From 65a83844e2f7e83db102d724c69b3ed34a1dc421 Mon Sep 17 00:00:00 2001 From: CyC2018 Date: Wed, 28 Oct 2026 22:51:04 +0800 Subject: [PATCH] =?UTF-8?q?Update=20Leetcode=20=E9=A2=98=E8=A7=A3=20-=20?= =?UTF-8?q?=E5=8A=A8=E6=80=81=E8=A7=84=E5=88=92.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 代码经力扣判题验证;413 示例中的子区间数、518 的链接与题名按力扣当前题面更新,母牛生产的题面按原题调整;重新绘制配图,难度标注按力扣更新,粗体小标题改为四级标题,示例代码块的语言标注改为 text。 --- notes/Leetcode 题解 - 动态规划.md | 104 ++++++++++++---------------- notes/pics/lc-dp-climb-stairs.png | Bin 0 -> 68759 bytes notes/pics/lc-dp-cow.png | Bin 0 -> 73595 bytes notes/pics/lc-dp-derangement.png | Bin 0 -> 77480 bytes notes/pics/lc-dp-house-robber.png | Bin 0 -> 82529 bytes notes/pics/lc-dp-knapsack-1d.png | Bin 0 -> 148116 bytes notes/pics/lc-dp-knapsack.png | Bin 0 -> 121790 bytes notes/pics/lc-dp-lcs.png | Bin 0 -> 135151 bytes notes/pics/lc-dp-lis.png | Bin 0 -> 107311 bytes notes/pics/lc-dp-stock-cooldown.png | Bin 0 -> 151441 bytes notes/pics/lc-dp-stock-fee.png | Bin 0 -> 159097 bytes notes/pics/lc-dp-unique-paths.png | Bin 0 -> 84425 bytes 12 files changed, 44 insertions(+), 60 deletions(-) create mode 100644 notes/pics/lc-dp-climb-stairs.png create mode 100644 notes/pics/lc-dp-cow.png create mode 100644 notes/pics/lc-dp-derangement.png create mode 100644 notes/pics/lc-dp-house-robber.png create mode 100644 notes/pics/lc-dp-knapsack-1d.png create mode 100644 notes/pics/lc-dp-knapsack.png create mode 100644 notes/pics/lc-dp-lcs.png create mode 100644 notes/pics/lc-dp-lis.png create mode 100644 notes/pics/lc-dp-stock-cooldown.png create mode 100644 notes/pics/lc-dp-stock-fee.png create mode 100644 notes/pics/lc-dp-unique-paths.png diff --git a/notes/Leetcode 题解 - 动态规划.md b/notes/Leetcode 题解 - 动态规划.md index a26d35ef..60c18dba 100644 --- a/notes/Leetcode 题解 - 动态规划.md +++ b/notes/Leetcode 题解 - 动态规划.md @@ -61,9 +61,7 @@ 第 i 个楼梯可以从第 i-1 和 i-2 个楼梯再走一步到达,走到第 i 个楼梯的方法数为走到第 i-1 和第 i-2 个楼梯的方法数之和。 - - -

+

考虑到 dp[i] 只与 dp[i - 1] 和 dp[i - 2] 有关,因此可以只用两个变量来存储 dp[i - 1] 和 dp[i - 2],使得原来的 O(N) 空间复杂度优化为 O(1) 复杂度。 @@ -84,7 +82,7 @@ public int climbStairs(int n) { ### 2. 强盗抢劫 -198\. House Robber (Easy) +198\. House Robber (Medium) [Leetcode](https://leetcode.com/problems/house-robber/description/) / [力扣](https://leetcode-cn.com/problems/house-robber/description/) @@ -94,9 +92,7 @@ public int climbStairs(int n) { 由于不能抢劫邻近住户,如果抢劫了第 i -1 个住户,那么就不能再抢劫第 i 个住户,所以 - - -

+

```java public int rob(int[] nums) { @@ -150,21 +146,17 @@ private int rob(int[] nums, int first, int last) { 综上所述,错误装信数量方式数量为: - - -

+

### 5. 母牛生产 程序员代码面试指南-P181 -题目描述:假设农场中成熟的母牛每年都会生 1 头小母牛,并且永远不会死。第一年有 1 只小母牛,从第二年开始,母牛开始生小母牛。每只小母牛 3 年之后成熟又可以生小母牛。给定整数 N,求 N 年后牛的数量。 +题目描述:假设农场中成熟的母牛每年都会生 1 头小母牛,并且永远不会死。第一年农场中有 1 只成熟的母牛,从第二年开始,母牛开始生小母牛。每只小母牛 3 年之后成熟又可以生小母牛。给定整数 N,求 N 年后牛的数量。 第 i 年牛的数量等于第 i-1 年牛的数量加上第 i-3 年牛的数量,这是因为第 i-3 年的牛到第 i 年都已成熟,每头都会生一头小牛: - - -

+

## 矩阵路径 @@ -174,7 +166,7 @@ private int rob(int[] nums, int first, int last) { [Leetcode](https://leetcode.com/problems/minimum-path-sum/description/) / [力扣](https://leetcode-cn.com/problems/minimum-path-sum/description/) -```html +```text [[1,3,1], [1,5,1], [4,2,1]] @@ -214,7 +206,7 @@ public int minPathSum(int[][] grid) { 题目描述:统计从矩阵左上角到右下角的路径总数,每次只能向右或者向下移动。 -

