From f4f15b57a966af32e3da4cafb8175e6c38da7bc5 Mon Sep 17 00:00:00 2001 From: CyC2018 Date: Fri, 30 Oct 2026 20:18:31 +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 补充最长有效括号、最长公共子序列等题目的空间优化解法与状态解析,新增编辑距离等配图并重新绘制股票交易配图,调整多处解题思路说明并补充参考资料。 --- notes/Leetcode 题解 - 动态规划.md | 251 +++++++++++++++++-------- notes/pics/lc-dp-edit-distance.png | Bin 0 -> 177237 bytes notes/pics/lc-dp-house-robber.png | Bin 82529 -> 83628 bytes notes/pics/lc-dp-lis-tails.png | Bin 0 -> 166422 bytes notes/pics/lc-dp-stock-cooldown.png | Bin 151441 -> 153016 bytes notes/pics/lc-dp-unique-paths.png | Bin 84425 -> 84849 bytes notes/pics/lc-dp-valid-parentheses.png | Bin 0 -> 107971 bytes 7 files changed, 168 insertions(+), 83 deletions(-) create mode 100644 notes/pics/lc-dp-edit-distance.png create mode 100644 notes/pics/lc-dp-lis-tails.png create mode 100644 notes/pics/lc-dp-valid-parentheses.png diff --git a/notes/Leetcode 题解 - 动态规划.md b/notes/Leetcode 题解 - 动态规划.md index 60c18dba..8118df1b 100644 --- a/notes/Leetcode 题解 - 动态规划.md +++ b/notes/Leetcode 题解 - 动态规划.md @@ -45,7 +45,9 @@ -递归和动态规划都是将原问题拆成多个子问题然后求解,它们之间最本质的区别是,动态规划保存了子问题的解,避免重复计算。 +递归和动态规划都是将原问题拆成多个子问题然后求解,区别是动态规划保存了子问题的解,避免重复计算,例如递归求斐波那契数 f(5) 时 f(3) 会被算 2 次。记忆化递归也属于动态规划,只是按自顶向下的顺序计算。 + +写动态规划要确定 dp 的含义、状态转移方程、初始值和计算顺序。 ## 斐波那契数列 @@ -90,7 +92,7 @@ public int climbStairs(int n) { 定义 dp 数组用来存储最大的抢劫量,其中 dp[i] 表示抢到第 i 个住户时的最大抢劫量。 -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 +抢第 i 个住户的话,第 i - 1 个住户就不能抢,因此取抢和不抢两种情况的较大值:

@@ -112,6 +114,10 @@ public int rob(int[] nums) { [Leetcode](https://leetcode.com/problems/house-robber-ii/description/) / [力扣](https://leetcode-cn.com/problems/house-robber-ii/description/) +题目描述:住户围成一圈,首尾两户相邻。 + +首尾两户不能同时抢,因此分别在 0 \~ n-2(不抢最后一户)和 1 \~ n-1(不抢第一户)中按上一题求解,取较大值。 + ```java public int rob(int[] nums) { if (nums == null || nums.length == 0) { @@ -139,12 +145,12 @@ private int rob(int[] nums, int first, int last) { 题目描述:有 N 个信和信封,它们被打乱,求每封信都装错信封的方式数量(错排数)。 -定义一个数组 dp 存储错误方式数量,dp[i] 表示前 i 个信和信封的错误方式数量。假设第 i 个信装到第 j 个信封里面,而第 j 个信装到第 k 个信封里面。根据 i 和 k 是否相等,有两种情况: +dp[i] 表示 i 封信都装错的方式数量。第 i 封信装进了第 j 个信封(j ≠ i,有 i - 1 种选法),再看第 j 封信装在哪里: -- i==k,交换 i 和 j 的信后,它们的信和信封在正确的位置,但是其余 i-2 封信有 dp[i-2] 种错误装信的方式。由于 j 有 i-1 种取值,因此共有 (i-1)\*dp[i-2] 种错误装信方式。 -- i != k,交换 i 和 j 的信后,第 i 个信和信封在正确的位置,其余 i-1 封信有 dp[i-1] 种错误装信方式。由于 j 有 i-1 种取值,因此共有 (i-1)\*dp[i-1] 种错误装信方式。 +- 第 j 封信装进第 i 个信封:i 和 j 互相装错,剩下的 i - 2 封信要全部装错,有 dp[i - 2] 种; +- 第 j 封信没有装进第 i 个信封:去掉第 i 封信和第 j 个信封,把第 i 个信封看成第 j 封信不能装的信封,问题就变成 i - 1 封信全部装错,有 dp[i - 1] 种。 -综上所述,错误装信数量方式数量为: +两种情况都要乘以 j 的 i - 1 种选法,因此:

