,21 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title=""
|
```python title=""
|
||||||
|
def algorithm(n):
|
||||||
|
a = 1 # +1
|
||||||
|
a = a + 1 # +1
|
||||||
|
a = a * 2 # +1
|
||||||
|
# 循环 n 次
|
||||||
|
for i in range(n): # +1
|
||||||
|
print(0) # +1
|
||||||
|
}
|
||||||
```
|
```
|
||||||
|
|
||||||
$T(n)$ 是个一次函数,说明时间增长趋势是线性的,因此易得时间复杂度是线性阶。
|
$T(n)$ 是个一次函数,说明时间增长趋势是线性的,因此易得时间复杂度是线性阶。
|
||||||
|
|
||||||
我们将线性阶的时间复杂度记为 $O(n)$ ,这个数学符号被称为「大 $O$ 记号 Big-$O$ Notation」,代表函数 $T(n)$ 的「渐进上界 asymptotic upper bound」。
|
我们将线性阶的时间复杂度记为 $O(n)$ ,这个数学符号被称为「大 $O$ 记号 Big-$O$ Notation」,代表函数 $T(n)$ 的「渐进上界 asymptotic upper bound」。
|
||||||
|
|
||||||
我们要推算时间复杂度,本质上是在计算「操作数量函数 $T(n)$ 」的渐进上界。下面我们先来看看函数渐进上界的数学定义。
|
我们要推算时间复杂度,本质上是在计算「操作数量函数 $T(n)$ 」的渐进上界。下面我们先来看看函数渐进上界的数学定义。
|
||||||
|
|
||||||
!!! abstract "函数渐进上界"
|
!!! abstract "函数渐进上界"
|
||||||
|
|
||||||
|
@ -174,6 +200,7 @@ $T(n)$ 是个一次函数,说明时间增长趋势是线性的,因此易得
|
||||||
3. **循环嵌套时使用乘法。** 总操作数量等于外层循环和内层循环操作数量之积,每一层循环依然可以分别套用上述 `1.` 和 `2.` 技巧。
|
3. **循环嵌套时使用乘法。** 总操作数量等于外层循环和内层循环操作数量之积,每一层循环依然可以分别套用上述 `1.` 和 `2.` 技巧。
|
||||||
|
|
||||||
根据以下示例,使用上述技巧前、后的统计结果分别为
|
根据以下示例,使用上述技巧前、后的统计结果分别为
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\begin{aligned}
|
\begin{aligned}
|
||||||
T(n) & = 2n(n + 1) + (5n + 1) + 2 & \text{完整统计 (-.-|||)} \newline
|
T(n) & = 2n(n + 1) + (5n + 1) + 2 & \text{完整统计 (-.-|||)} \newline
|
||||||
|
@ -181,6 +208,7 @@ T(n) & = 2n(n + 1) + (5n + 1) + 2 & \text{完整统计 (-.-|||)} \newline
|
||||||
T(n) & = n^2 + n & \text{偷懒统计 (o.O)}
|
T(n) & = n^2 + n & \text{偷懒统计 (o.O)}
|
||||||
\end{aligned}
|
\end{aligned}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
最终,两者都能推出相同的时间复杂度结果,即 $O(n^2)$ 。
|
最终,两者都能推出相同的时间复杂度结果,即 $O(n^2)$ 。
|
||||||
|
|
||||||
=== "Java"
|
=== "Java"
|
||||||
|
@ -211,7 +239,16 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title=""
|
```python title=""
|
||||||
|
def algorithm(n):
|
||||||
|
a = 1 # +0(技巧 1)
|
||||||
|
a = a + n # +0(技巧 1)
|
||||||
|
# +n(技巧 2)
|
||||||
|
for i in range(5 * n + 1):
|
||||||
|
print(0)
|
||||||
|
# +n*n(技巧 3)
|
||||||
|
for i in range(2 * n):
|
||||||
|
for j in range(n + 1):
|
||||||
|
print(0)
|
||||||
```
|
```
|
||||||
|
|
||||||
### 2. 判断渐进上界
|
### 2. 判断渐进上界
|
||||||
|
@ -279,7 +316,13 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 常数阶 """
|
||||||
|
def constant(n):
|
||||||
|
count = 0
|
||||||
|
size = 100000
|
||||||
|
for _ in range(size):
|
||||||
|
count += 1
|
||||||
|
return count
|
||||||
```
|
```
|
||||||
|
|
||||||
### 线性阶 $O(n)$
|
### 线性阶 $O(n)$
|
||||||
|
@ -307,7 +350,12 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 线性阶 """
