【leetcode】115. Distinct Subsequences【java】

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.

遇到这种两个串的问题，很容易想到DP。但是这道题的递推关系不明显。可以先尝试做一个二维的表int[][] dp，用来记录匹配子序列的个数（以S="rabbbit",T = "rabbit"为例）：

    r a b b b i t

  1 1 1 1 1 1 1 1

r 0 1 1 1 1 1 1 1

a 0 0 1 1 1 1 1 1

b 0 0 0 1 2 3 3 3

b 0 0 0 0 1 3 3 3

i 0 0 0 0 0 0 3 3

t 0 0 0 0 0 0 0 3  

从这个表可以看出，无论T的字符与S的字符是否匹配，dp[i][j] = dp[i][j - 1].就是说，假设S已经匹配了j - 1个字符，得到匹配个数为dp[i][j - 1].现在无论S[j]是不是和T[i]匹配，匹配的个数至少是dp[i][j - 1]。除此之外，当S[j]和T[i]相等时，我们可以让S[j]和T[i]匹配，然后让S[j - 1]和T[i - 1]去匹配。所以递推关系为：

dp[0][0] = 1; // T和S都是空串.

dp[0][1 ... S.length() - 1] = 1; // T是空串，S只有一种子序列匹配。

dp[1 ... T.length() - 1][0] = 0; // S是空串，T不是空串，S没有子序列匹配。

dp[i][j] = dp[i][j - 1] + (T[i - 1] == S[j - 1] ? dp[i - 1][j - 1] : 0).1 <= i <= T.length(), 1 <= j <= S.length()

Java代码：

public class Solution {    public int numDistinct(String s, String t) {        int[][] dp = new int[t.length() + 1][s.length() + 1];        dp[0][0] = 1;        for (int i = 1; i <= t.length(); i++) {            dp[i][0] = 0;        }        for (int j = 1; j <= s.length(); j++) {            dp[0][j] = 1;        }        for (int i = 1; i <= t.length(); i++) {            for (int j = 1; j <= s.length(); j++) {                dp[i][j] = dp[i][j - 1];                if (s.charAt(j - 1) == t.charAt(i - 1)){                    dp[i][j] += dp[i - 1][j - 1];                }            }        }        return dp[t.length()][s.length()];    }}