+

```java public int uniquePaths(int m, int n) { @@ -285,7 +277,7 @@ public int maximalSquare(char[][] matrix) { [Leetcode](https://leetcode.com/problems/range-sum-query-immutable/description/) / [力扣](https://leetcode-cn.com/problems/range-sum-query-immutable/description/) -```html +```text Given nums = [-2, 0, 3, -5, 2, -1] sumRange(0, 2) -> 1 @@ -319,16 +311,16 @@ class NumArray { [Leetcode](https://leetcode.com/problems/arithmetic-slices/description/) / [力扣](https://leetcode-cn.com/problems/arithmetic-slices/description/) -```html +```text A = [0, 1, 2, 3, 4] -return: 6, for 3 arithmetic slices in A: +return: 6, for 6 arithmetic slices in A: [0, 1, 2], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4], -[ 1, 2, 3, 4], +[1, 2, 3, 4], [2, 3, 4] ``` @@ -336,7 +328,7 @@ dp[i] 表示以 A[i] 为结尾的等差递增子区间的个数。 当 A[i] - A[i-1] == A[i-1] - A[i-2],那么 [A[i-2], A[i-1], A[i]] 构成一个等差递增子区间。而且在以 A[i-1] 为结尾的递增子区间的后面再加上一个 A[i],一样可以构成新的递增子区间。 -```html +```text dp[2] = 1 [0, 1, 2] dp[3] = dp[2] + 1 = 2 @@ -412,7 +404,7 @@ public int longestValidParentheses(String s) { ### 1. 分割整数的最大乘积 -343\. Integer Break (Medim) +343\. Integer Break (Medium) [Leetcode](https://leetcode.com/problems/integer-break/description/) / [力扣](https://leetcode-cn.com/problems/integer-break/description/) @@ -433,7 +425,7 @@ public int integerBreak(int n) { ### 2. 按平方数来分割整数 -279\. Perfect Squares(Medium) +279\. Perfect Squares (Medium) [Leetcode](https://leetcode.com/problems/perfect-squares/description/) / [力扣](https://leetcode-cn.com/problems/perfect-squares/description/) @@ -513,9 +505,7 @@ public int numDecodings(String s) { 因为在求 dp[n] 时可能无法找到一个满足条件的递增子序列,此时 {Sn} 就构成了递增子序列,需要对前面的求解方程做修改,令 dp[n] 最小为 1,即: - - -

+

对于一个长度为 N 的序列,最长递增子序列并不一定会以 SN 为结尾,因此 dp[N] 不是序列的最长递增子序列的长度,需要遍历 dp 数组找出最大值才是所要的结果,max{ dp[i] | 1 <= i <= N} 即为所求。 @@ -561,7 +551,7 @@ return ret; 例如对于数组 [4,3,6,5],有: -```html +```text tails len num [] 0 4 [4] 1 3 @@ -609,7 +599,7 @@ private int binarySearch(int[] tails, int len, int key) { [Leetcode](https://leetcode.com/problems/maximum-length-of-pair-chain/description/) / [力扣](https://leetcode-cn.com/problems/maximum-length-of-pair-chain/description/) -```html +```text Input: [[1,2], [2,3], [3,4]] Output: 2 Explanation: The longest chain is [1,2] -> [3,4] @@ -643,7 +633,7 @@ public int findLongestChain(int[][] pairs) { [Leetcode](https://leetcode.com/problems/wiggle-subsequence/description/) / [力扣](https://leetcode-cn.com/problems/wiggle-subsequence/description/) -```html +```text Input: [1,7,4,9,2,5] Output: 6 The entire sequence is a wiggle sequence. @@ -686,9 +676,7 @@ public int wiggleMaxLength(int[] nums) { 综上,最长公共子序列的状态转移方程为: - - -