|
||||||
|
def linear(n):
|
||||||
|
count = 0
|
||||||
|
for _ in range(n):
|
||||||
|
count += 1
|
||||||
|
return count
|
||||||
```
|
```
|
||||||
|
|
||||||
「遍历数组」和「遍历链表」等操作,时间复杂度都为 $O(n)$ ,其中 $n$ 为数组或链表的长度。
|
「遍历数组」和「遍历链表」等操作,时间复杂度都为 $O(n)$ ,其中 $n$ 为数组或链表的长度。
|
||||||
|
@ -339,7 +387,13 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 线性阶(遍历数组)"""
|
||||||
|
def array_traversal(nums):
|
||||||
|
count = 0
|
||||||
|
# 循环次数与数组长度成正比
|
||||||
|
for num in nums:
|
||||||
|
count += 1
|
||||||
|
return count
|
||||||
```
|
```
|
||||||
|
|
||||||
### 平方阶 $O(n^2)$
|
### 平方阶 $O(n^2)$
|
||||||
|
@ -352,6 +406,7 @@ $$
|
||||||
/* 平方阶 */
|
/* 平方阶 */
|
||||||
int quadratic(int n) {
|
int quadratic(int n) {
|
||||||
int count = 0;
|
int count = 0;
|
||||||
|
// 循环次数与数组长度成平方关系
|
||||||
for (int i = 0; i < n; i++) {
|
for (int i = 0; i < n; i++) {
|
||||||
for (int j = 0; j < n; j++) {
|
for (int j = 0; j < n; j++) {
|
||||||
count++;
|
count++;
|
||||||
|
@ -370,7 +425,14 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 平方阶 """
|
||||||
|
def quadratic(n):
|
||||||
|
count = 0
|
||||||
|
# 循环次数与数组长度成平方关系
|
||||||
|
for i in range(n):
|
||||||
|
for j in range(n):
|
||||||
|
count += 1
|
||||||
|
return count
|
||||||
```
|
```
|
||||||
|
|
||||||
![time_complexity_constant_linear_quadratic](time_complexity.assets/time_complexity_constant_linear_quadratic.png)
|
![time_complexity_constant_linear_quadratic](time_complexity.assets/time_complexity_constant_linear_quadratic.png)
|
||||||
|
@ -387,18 +449,22 @@ $$
|
||||||
|
|
||||||
```java title="" title="time_complexity_types.java"
|
```java title="" title="time_complexity_types.java"
|
||||||
/* 平方阶(冒泡排序) */
|
/* 平方阶(冒泡排序) */
|
||||||
void bubbleSort(int[] nums) {
|
int bubbleSort(int[] nums) {
|
||||||
int n = nums.length;
|
int count = 0; // 计数器
|
||||||
for (int i = 0; i < n - 1; i++) {
|
// 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
||||||
for (int j = 0; j < n - 1 - i; j++) {
|
for (int i = nums.length - 1; i > 0; i--) {
|
||||||
|
// 内循环:冒泡操作
|
||||||
|
for (int j = 0; j < i; j++) {
|
||||||
if (nums[j] > nums[j + 1]) {
|
if (nums[j] > nums[j + 1]) {
|
||||||
// 交换 nums[j] 和 nums[j + 1]
|
// 交换 nums[j] 与 nums[j + 1]
|
||||||
int tmp = nums[j];
|
int tmp = nums[j];
|
||||||
nums[j] = nums[j + 1];
|
nums[j] = nums[j + 1];
|
||||||
nums[j + 1] = tmp;
|
nums[j + 1] = tmp;
|
||||||
|
count += 3; // 元素交换包含 3 个单元操作
|
||||||
}
|
}
|
||||||
}
|
}
|
||||||
}
|
}
|
||||||
|
return count;
|
||||||
}
|
}
|
||||||
```
|
```
|
||||||
|
|
||||||
|
@ -411,7 +477,20 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 平方阶(冒泡排序)"""
