379 lines
8.2 KiB
Markdown
379 lines
8.2 KiB
Markdown
# 数据结构与算法
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## 1. 链表
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> **场景**：操作历史记录（浏览器前进后退）、LRU 缓存淘汰、撤销重做功能、音乐播放列表。
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> **解决**：频繁插入/删除场景下数组性能差的问题，O(1) 插入删除。
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### 反转链表
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```js
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function reverseList(head) {
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let prev = null, curr = head;
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while (curr) {
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const next = curr.next;
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curr.next = prev;
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prev = curr;
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curr = next;
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}
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return prev;
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}
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```
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### 环形链表判断
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```js
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function hasCycle(head) {
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let slow = head, fast = head;
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while (fast?.next) {
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slow = slow.next;
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fast = fast.next.next;
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if (slow === fast) return true;
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}
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return false;
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}
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```
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### 合并有序链表
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```js
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function mergeTwoLists(l1, l2) {
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const dummy = { next: null };
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let curr = dummy;
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while (l1 && l2) {
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if (l1.val <= l2.val) {
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curr.next = l1;
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l1 = l1.next;
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} else {
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curr.next = l2;
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l2 = l2.next;
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}
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curr = curr.next;
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}
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curr.next = l1 || l2;
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return dummy.next;
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}
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```
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---
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## 2. 二叉树
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> **场景**：DOM 树遍历、文件目录结构、组织架构图、表达式解析、前端路由匹配。
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> **解决**：层级数据的存储与高效查找，遍历、搜索与路径问题。
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### 遍历（前中后序）
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```js
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// 前序：根-左-右
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const preorder = (root, res = []) => {
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if (!root) return res;
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res.push(root.val);
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preorder(root.left, res);
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preorder(root.right, res);
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return res;
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};
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// 中序：左-根-右
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const inorder = (root, res = []) => {
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if (!root) return res;
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inorder(root.left, res);
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res.push(root.val);
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inorder(root.right, res);
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return res;
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};
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// 后序：左-右-根
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const postorder = (root, res = []) => {
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if (!root) return res;
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postorder(root.left, res);
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postorder(root.right, res);
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res.push(root.val);
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return res;
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};
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```
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### 求最大深度
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```js
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const maxDepth = root => {
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if (!root) return 0;
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return 1 + Math.max(maxDepth(root.left), maxDepth(root.right));
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};
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```
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### 路径和
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```js
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function hasPathSum(root, targetSum) {
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if (!root) return false;
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if (!root.left && !root.right) return targetSum === root.val;
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return hasPathSum(root.left, targetSum - root.val)
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|| hasPathSum(root.right, targetSum - root.val);
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}
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```
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---
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## 3. 栈与队列
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> **场景**：栈用于括号匹配、撤销操作、函数调用栈；队列用于任务调度、消息队列、BFS 广度优先搜索。
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> **解决**：先进后出/先进先出的数据操作顺序控制。
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### 用栈实现队列
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```js
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class MyQueue {
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constructor() {
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this.inStack = [];
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this.outStack = [];
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}
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push(x) { this.inStack.push(x); }
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pop() {
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if (!this.outStack.length) {
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while (this.inStack.length) this.outStack.push(this.inStack.pop());
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}
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return this.outStack.pop();
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}
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peek() {
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if (!this.outStack.length) {
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while (this.inStack.length) this.outStack.push(this.inStack.pop());
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}
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return this.outStack[this.outStack.length - 1];
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}
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empty() { return !this.inStack.length && !this.outStack.length; }
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}
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```
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### 有效的括号
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```js
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function isValid(s) {
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const map = { ')': '(', ']': '[', '}': '{' };
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const stack = [];
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for (const c of s) {
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if (map[c]) {
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if (stack.pop() !== map[c]) return false;
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} else {
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stack.push(c);
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}
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}
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return !stack.length;
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}
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```
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---
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## 4. 哈希表
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> **场景**：快速查找（用户ID查询）、统计词频、数据去重、分组操作。
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> **解决**：O(1) 时间复杂度的键值对存储和查找，避免线性遍历。
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### 两数之和
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```js
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function twoSum(nums, target) {
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const map = new Map();
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for (let i = 0; i < nums.length; i++) {
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const diff = target - nums[i];
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if (map.has(diff)) return [map.get(diff), i];
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map.set(nums[i], i);
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}
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return [];
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}
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```
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### 字母异位词分组
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```js
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function groupAnagrams(strs) {
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const map = new Map();
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for (const s of strs) {
