[LeetCode] Distinct Subsequences 不同的子序列

时间：2023-03-08 17:10:33

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

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).

Example 1:

Input: S = "rabbbit", T = "rabbit"

Output: 3
Explanation:

As shown below, there are 3 ways you can generate "rabbit" from S.

(The caret symbol ^ means the chosen letters)

rabbbit

^^^^ ^^
rabbbit

^^ ^^^^
rabbbit

^^^ ^^^

Example 2:

Input: S = "babgbag", T = "bag"

Output: 5
Explanation:

As shown below, there are 5 ways you can generate "bag" from S.

(The caret symbol ^ means the chosen letters)

babgbag

^^ ^
babgbag

^^    ^
babgbag

^    ^^
babgbag

  ^  ^^
babgbag

    ^^^

看到有关字符串的子序列或者配准类的问题，首先应该考虑的就是用动态规划 Dynamic Programming 来求解，这个应成为条件反射。而所有 DP 问题的核心就是找出状态转移方程，想这道题就是递推一个二维的 dp 数组，其中 dp[i][j] 表示s中范围是 [0, i] 的子串中能组成t中范围是 [0, j] 的子串的子序列的个数。下面我们从题目中给的例子来分析，这个二维 dp 数组应为：

  Ø r a b b b i t

Ø 1 1 1 1 1 1 1 1

r  1 1

a   1 1

b 0   1 2

b     1 3

i      0 3

t       0 3

首先，若原字符串和子序列都为空时，返回1，因为空串也是空串的一个子序列。若原字符串不为空，而子序列为空，也返回1，因为空串也是任意字符串的一个子序列。而当原字符串为空，子序列不为空时，返回0，因为非空字符串不能当空字符串的子序列。理清这些，二维数组 dp 的边缘便可以初始化了，下面只要找出状态转移方程，就可以更新整个 dp 数组了。我们通过观察上面的二维数组可以发现，当更新到 dp[i][j] 时，dp[i][j] >= dp[i][j - 1] 总是成立，再进一步观察发现，当 T[i - 1] == S[j - 1] 时，dp[i][j] = dp[i][j - 1] + dp[i - 1][j - 1]，若不等， dp[i][j] = dp[i][j - 1]，所以，综合以上，递推式为：

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

根据以上分析，可以写出代码如下：

class Solution {

public:

    int numDistinct(string s, string t) {

        int m = s.size(), n = t.size();

        vector<vector<long>> dp(n + , vector<long>(m + ));

        for (int j = ; j <= m; ++j) dp[][j] = ;

        for (int i = ; i <= n; ++i) {

            for (int j = ; j <= m; ++j) {

                dp[i][j] = dp[i][j - ] + (t[i - ] == s[j - ] ? dp[i - ][j - ] : );

            }

        }

        return dp[n][m];

    }

};

Github 同步地址：

https://github.com/grandyang/leetcode/issues/115

参考资料：

https://leetcode.com/problems/distinct-subsequences/

https://leetcode.com/problems/distinct-subsequences/discuss/37327/Easy-to-understand-DP-in-Java

https://leetcode.com/problems/distinct-subsequences/discuss/37412/Any-better-solution-that-takes-less-than-O(n2)-space-while-in-O(n2)-time

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