|
||||||
|
def bubble_sort(nums):
|
||||||
|
count = 0 # 计数器
|
||||||
|
# 外循环:待排序元素数量为 n-1, n-2, ..., 1
|
||||||
|
for i in range(len(nums) - 1, 0, -1):
|
||||||
|
# 内循环:冒泡操作
|
||||||
|
for j in range(i):
|
||||||
|
if nums[j] > nums[j + 1]:
|
||||||
|
# 交换 nums[j] 与 nums[j + 1]
|
||||||
|
tmp = nums[j]
|
||||||
|
nums[j] = nums[j + 1]
|
||||||
|
nums[j + 1] = tmp
|
||||||
|
count += 3 # 元素交换包含 3 个单元操作
|
||||||
|
return count
|
||||||
```
|
```
|
||||||
|
|
||||||
### 指数阶 $O(2^n)$
|
### 指数阶 $O(2^n)$
|
||||||
|
@ -425,7 +504,7 @@ $$
|
||||||
=== "Java"
|
=== "Java"
|
||||||
|
|
||||||
```java title="" title="time_complexity_types.java"
|
```java title="" title="time_complexity_types.java"
|
||||||
/* 指数阶(遍历实现) */
|
/* 指数阶(循环实现) */
|
||||||
int exponential(int n) {
|
int exponential(int n) {
|
||||||
int count = 0, base = 1;
|
int count = 0, base = 1;
|
||||||
// cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
|
// cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
|
||||||
|
@ -449,7 +528,16 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 指数阶(循环实现)"""
|
||||||
|
def exponential(n):
|
||||||
|
count, base = 0, 1
|
||||||
|
# cell 每轮一分为二,形成数列 1, 2, 4, 8, ..., 2^(n-1)
|
||||||
|
for _ in range(n):
|
||||||
|
for _ in range(base):
|
||||||
|
count += 1
|
||||||
|
base *= 2
|
||||||
|
# count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1
|
||||||
|
return count
|
||||||
```
|
```
|
||||||
|
|
||||||
![time_complexity_exponential](time_complexity.assets/time_complexity_exponential.png)
|
![time_complexity_exponential](time_complexity.assets/time_complexity_exponential.png)
|
||||||
|
@ -477,7 +565,10 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 指数阶(递归实现)"""
|
||||||
|
def exp_recur(n):
|
||||||
|
if n == 1: return 1
|
||||||
|
return exp_recur(n - 1) + exp_recur(n - 1) + 1
|
||||||
```
|
```
|
||||||
|
|
||||||
### 对数阶 $O(\log n)$
|
### 对数阶 $O(\log n)$
|
||||||
|
@ -511,7 +602,13 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 对数阶(循环实现)"""
|
||||||
|
def logarithmic(n):
|
||||||
|
count = 0
|
||||||
|
while n > 1:
|
||||||
|
n = n / 2
|
||||||
|
count += 1
|
||||||
|
return count
|
||||||
```
|
```
|
||||||
|
|
||||||
![time_complexity_logarithmic](time_complexity.assets/time_complexity_logarithmic.png)
|
![time_complexity_logarithmic](time_complexity.assets/time_complexity_logarithmic.png)
|
||||||
|
@ -539,7 +636,10 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 对数阶(递归实现)"""
|
||||||
|
def log_recur(n):
|
||||||
|
if n <= 1: return 0
|
||||||
|
return log_recur(n / 2) + 1
|
||||||
```