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const key = [...s].sort().join('');
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map.set(key, (map.get(key) || []).concat(s));
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}
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return [...map.values()];
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}
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```
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---
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## 5. 排序算法
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> **场景**：商品价格/销量排序、排行榜、数据可视化预处理。
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> **解决**：无序数据有序化，理解分治思想（快排、归并）是算法基础。
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### 快速排序
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```js
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function quickSort(arr) {
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if (arr.length <= 1) return arr;
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const pivot = arr[Math.floor(arr.length / 2)];
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const left = arr.filter(x => x < pivot);
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const middle = arr.filter(x => x === pivot);
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const right = arr.filter(x => x > pivot);
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return [...quickSort(left), ...middle, ...quickSort(right)];
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}
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```
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### 归并排序
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```js
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function mergeSort(arr) {
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if (arr.length <= 1) return arr;
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const mid = Math.floor(arr.length / 2);
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const left = mergeSort(arr.slice(0, mid));
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const right = mergeSort(arr.slice(mid));
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return merge(left, right);
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}
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function merge(left, right) {
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const result = [];
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let i = 0, j = 0;
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while (i < left.length && j < right.length) {
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result.push(left[i] < right[j] ? left[i++] : right[j++]);
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}
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return result.concat(left.slice(i), right.slice(j));
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}
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```
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---
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## 6. 二分查找
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> **场景**：有序列表快速定位（分页跳转）、版本号查找、IP 地址归属地查询、猜数字游戏。
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> **解决**：在有序数据中 O(log n) 时间复杂度快速查找，比线性查找效率高很多。
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### 基础二分查找
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```js
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function binarySearch(arr, target) {
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let left = 0, right = arr.length - 1;
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while (left <= right) {
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const mid = Math.floor((left + right) / 2);
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if (arr[mid] === target) return mid;
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arr[mid] < target ? (left = mid + 1) : (right = mid - 1);
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}
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return -1;
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}
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```
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### 旋转数组查找
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```js
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function searchRotated(nums, target) {
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let left = 0, right = nums.length - 1;
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while (left <= right) {
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const mid = Math.floor((left + right) / 2);
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if (nums[mid] === target) return mid;
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// 左半边有序
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if (nums[left] <= nums[mid]) {
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if (target >= nums[left] && target < nums[mid]) right = mid - 1;
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else left = mid + 1;
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} else {
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// 右半边有序
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if (target > nums[mid] && target <= nums[right]) left = mid + 1;
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else right = mid - 1;
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}
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}
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return -1;
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}
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```
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---
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## 7. 斐波那契数列
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> **场景**：算法入门经典题、理解递归与动态规划思想、缓存优化演示、爬楼梯问题变种。
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> **解决**：求第 n 位斐波那契数（F(0)=0, F(1)=1, F(n)=F(n-1)+F(n-2)）。
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### 方法一：普通递归（不推荐）
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```js
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// 时间复杂度 O(2^n)，存在大量重复计算
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function fib(n) {
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if (n <= 1) return n;
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return fib(n - 1) + fib(n - 2);
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}
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```
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### 方法二：记忆化递归
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```js
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// 时间复杂度 O(n)，空间复杂度 O(n)
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function fib(n, memo = {}) {
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if (n <= 1) return n;
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if (memo[n]) return memo[n];
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return memo[n] = fib(n - 1, memo) + fib(n - 2, memo);
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}
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```
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### 方法三：动态规划
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```js
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// 时间复杂度 O(n)，空间复杂度 O(n)
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function fib(n) {
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if (n <= 1) return n;
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const dp = [0, 1];
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for (let i = 2; i <= n; i++) {
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dp[i] = dp[i - 1] + dp[i - 2];
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}
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return dp[n];
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}
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```
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### 方法四：空间优化（推荐）
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```js
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// 时间复杂度 O(n)，空间复杂度 O(1)
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function fib(n) {
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if (n <= 1) return n;
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let prev = 0, curr = 1;
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for (let i = 2; i <= n; i++) {
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[prev, curr] = [curr, prev + curr];
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}
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return curr;
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}
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```
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### 方法五：矩阵快速幂（大数优化）
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```js
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// 时间复杂度 O(log n)，适合求极大位数
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function fib(n) {
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if (n <= 1) return n;
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const multiply = (a, b) => [
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[a[0][0] * b[0][0] + a[0][1] * b[1][0], a[0][0] * b[0][1] + a[0][1] * b[1][1]],
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[a[1][0] * b[0][0] + a[1][1] * b[1][0], a[1][0] * b[0][1] + a[1][1] * b[1][1]]
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];
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const power = (m, p) => {
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let result = [[1, 0], [0, 1]]; // 单位矩阵
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while (p > 0) {
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if (p & 1) result = multiply(result, m);
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m = multiply(m, m);
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p >>= 1;
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}
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return result;
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};
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const matrix = [[1, 1], [1, 0]];
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return power(matrix, n - 1)[0][0];
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}
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```
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| 方法 | 时间复杂度 | 空间复杂度 | 适用场景 |
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|------|-----------|-----------|----------|
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| 普通递归 | O(2^n) | O(n) | 仅理解思想 |
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| 记忆化递归 | O(n) | O(n) | 面试常考 |
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| 动态规划 | O(n) | O(n) | 面试常考 |
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| 空间优化 | O(n) | O(1) | **推荐** |
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| 矩阵快速幂 | O(log n) | O(1) | 大数/竞赛 |
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