@@ -175,6 +181,8 @@ Given the above grid map, return 7. Because the path 1→3→1→1→1 minimizes 题目描述:求从矩阵的左上角到右下角的最小路径和,每次只能向右和向下移动。 +dp[j] 表示走到当前行第 j 列的最小路径和,只用一行滚动更新:更新前的 dp[j] 是上一行的值(从上侧走来),dp[j - 1] 已经是本行的值(从左侧走来)。 + ```java public int minPathSum(int[][] grid) { if (grid.length == 0 || grid[0].length == 0) { @@ -243,13 +251,13 @@ public int uniquePaths(int m, int n) { 题目描述:在由 0 和 1 组成的矩阵中,找出只包含 1 的最大正方形的面积。 -dp[i][j] 表示以 (i, j) 为右下角、只包含 1 的最大正方形的边长。它的边长受左边、上边和左上角三个方向的限制,取三者的最小值再加 1: +dp[i][j] 表示以 (i, j) 为右下角、只包含 1 的最大正方形的边长。当前格为 1 时,边长受左边、上边和左上角的限制,取三者的最小值再加 1。左上角不能省略,例如下面右边的例子只能得到边长 1。 ```text dp[i][j] = min(dp[i - 1][j], dp[i][j - 1], dp[i - 1][j - 1]) + 1 -matrix: 1 1 1 dp: 1 1 1 - 1 1 1 1 2 2 +matrix: 1 1 1 dp: 1 1 1 matrix: 0 1 dp: 0 1 + 1 1 1 1 2 2 1 1 1 1 1 1 1 1 2 3 ``` @@ -324,25 +332,27 @@ return: 6, for 6 arithmetic slices in A: [2, 3, 4] ``` -dp[i] 表示以 A[i] 为结尾的等差递增子区间的个数。 +子区间要求连续、长度至少为 3,公差可以为 0 或负数,例如 [7, 7, 7]、[3, 2, 1]。 -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。 +dp[i] 表示以 A[i] 为结尾的等差子区间的个数。 + +当 A[i] - A[i-1] == A[i-1] - A[i-2],那么 [A[i-2], A[i-1], A[i]] 构成一个等差子区间。而且在以 A[i-1] 为结尾的等差子区间的后面再加上一个 A[i],一样可以构成新的等差子区间。 ```text dp[2] = 1 [0, 1, 2] dp[3] = dp[2] + 1 = 2 [0, 1, 2, 3], // [0, 1, 2] 之后加一个 3 - [1, 2, 3] // 新的递增子区间 + [1, 2, 3] // 新的等差子区间 dp[4] = dp[3] + 1 = 3 [0, 1, 2, 3, 4], // [0, 1, 2, 3] 之后加一个 4 [1, 2, 3, 4], // [1, 2, 3] 之后加一个 4 - [2, 3, 4] // 新的递增子区间 + [2, 3, 4] // 新的等差子区间 ``` 综上,在 A[i] - A[i-1] == A[i-1] - A[i-2] 时,dp[i] = dp[i-1] + 1。 -因为递增子区间不一定以最后一个元素为结尾,可以是任意一个元素结尾,因此需要返回 dp 数组累加的结果。 +因为等差子区间不一定以最后一个元素为结尾,可以是任意一个元素结尾,因此需要返回 dp 数组累加的结果。 ```java public int numberOfArithmeticSlices(int[] A) { @@ -375,12 +385,9 @@ Input: ")()())" Output: 4 ``` -dp[i] 表示以 s[i] 结尾的最长有效括号的长度,只有 s[i] 为 ) 时才可能不为 0。以 s[i - 1] 结尾的有效括号长度为 dp[i - 1],跳过这一段之后的位置 j = i - dp[i - 1] - 1 如果是 (,就能和 s[i] 配对,长度为 dp[i - 1] + 2,再加上 j 前面紧挨着的有效括号长度 dp[j - 1]。 +dp[i] 表示以 s[i] 结尾的最长有效括号的长度,只有 s[i] 为 ) 时才可能不为 0。以 s[i - 1] 结尾的有效括号长度为 dp[i - 1],跳过这一段之后的位置 j = i - dp[i - 1] - 1 如果是 (,就能和 s[i] 配对,长度为 dp[i - 1] + 2,再加上 j 前面紧挨着的有效括号长度 dp[j - 1]。以 "()(())" 为例: -```text -s : ( ) ( ( ) ) -dp : 0 2 0 0 2 6 -``` +

