[LeetCode]Distinct Subsequences,解题报告

Posted 2023-02-24 低调小一

题目

Given a string S and a string T, count the number of distinct subsequences of T in S.

A subsequence of a string is a new string which is formed from the original string by deleting some (can be none) of the characters without disturbing the relative positions of the remaining characters. (ie, "ACE" is a subsequence of "ABCDE" while "AEC" is not).

Here is an example:
S = "rabbbit", T = "rabbit"

Return 3.

思路1

开始很容易想到深搜，通过flags数组做标记位，得到子串的个数
代码：

import java.util.Scanner;

public class DistinctSubsequences 
    private static int disNum = 0;

    public static int numDistinct(String S, String T) 
        int[] flags = new int[S.length()];
        int num = 0;

        dfs(num, flags, 0, 0, S, T);

        return disNum;
    

    public static void dfs(int num, int[] flags, int indexS, int indexT, String S, String T) 
        if (num == T.length()) 
            disNum++;
         else 
            for (int i = indexS; i < S.length(); i ++)                 
                if (S.charAt(i) == T.charAt(indexT) && flags[i] == 0) 
                    flags[i] = 1;
                    num++;
                    dfs(num, flags, i + 1, indexT + 1, S, T);
                    flags[i] = 0;
                    num--;
                
            
        
    

    public static void main(String[] args) 
        Scanner cin = new Scanner(System.in);

        while (cin.hasNext()) 
            String S = cin.nextLine();
            String T = cin.nextLine();
            disNum = 0;

            int res = numDistinct(S, T);

            System.out.println(res);
        

        cin.close();
    

但是在大集合的时候 Time Limit Exceeded

思路2

既然简单的深搜超时，只能考虑稍微复杂一点的DP了。可以参考动态规划经典的例子，最长公共子序列。
这里我采用二维数组int[][] dp来记录匹配子序列的个数，则状态方程为：
dp[0][0] = 1, T和S均为空串 dp[0][1..S.length() - 1] = 1, T为空串，S只有一种子序列匹配 dp[1..T.length() - 1][0] = 0, S为空串 dp[i][j] = dp[i][j - 1] + (T[i - 1] == S[j - 1] ? dp[i - 1][j - 1] : 0)
代码：

public class Solution 
    public static int numDistinct(String S, String T) 
        if (S == null || S.length() == 0) 
            return 0;
        

        int[][] dp = new int[T.length() + 1][S.length() + 1];

        dp[0][0] = 1;

        for (int i = 1; i <= S.length(); i++) 
            dp[0][i] = 1;
        

        for (int i = 1; i <= T.length(); i++) 
            dp[i][0] = 0;
        

        for (int i = 1; i <= T.length(); i++) 
            for (int j = 1; j <= S.length(); j++) 
                if (T.charAt(i - 1) == S.charAt(j - 1)) 
                    dp[i][j] = dp[i][j - 1] + dp[i - 1][j - 1];
                 else 
                    dp[i][j] = dp[i][j - 1];
                
            
        

        return dp[T.length()][S.length()];