Distinct Subsequences

描述

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.

分析

设状态为f(i,j)，表示T[0,j]在S[0,i]里出现的次数。首先，无论S[i]和T[j]是否相等，若不使用S[i]，则f(i,j)=f(i-1,j)；若S[i]==T[j]，则可以使用S[i]，此时f(i,j)=f(i-1,j)+f(i-1, j-1)。

代码

  1. // Distinct Subsequences
  2. // 二维动规+滚动数组
  3. // 时间复杂度O(m*n)，空间复杂度O(n)
  4. class Solution {
  5. public:
  6.     int numDistinct(const string &S, const string &T) {
  7.         vector<int> f(T.size() + 1);
  8.         f[0] = 1;
  9.         for (int i = 0; i < S.size(); ++i) {
  10.             for (int j = T.size() - 1; j >= 0; --j) {
  11.                 f[j + 1] += S[i] == T[j] ? f[j] : 0;
  12.             }
  13.         }
  15.         return f[T.size()];
  16.     }
  17. };

原文: https://soulmachine.gitbooks.io/algorithm-essentials/content/cpp/dp/distinct-subsequences.html