|
```
|
||||||
|
|
||||||
### 线性对数阶 $O(n \log n)$
|
### 线性对数阶 $O(n \log n)$
|
||||||
|
@ -572,7 +672,14 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 线性对数阶 """
|
||||||
|
def linear_log_recur(n):
|
||||||
|
if n <= 1: return 1
|
||||||
|
count = linear_log_recur(n // 2) + \
|
||||||
|
linear_log_recur(n // 2)
|
||||||
|
for _ in range(n):
|
||||||
|
count += 1
|
||||||
|
return count
|
||||||
```
|
```
|
||||||
|
|
||||||
![time_complexity_logarithmic_linear](time_complexity.assets/time_complexity_logarithmic_linear.png)
|
![time_complexity_logarithmic_linear](time_complexity.assets/time_complexity_logarithmic_linear.png)
|
||||||
|
@ -613,7 +720,14 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="time_complexity_types.py"
|
```python title="time_complexity_types.py"
|
||||||
|
""" 阶乘阶(递归实现)"""
|
||||||
|
def factorial_recur(n):
|
||||||
|
if n == 0: return 1
|
||||||
|
count = 0
|
||||||
|
# 从 1 个分裂出 n 个
|
||||||
|
for _ in range(n):
|
||||||
|
count += factorial_recur(n - 1)
|
||||||
|
return count
|
||||||
```
|
```
|
||||||
|
|
||||||
![time_complexity_factorial](time_complexity.assets/time_complexity_factorial.png)
|
![time_complexity_factorial](time_complexity.assets/time_complexity_factorial.png)
|
||||||
|
@ -681,7 +795,29 @@ $$
|
||||||
=== "Python"
|
=== "Python"
|
||||||
|
|
||||||
```python title="worst_best_time_complexity.py"
|
```python title="worst_best_time_complexity.py"
|
||||||
|
""" 生成一个数组,元素为: 1, 2, ..., n ,顺序被打乱 """
|
||||||
|
def random_numbers(n):
|
||||||
|
# 生成数组 nums =: 1, 2, 3, ..., n
|
||||||
|
nums = [i for i in range(1, n + 1)]
|
||||||
|
# 随机打乱数组元素
|
||||||
|
random.shuffle(nums)
|
||||||
|
return nums
|
||||||
|
|
||||||
|
""" 查找数组 nums 中数字 1 所在索引 """
|
||||||
|
def find_one(nums):
|
||||||
|
for i in range(len(nums)):
|
||||||
|
if nums[i] == 1:
|
||||||
|
return i
|
||||||
|
return -1
|
||||||
|
|
||||||
|
""" Driver Code """
|
||||||
|
if __name__ == "__main__":
|
||||||
|
for i in range(10):
|
||||||
|
n = 100
|
||||||
|
nums = random_numbers(n)
|
||||||
|
index = find_one(nums)
|
||||||
|
print("\n数组 [ 1, 2, ..., n ] 被打乱后 =", nums)
|
||||||
|
print("数字 1 的索引为", index)
|
||||||
```
|
```
|
||||||
|
|
||||||
!!! tip
|
!!! tip
|
||||||
|
|
|
@ -111,7 +111,7 @@ comments: true
|
||||||
|
|
||||||
## 致谢
|
## 致谢
|
||||||
|
|
||||||
感谢本开源书的每一位撰稿人,是他们的无私奉献让这本书变得更好,他们的 GitHub ID(按首次提交时间排序)为:krahets, *(等待下一位创作者)*
|
感谢本开源书的每一位撰稿人,是他们的无私奉献让这本书变得更好,他们的 GitHub ID(按首次提交时间排序)为:krahets, Reanon.
|
||||||
|
|
||||||
本书的成书过程中,我获得了许多人的帮助,包括但不限于:
|
本书的成书过程中,我获得了许多人的帮助,包括但不限于:
|
||||||
|
|
||||||
|
|
Loading…
Reference in a new issue