```java public int longestValidParentheses(String s) { @@ -400,6 +407,40 @@ public int longestValidParentheses(String s) { } ``` +只用计数可以把空间降到 O(1):从左往右数左右括号,相等时得到一段有效括号,右括号更多时清零;"(()" 这类左括号一直更多的情况会被漏掉,再从右往左对称地扫一遍。 + +```java +public int longestValidParentheses(String s) { + int left = 0, right = 0, ret = 0; + for (int i = 0; i < s.length(); i++) { + if (s.charAt(i) == '(') { + left++; + } else { + right++; + } + if (left == right) { + ret = Math.max(ret, 2 * right); + } else if (right > left) { + left = right = 0; + } + } + left = right = 0; + for (int i = s.length() - 1; i >= 0; i--) { + if (s.charAt(i) == '(') { + left++; + } else { + right++; + } + if (left == right) { + ret = Math.max(ret, 2 * left); + } else if (left > right) { + left = right = 0; + } + } + return ret; +} +``` + ## 分割整数 ### 1. 分割整数的最大乘积 @@ -408,7 +449,9 @@ public int longestValidParentheses(String s) { [Leetcode](https://leetcode.com/problems/integer-break/description/) / [力扣](https://leetcode-cn.com/problems/integer-break/description/) -题目描述:For example, given n = 2, return 1 (2 = 1 + 1); given n = 10, return 36 (10 = 3 + 3 + 4). +题目描述:把正整数 n 拆成至少两个正整数的和,求这些整数乘积的最大值。For example, given n = 2, return 1 (2 = 1 + 1); given n = 10, return 36 (10 = 3 + 3 + 4). + +dp[i] 表示把 i 拆分后的最大乘积。枚举拆出的第一个数 j,剩下的 i - j 可以继续拆分(j \* dp[i - j]),也可以不拆(j \* (i - j)),不拆可能更大,例如 dp[3] = 2 小于 3。 ```java public int integerBreak(int n) { @@ -429,7 +472,9 @@ public int integerBreak(int n) { [Leetcode](https://leetcode.com/problems/perfect-squares/description/) / [力扣](https://leetcode-cn.com/problems/perfect-squares/description/) -题目描述:For example, given n = 12, return 3 because 12 = 4 + 4 + 4; given n = 13, return 2 because 13 = 4 + 9. +题目描述:把 n 表示成若干个完全平方数的和,求最少需要几个。For example, given n = 12, return 3 because 12 = 4 + 4 + 4; given n = 13, return 2 because 13 = 4 + 9. + +dp[i] 表示和为 i 的最少平方数个数,枚举最后一个平方数 square,dp[i] = min(dp[i - square] + 1)。 ```java public int numSquares(int n) { @@ -467,7 +512,9 @@ private List generateSquareList(int n) { [Leetcode](https://leetcode.com/problems/decode-ways/description/) / [力扣](https://leetcode-cn.com/problems/decode-ways/description/) -题目描述:Given encoded message "12", it could be decoded as "AB" (1 2) or "L" (12). +题目描述:字母 A \~ Z 分别编码为 1 \~ 26,求一个数字串有多少种解码方法。