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一、类型描述

Two Sequences 的题目一般会提供两个sequence，一般问最大/最小、true/false、count(*)这几类的问题。
其中，Two Sequences的动态规划题目的四要素：

1. state：dp[i][j] 一般表示 第一个sequence的前i个字符 和 第二个sequence的前j个字符 怎么怎么样。
2. Initialization： 这类型动态规划一般初始化dp数组的方式是根据题目的含义初始化第一行和第一列。
3. function：解决动态规划的function问题，就是找到当前待解决问题dp[i][j] 和 前面解决过的问（例如，dp[i - 1][j]、dp[i][j - 1]）的之间的关系。
4. answer：dp[s1.length()][s2.length()]

二、Leetcode题目举例

97. Interleaving String hard

Given s1, s2, s3, find whether s3 is formed by the interleaving of s1 and s2.
Example 1:

Input: s1 = "aabcc", s2 = "dbbca", s3 = "aadbbcbcac"
Output: true

Example 2:

Input: s1 = "aabcc", s2 = "dbbca", s3 = "aadbbbaccc"
Output: false

题目中，提供了两个source string，一个目标string，问的也是true or false的问题，因此可以考虑two sequences的解法。

1. 取状态函数为dp[i][j] 表示 s1的前i个字符 和 s2的前j个字符 能否组成 s3的前 i + j 个字符。
2. dp[i][j] 和sub question之间的关系：
   需要求的：dp[i][j]
   之前求得结果的子问题：dp[i - 1][j]、dp[i][j - 1] .......
   优先考虑能否从靠近dp[i][j]的子问题求解：可以发现，可以更具s3的前（i+j）个字符的最后一个字符 来自于 s1 还是 s2 来分类：
   因此 dp[i][j] = (dp[i - 1][j] && s1.charAt(i - 1) == s3.charAt(i + j - 1)) || (dp[i][j - 1] && s2.charAt(j - 1) == s3.charAt(i + j - 1));
   Java Solution:
   

class Solution {
    public boolean isInterleave(String s1, String s2, String s3) {
        if (s3.length() != s1.length() + s2.length()) { return false; }
        
        // state dp[i][j] 表示s1的前i个字符 和 s2的前j个字符 能否组成s3的前(i + j) 个字符
        boolean[][] dp = new boolean[s1.length() + 1][s2.length() + 1];
        
        // initialization
        // dp[0][j] ：s1的前0个字符 和 s2的前j个字符 能否组成 s3的前j个字符？
        for (int j = 0; j < dp[0].length; j++) {
            // 要判断两个字符串是否相等， 必须用字符串的equals方法来比较
            dp[0][j] = s2.substring(0, j).equals(s3.substring(0, j));
            
            //dp[0][j] = s2.substring(0, j) == s3.substring(0, j);  // 比较的是两个字符串的reference，并不是实际是否相等，因此这么写是错的
        }

        // dp[i][0] 同理
        for (int i = 0; i < dp.length; i++) {
            dp[i][0] = s1.substring(0, i).equals(s3.substring(0, i));
            //dp[i][0] = s1.substring(0, i) == s3.substring(0, i);  // 错误
        }
        
        // function
        for (int i = 1; i < dp.length; i++) {
            for (int j = 1; j < dp[0].length; j++) {
                dp[i][j] = (dp[i - 1][j] && (s1.charAt(i - 1) == s3.charAt(i + j - 1))) || (dp[i][j - 1] && (s2.charAt(j - 1) == s3.charAt(i + j - 1)));
            }
        }
        
        // answer
        return dp[s1.length()][s2.length()];
    } 
}

2. 1143. Longest Common Subsequence Medium

Given two strings text1 and text2, return the length of their longest common subsequence.
A subsequence of a string is a new string generated from the original string with some characters(can be none) deleted without changing the relative order of the remaining characters. (eg, "ace" is a subsequence of "abcde" while "aec" is not). A common subsequence of two strings is a subsequence that is common to both strings.
If there is no common subsequence, return 0.
Example1

Input: text1 = "abcde", text2 = "ace"
Output: 3
Explanation: The longest common subsequence is "ace" and its length is 3.
Example2
Input: text1 = "abc", text2 = "abc"
Output: 3
Explanation: The longest common subsequence is "abc" and its length is 3.

<LeetCode>动态规划——Two Sequences 类型题目

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原文地址：https://www.cnblogs.com/isguoqiang/p/11529981.html