Leetcode: Distinct Subsequence

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.

Solution:
Another DP problem.
可以这样想，设母串的长度为 j， 子串的长度为 i，我们要求的就是长度为 i 的字串在长度为 j 的母串中出现的次数，设为 t[i][j]，若母串的最后一个字符与子串的最后一个字符不同，则长度为 i 的子串在长度为 j 的母串中出现的次数就是母串的前 j - 1 个字符中子串出现的次数，即 t[i][j] = t[i][j - 1]，若母串的最后一个字符与子串的最后一个字符相同，那么除了前 j - 1 个字符出现字串的次数外，还要加上子串的前 i - 1 个字符在母串的前 j - 1 个字符中出现的次数，即 t[i][j] = t[i][j - 1] + t[i - 1][j - 1]。

It is very much the same as how we solve C(n, m) or the knapsack problem.

  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