Given encoded message "12", it could be decoded as "AB" (1 2) or "L" (12). + +dp[i] 表示前 i 个字符的解码方法数。第 i 个字符不为 0 时可以单独解码,加上 dp[i - 1];最后两个字符在 10 \~ 26 之间时可以一起解码,加上 dp[i - 2]。例如 "226" 的 dp 为 [1, 1, 2, 3]。 ```java public int numDecodings(String s) { @@ -497,9 +544,9 @@ public int numDecodings(String s) { ## 最长递增子序列 -已知一个序列 {S1, S2,...,Sn},取出若干数组成新的序列 {Si1, Si2,..., Sim},其中 i1、i2 ... im 保持递增,即新序列中各个数仍然保持原数列中的先后顺序,称新序列为原序列的一个 **子序列** 。 +已知一个序列 {S1, S2,...,Sn},取出若干数组成新的序列 {Si1, Si2,..., Sim},其中 i1、i2 ... im 保持递增,即新序列中各个数仍然保持原数列中的先后顺序,称新序列为原序列的一个 **子序列**。 -如果在子序列中,当下标 ix > iy 时,Six > Siy,称子序列为原序列的一个 **递增子序列** 。 +如果在子序列中,当下标 ix > iy 时,Six > Siy,称子序列为原序列的一个 **递增子序列**。 定义一个数组 dp 存储最长递增子序列的长度,dp[n] 表示以 Sn 结尾的序列的最长递增子序列长度。对于一个递增子序列 {Si1, Si2,...,Sim},如果 im < n 并且 Sim < Sn,此时 {Si1, Si2,..., Sim, Sn} 为一个递增子序列,递增子序列的长度增加 1。满足上述条件的递增子序列中,长度最长的那个递增子序列就是要找的,在长度最长的递增子序列上加上 Sn 就构成了以 Sn 为结尾的最长递增子序列。因此 dp[n] = max{ dp[i]+1 | Si < Sn && i < n} 。 @@ -544,23 +591,16 @@ return ret; 以上解法的时间复杂度为 O(N2),可以使用二分查找将时间复杂度降低为 O(NlogN)。 -定义一个 tails 数组,其中 tails[i] 存储长度为 i + 1 的最长递增子序列的最后一个元素。对于一个元素 x, +定义一个 tails 数组,其中 tails[i] 存储所有长度为 i + 1 的递增子序列中最小的末尾元素。对于一个元素 x, - 如果它大于 tails 数组所有的值,那么把它添加到 tails 后面,表示最长递增子序列长度加 1; - 如果 tails[i-1] \< x \<= tails[i],那么更新 tails[i] = x。 -例如对于数组 [4,3,6,5],有: +例如对于数组 [2,5,3,7,1]: -```text -tails len num -[] 0 4 -[4] 1 3 -[3] 1 6 -[3,6] 2 5 -[3,5] 2 null -``` +

-可以看出 tails 数组保持有序,因此在查找 Si 位于 tails 数组的位置时就可以使用二分查找。 +tails 数组保持有序,因此可以用二分查找 x 的位置。需要注意的是,tails 不一定是真实的递增子序列,只有它的长度是答案。 ```java public int lengthOfLIS(int[] nums) { @@ -593,6 +633,8 @@ private int binarySearch(int[] tails, int len, int key) { } ``` +要输出一个最长递增子序列,可以在 O(N2) 解法中用 prev[i] 记录 dp[i] 由哪个 j 转移而来,再从 dp 最大的位置沿 prev 往回走。 + ### 2. 一组整数对能够构成的最长链 646\. Maximum Length of Pair Chain (Medium) @@ -607,6 +649,8 @@ Explanation: The longest chain is [1,2] -> [3,4] 题目描述:对于 (a, b) 和 (c, d) ,如果 b \< c,则它们可以构成一条链。 +整数对可以按任意顺序选取,链上的左端点一定递增,因此先按左端点排序,再按最长递增子序列的方法求解。 + ```java public int findLongestChain(int[][] pairs) { if (pairs == null || pairs.length == 0) { @@ -648,6 +692,8 @@ Output: 2 要求:使用 O(N) 时间复杂度求解。 +up、down 分别表示到当前元素为止、最后一步上升和下降的最长摆动子序列长度。nums[i] > nums[i - 1] 时,nums[i] 可以接在最后一步下降的序列后面,up = down + 1;下降时 down = up + 1;相等时都不变。 + ```java public int wiggleMaxLength(int[] nums) { if (nums == null || nums.length == 0) { @@ -667,7 +713,7 @@ public int wiggleMaxLength(int[] nums) { ## 最长公共子序列 -对于两个子序列 S1 和 S2,找出它们最长的公共子序列。 +对于两个序列 S1 和 S2,找出它们最长的公共子序列。 定义一个二维数组 dp 用来存储最长公共子序列的长度,其中 dp[i][j] 表示 S1 的前 i 个字符与 S2 的前 j 个字符最长公共子序列的长度。考虑 S1i 与 S2j 值是否相等,分为两种情况: @@ -682,8 +728,8 @@ public int wiggleMaxLength(int[] nums) { 与最长递增子序列相比,最长公共子序列有以下不同点: -- 针对的是两个序列,求它们的最长公共子序列。 -- 在最长递增子序列中,dp[i] 表示以 Si 为结尾的最长递增子序列长度,子序列必须包含 Si ;在最长公共子序列中,dp[i][j] 表示 S1 中前 i 个字符与 S2 中前 j 个字符的最长公共子序列长度,不一定包含 S1i 和 S2j。 +- 针对的是两个序列,求它们的最长公共子序列; +- 在最长递增子序列中,dp[i] 表示以 Si 为结尾的最长递增子序列长度,子序列必须包含 Si ;在最长公共子序列中,dp[i][j] 表示 S1 中前 i 个字符与 S2 中前 j 个字符的最长公共子序列长度,不一定包含 S1i 和 S2j; - 在求最终解时,最长公共子序列中 dp[N][M] 就是最终解,而最长递增子序列中 dp[N] 不是最终解,因为以 SN 为结尾的最长递增子序列不一定是整个序列最长递增子序列,需要遍历一遍 dp 数组找到最大者。 ### 1. 最长公共子序列 @@ -693,25 +739,52 @@ public int wiggleMaxLength(int[] nums) { [Leetcode](https://leetcode.com/problems/longest-common-subsequence/) / [力扣](https://leetcode-cn.com/problems/longest-common-subsequence/) ```java - public int longestCommonSubsequence(String text1, String text2) { - int n1 = text1.length(), n2 = text2.length(); - int[][] dp = new int[n1 + 1][n2 + 1]; - for (int i = 1; i <= n1; i++) { - for (int j = 1; j <= n2; j++) { - if (text1.charAt(i - 1) == text2.charAt(j - 1)) { - dp[i][j] = dp[i - 1][j - 1] + 1; - } else { - dp[i][j] = Math.max(dp[i - 1][j], dp[i][j - 1]); - } +public int longestCommonSubsequence(String text1, String text2) { + int n1 = text1.length(), n2 = text2.length(); + int[][] dp = new int[n1 + 1][n2 + 1]; + for (int i = 1; i <= n1; i++) { + for (int j = 1; j <= n2; j++) { + if (text1.charAt(i - 1) == text2.charAt(j - 1)) { + dp[i][j] = dp[i - 1][j - 1] + 1; + } else { + dp[i][j] = Math.max(dp[i - 1][j], dp[i][j - 1]); } } - return dp[n1][n2]; } + return dp[n1][n2]; +} +``` + +可以像[矩阵的最小路径和](#1-矩阵的最小路径和)一样只用一行滚动更新,空间降为 O(min(M, N))。左上角 dp[i - 1][j - 1] 在算 dp[j - 1] 时已被覆盖,要提前用变量保存: + +```java +public int longestCommonSubsequence(String text1, String text2) { + if (text1.length() < text2.length()) { // 让 text2 是较短的那个 + String t = text1; + text1 = text2; + text2 = t; + } + int n = text2.length(); + int[] dp = new int[n + 1]; + for (int i = 1; i <= text1.length(); i++) { + int leftUp = 0; // dp[i - 1][j - 1] + for (int j = 1; j <= n; j++) { + int up = dp[j]; // dp[i - 1][j],被覆盖前先保存 + if (text1.charAt(i - 1) == text2.charAt(j - 1)) { + dp[j] = leftUp + 1; + } else { + dp[j] = Math.max(up, dp[j - 1]); + } + leftUp = up; + } + } + return dp[n]; +} ``` ## 0-1 背包 -有一个容量为 N 的背包,要用这个背包装下物品的价值最大,这些物品有两个属性:体积 w 和价值 v。 +有一个容量为 W 的背包和 N 件物品,要用这个背包装下物品的价值最大,这些物品有两个属性:体积 w 和价值 v。 定义一个二维数组 dp 存储最大价值,其中 dp[i][j] 表示前 i 件物品体积不超过 j 的情况下能达到的最大价值。设第 i 件物品体积为 w,价值为 v,根据第 i 件物品是否添加到背包中,可以分两种情况讨论: @@ -778,13 +851,10 @@ public int knapsack(int W, int N, int[] weights, int[] values) { #### 变种 -- 完全背包:物品数量为无限个 - -- 多重背包:物品数量有限制 - -- 多维费用背包:物品不仅有重量,还有体积,同时考虑这两种限制 - -- 其它:物品之间相互约束或者依赖 +- 完全背包:每种物品的数量无限,例如[找零钱的最少硬币数](#4-找零钱的最少硬币数); +- 多重背包:每种物品的数量有限制; +- 多维费用背包:物品不仅有重量,还有体积,同时考虑这两种限制,例如 [01 字符构成最多的字符串](#3-01-字符构成最多的字符串); +- 其它:物品之间相互约束或者依赖。 ### 1. 划分数组为和相等的两部分 @@ -858,7 +928,7 @@ sum(P) + sum(N) + sum(P) - sum(N) = target + sum(P) + sum(N) 2 * sum(P) = target + sum(nums) ``` -因此只要找到一个子集,令它们都取正号,并且和等于 (target + sum(nums))/2,就证明存在解。 +每种添加符号的方式对应一个取正号的子集 P,因此答案是和为 W = (target + sum(nums)) / 2 的子集个数。dp[j] 表示和为 j 的子集个数,dp[0] = 1 对应空集。 ```java public int findTargetSumWays(int[] nums, int S) { @@ -964,25 +1034,21 @@ return -1. - 物品大小:面额 - 物品价值:数量 -因为硬币可以重复使用,因此这是一个完全背包问题。完全背包只需要将 0-1 背包的逆序遍历 dp 数组改为正序遍历即可。 +因为硬币可以重复使用,因此这是一个完全背包问题。完全背包只需要将 0-1 背包的逆序遍历 dp 数组改为正序遍历即可(见[空间优化](#空间优化)),正序时 dp[i - coin] 可能已经放过本轮的硬币,相当于同一种硬币再放一次。dp[i] 表示凑出金额 i 的最少硬币数。 + +每次选面额最大的硬币不一定最优,例如 coins = [1, 3, 4]、amount = 6,贪心得到 4 + 1 + 1,最优是 3 + 3。 ```java public int coinChange(int[] coins, int amount) { - if (amount == 0 || coins == null) return 0; int[] dp = new int[amount + 1]; + Arrays.fill(dp, amount + 1); // amount + 1 表示凑不出 + dp[0] = 0; for (int coin : coins) { - for (int i = coin; i <= amount; i++) { //将逆序遍历改为正序遍历 - if (i == coin) { - dp[i] = 1; - } else if (dp[i] == 0 && dp[i - coin] != 0) { - dp[i] = dp[i - coin] + 1; - - } else if (dp[i - coin] != 0) { - dp[i] = Math.min(dp[i], dp[i - coin] + 1); - } + for (int i = coin; i <= amount; i++) { // 将逆序遍历改为正序遍历 + dp[i] = Math.min(dp[i], dp[i - coin] + 1); } } - return dp[amount] == 0 ? -1 : dp[amount]; + return dp[amount] > amount ? -1 : dp[amount]; } ``` @@ -1002,7 +1068,7 @@ Explanation: there are four ways to make up the amount: 5=1+1+1+1+1 ``` -完全背包问题,使用 dp 记录可达成目标的组合数目。 +完全背包问题,dp[i] 记录凑出金额 i 的组合数目。外层遍历硬币、内层遍历金额,硬币按固定的顺序加入,1+2 和 2+1 只算一种。 ```java public int change(int amount, int[] coins) { @@ -1032,15 +1098,9 @@ dict = ["leet", "code"]. Return true because "leetcode" can be segmented as "leet code". ``` -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题。 +dp[i] 表示 s 的前 i 个字符能否拆分成 dict 中的单词。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题。 -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": - -```text -["lee", "tc", "cod"] -``` - -求解顺序的完全背包问题时,对物品的迭代应该放在最里层,对背包的迭代放在外层,只有这样才能让物品按一定顺序放入背包中。 +与上一题不同,单词在 s 中的先后顺序任意,因此外层遍历位置 i、内层遍历单词。交换两层循环时单词只能按 dict 中的顺序出现,例如 s = "codeleet"、dict = ["leet", "code"] 会得到 false。 ```java public boolean wordBreak(String s, List wordDict) { @@ -1083,7 +1143,12 @@ Note that different sequences are counted as different combinations. Therefore the output is 7. ``` -涉及顺序的完全背包。 +涉及顺序的完全背包。外层遍历 target、内层遍历 nums,每个金额的最后一个数可以是任意一个 num,(1, 2) 和 (2, 1) 分别计数。以 nums = [1, 2]、target = 3 对比[找零钱的硬币数组合](#5-找零钱的硬币数组合): + +```text +外层遍历硬币(518):处理 1 后 dp = [1, 1, 1, 1],处理 2 后 dp = [1, 1, 2, 2] 2 种:1+1+1、1+2 +外层遍历金额(377):dp = [1, 1, 2, 3] 3 种:1+1+1、1+2、2+1 +``` ```java public int combinationSum4(int[] nums, int target) { @@ -1110,7 +1175,9 @@ public int combinationSum4(int[] nums, int target) { [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。 +题目描述:可以进行多次交易,但卖出之后需要有一天的冷却时间才能再次买入。 + +buy、s1、sell、s2 分别表示当天买入、持有且当天没买、当天卖出、不持有且当天没卖,值为到这一天为止的最大收益。冷却期体现在 buy 只能从前一天的 s2 转移过来。第 0 天的 s1、sell 只是占位初值,不会得到比合法交易更大的收益。

@@ -1155,6 +1222,8 @@ 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。 +状态与[需要冷却期的股票交易](#1-需要冷却期的股票交易)相同,区别是 buy 也可以从前一天的 sell 转移过来,卖出时扣除手续费。 +

```java @@ -1176,13 +1245,14 @@ public int maxProfit(int[] prices, int fee) { } ``` - ### 3. 只能进行两次的股票交易 123\. Best Time to Buy and Sell Stock III (Hard) [Leetcode](https://leetcode.com/problems/best-time-to-buy-and-sell-stock-iii/description/) / [力扣](https://leetcode-cn.com/problems/best-time-to-buy-and-sell-stock-iii/description/) +四个变量依次是到当天为止完成第一次买入、第一次卖出、第二次买入、第二次卖出后的最大收益,第二次买入以第一次卖出的收益为起点。 + ```java public int maxProfit(int[] prices) { int firstBuy = Integer.MIN_VALUE, firstSell = 0; @@ -1211,6 +1281,8 @@ public int maxProfit(int[] prices) { [Leetcode](https://leetcode.com/problems/best-time-to-buy-and-sell-stock-iv/description/) / [力扣](https://leetcode-cn.com/problems/best-time-to-buy-and-sell-stock-iv/description/) +maxProfit[i][j] 表示最多交易 i 次、到第 j 天为止的最大收益。