+

对于长度为 N 的序列 S1 和长度为 M 的序列 S2,dp[N][M] 就是序列 S1 和序列 S2 的最长公共子序列长度。 @@ -732,9 +720,7 @@ public int wiggleMaxLength(int[] nums) { 第 i 件物品可添加也可以不添加,取决于哪种情况下最大价值更大。因此,0-1 背包的状态转移方程为: - - -

+

```java // W 为背包总体积 @@ -757,13 +743,11 @@ public int knapsack(int W, int N, int[] weights, int[] values) { } ``` -**空间优化** +#### 空间优化 在程序实现时可以对 0-1 背包做优化。观察状态转移方程可以知道,前 i 件物品的状态仅与前 i-1 件物品的状态有关,因此可以将 dp 定义为一维数组,其中 dp[j] 既可以表示 dp[i-1][j] 也可以表示 dp[i][j]。此时, - - -

+

因为 dp[j-w] 表示 dp[i-1][j-w],因此不能先求 dp[i][j-w],防止将 dp[i-1][j-w] 覆盖。也就是说要先计算 dp[i][j] 再计算 dp[i][j-w],在程序实现时需要按倒序来循环求解。 @@ -782,9 +766,9 @@ public int knapsack(int W, int N, int[] weights, int[] values) { } ``` -**无法使用贪心算法的解释** +#### 无法使用贪心算法的解释 -0-1 背包问题无法使用贪心算法来求解,也就是说不能按照先添加性价比最高的物品来达到最优,这是因为这种方式可能造成背包空间的浪费,从而无法达到最优。考虑下面的物品和一个容量为 5 的背包,如果先添加物品 0 再添加物品 1,那么只能存放的价值为 16,浪费了大小为 2 的空间。最优的方式是存放物品 1 和物品 2,价值为 22. +0-1 背包问题无法使用贪心算法来求解,也就是说不能按照先添加性价比最高的物品来达到最优,这是因为这种方式可能造成背包空间的浪费,从而无法达到最优。考虑下面的物品和一个容量为 5 的背包,如果先添加物品 0 再添加物品 1,那么只能存放的价值为 16,浪费了大小为 2 的空间。最优的方式是存放物品 1 和物品 2,价值为 22。 | id | w | v | v/w | | --- | --- | --- | --- | @@ -792,7 +776,7 @@ public int knapsack(int W, int N, int[] weights, int[] values) { | 1 | 2 | 10 | 5 | | 2 | 3 | 12 | 4 | -**变种** +#### 变种 - 完全背包:物品数量为无限个 @@ -808,7 +792,7 @@ public int knapsack(int W, int N, int[] weights, int[] values) { [Leetcode](https://leetcode.com/problems/partition-equal-subset-sum/description/) / [力扣](https://leetcode-cn.com/problems/partition-equal-subset-sum/description/) -```html +```text Input: [1, 5, 11, 5] Output: true @@ -850,7 +834,7 @@ private int computeArraySum(int[] nums) { [Leetcode](https://leetcode.com/problems/target-sum/description/) / [力扣](https://leetcode-cn.com/problems/target-sum/description/) -```html +```text Input: nums is [1, 1, 1, 1, 1], S is 3. Output: 5 Explanation: @@ -868,7 +852,7 @@ There are 5 ways to assign symbols to make the sum of nums be target 3. 可以将这组数看成两部分,P 和 N,其中 P 使用正号,N 使用负号,有以下推导: -```html +```text sum(P) - sum(N) = target sum(P) + sum(N) + sum(P) - sum(N) = target + sum(P) + sum(N) 2 * sum(P) = target + sum(nums) @@ -924,7 +908,7 @@ private int findTargetSumWays(int[] nums, int start, int S) { [Leetcode](https://leetcode.com/problems/ones-and-zeroes/description/) / [力扣](https://leetcode-cn.com/problems/ones-and-zeroes/description/) -```html +```text Input: Array = {"10", "0001", "111001", "1", "0"}, m = 5, n = 3 Output: 4 @@ -964,7 +948,7 @@ public int findMaxForm(String[] strs, int m, int n) { [Leetcode](https://leetcode.com/problems/coin-change/description/) / [力扣](https://leetcode-cn.com/problems/coin-change/description/) -```html +```text Example 1: coins = [1, 2, 5], amount = 11 return 3 (11 = 5 + 5 + 1) @@ -1004,9 +988,9 @@ public int coinChange(int[] coins, int amount) { ### 5. 找零钱的硬币数组合 -518\. Coin Change 2 (Medium) +518\. Coin Change II (Medium) -[Leetcode](https://leetcode.com/problems/coin-change-2/description/) / [力扣](https://leetcode-cn.com/problems/coin-change-2/description/) +[Leetcode](https://leetcode.com/problems/coin-change-ii/description/) / [力扣](https://leetcode-cn.com/problems/coin-change-ii/description/) ```text Input: amount = 5, coins = [1, 2, 5] @@ -1042,7 +1026,7 @@ public int change(int amount, int[] coins) { [Leetcode](https://leetcode.com/problems/word-break/description/) / [力扣](https://leetcode-cn.com/problems/word-break/description/) -```html +```text s = "leetcode", dict = ["leet", "code"]. Return true because "leetcode" can be segmented as "leet code". @@ -1052,7 +1036,7 @@ dict 中的单词没有使用次数的限制,因此这是一个完全背包问 该问题涉及到字典中单词的使用顺序,也就是说物品必须按一定顺序放入背包中,例如下面的 dict 就不够组成字符串 "leetcode": -```html +```text ["lee", "tc", "cod"] ``` @@ -1081,7 +1065,7 @@ public boolean wordBreak(String s, List wordDict) { [Leetcode](https://leetcode.com/problems/combination-sum-iv/description/) / [力扣](https://leetcode-cn.com/problems/combination-sum-iv/description/) -```html +```text nums = [1, 2, 3] target = 4 @@ -1122,13 +1106,13 @@ public int combinationSum4(int[] nums, int target) { ### 1. 需要冷却期的股票交易 -309\. Best Time to Buy and Sell Stock with Cooldown(Medium) +309\. Best Time to Buy and Sell Stock with Cooldown (Medium) [Leetcode](https://leetcode.com/problems/best-time-to-buy-and-sell-stock-with-cooldown/description/) / [力扣](https://leetcode-cn.com/problems/best-time-to-buy-and-sell-stock-with-cooldown/description/) 题目描述:交易之后需要有一天的冷却时间。 -

