Distinct Subsequences | Training dragons the hard way - Programming Every Day!

Distinct Subsequences | Training dragons the hard way - Programming Every Day!

Problem Description

Given a string S and a string T, count the number of distinct sub-sequences of T in S.

A sub-sequence 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.

NOTE: The above problem statement is not very clear. We can make it "cleaner" by re-stating the problem as:
Given a string S and a string T, counting the number of ways that we remove some (or none) characters in S to get the remaining string equal to T.

Solutions

For a string str, we denote str[0,j] is the sub-string of str from index 0 to j inclusively. We easily guess that this solution can be solved by Dynamic Programming.
If we call dp[i][j] is the number of ways to remove some characters from S[0,i] to get T[0,j], we have the recursive formula:
dp [i][j] = dp[i-1][j] if S[i] != T[j] , or
dp [i][j] = dp[i-1][j] + dp[i-1][j-1] if S[i] ==T[j]

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