第 j 天要么不卖出,收益为 maxProfit[i][j - 1];要么在之前的某天 m 买入、第 j 天卖出,收益为 prices[j] + max(maxProfit[i - 1][m] - prices[m]),localMax 在遍历 j 时维护这个最大值。k >= n / 2 时次数不构成限制,累加所有上涨的差价即可。 + ```java public int maxProfit(int k, int[] prices) { int n = prices.length; @@ -1249,7 +1321,7 @@ Output: 2 Explanation: You need one step to make "sea" to "ea" and another step to make "eat" to "ea". ``` -可以转换为求两个字符串的最长公共子序列问题。 +可以转换为求两个字符串的最长公共子序列问题:两个字符串都只保留最长公共子序列,其余字符删掉,删除次数为 m + n - 2 \* LCS。 ```java public int minDistance(String word1, String word2) { @@ -1297,6 +1369,10 @@ exection -> execution (insert 'u') 题目描述:修改一个字符串成为另一个字符串,使得修改次数最少。一次修改操作包括:插入一个字符、删除一个字符、替换一个字符。 +dp[i][j] 表示 word1 的前 i 个字符变成 word2 的前 j 个字符的最少操作数。最后一个字符相等时 dp[i][j] = dp[i - 1][j - 1];不相等时,最后一步是替换、删除或插入,取三者的最小值再加 1: + +

+ ```java public int minDistance(String word1, String word2) { if (word1 == null || word2 == null) { @@ -1323,6 +1399,8 @@ public int minDistance(String word1, String word2) { } ``` +空间复杂度可以和[最长公共子序列](#1-最长公共子序列)一样用一维数组降到 O(N)。 + ### 3. 复制粘贴字符 650\. 2 Keys Keyboard (Medium) @@ -1341,6 +1419,8 @@ In step 2, we use Paste operation to get 'AA'. In step 3, we use Paste operation to get 'AAA'. ``` +如果 n 有因子 i,可以先得到 n / i 个 A,再复制一次、粘贴 i - 1 次,用 i 步变成 n 个 A。因此总步数是 n 的质因数之和,例如 12 = 2 × 2 × 3,需要 7 步。 + ```java public int minSteps(int n) { if (n == 1) return 0; @@ -1367,3 +1447,8 @@ public int minSteps(int n) { return dp[n]; } ``` + +## 参考资料 + +- [LeetCode 题库](https://leetcode.com/problemset/) +- 左程云. 程序员代码面试指南:IT 名企算法与数据结构题目最优解 [M]. 电子工业出版社, 2015. diff --git a/notes/pics/lc-dp-edit-distance.png b/notes/pics/lc-dp-edit-distance.png new file mode 100644 index 0000000000000000000000000000000000000000..3c3c502b21144d292b02262a9af7ab58175487e7 GIT binary patch literal 177237 zcmXVYWk8kP*6mglBow7f0qG8r77%HW7LgRBLAp~#kdO{(Ub;cLK^jyVq`SMj?%e0Q z{5Y>ik^MYt%{AtTx%5|*m%NWbf`LFF?n_IFDI*Yf=@E$AllN}J@0h>s(t-axvKLpk zf2(J2X!}vm)*SK56U28LamFG< z()zQ9&EVJN8hcL))bdmrKkE_sN|Qc&!-$WFk)tgg%U90fA_)>rvEIkY8#ZGn+N}^x zQG{QW$AwV0uw%8=Soq&f+K4W*@_`qEjVt&=e3c?)8U>6| zzl}=YF%xL81!Su-<_$9@F=6=J-rs!A)kAm}l<`kELgO504k|X|NbRU7b0*Wy{Q}TqB50i7_gbR+%x1Ke5MvbIgJ} 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