+

```java public int maxProfit(int[] prices) { @@ -1158,7 +1142,7 @@ public int maxProfit(int[] prices) { [Leetcode](https://leetcode.com/problems/best-time-to-buy-and-sell-stock-with-transaction-fee/description/) / [力扣](https://leetcode-cn.com/problems/best-time-to-buy-and-sell-stock-with-transaction-fee/description/) -```html +```text Input: prices = [1, 3, 2, 8, 4, 9], fee = 2 Output: 8 Explanation: The maximum profit can be achieved by: @@ -1171,7 +1155,7 @@ The total profit is ((8 - 1) - 2) + ((9 - 4) - 2) = 8. 题目描述:每交易一次,都要支付一定的费用。 -

+

```java public int maxProfit(int[] prices, int fee) { @@ -1259,7 +1243,7 @@ public int maxProfit(int k, int[] prices) { [Leetcode](https://leetcode.com/problems/delete-operation-for-two-strings/description/) / [力扣](https://leetcode-cn.com/problems/delete-operation-for-two-strings/description/) -```html +```text Input: "sea", "eat" Output: 2 Explanation: You need one step to make "sea" to "ea" and another step to make "eat" to "ea". @@ -1286,11 +1270,11 @@ public int minDistance(String word1, String word2) { ### 2. 编辑距离 -72\. Edit Distance (Hard) +72\. Edit Distance (Medium) [Leetcode](https://leetcode.com/problems/edit-distance/description/) / [力扣](https://leetcode-cn.com/problems/edit-distance/description/) -```html +```text Example 1: Input: word1 = "horse", word2 = "ros" @@ -1351,7 +1335,7 @@ public int minDistance(String word1, String word2) { Input: 3 Output: 3 Explanation: -Intitally, we have one character 'A'. +Initially, we have one character 'A'. In step 1, we use Copy All operation. In step 2, we use Paste operation to get 'AA'. 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