how to find the number of distinct subsequences of a string?

Question

Here is another spoj problem that asks how to find the number of distinct subsequences of a string ?

For example,

Input
AAA
ABCDEFG
CODECRAFT

Output
4
128
496

How can I solve this problem ?

Answer 1

It's a classic dynamic programming problem.

Let:

dp[i] = number of distinct subsequences ending with a[i]
sum[i] = dp[1] + dp[2] + ... + dp[i]. So sum[n] will be your answer.
last[i] = last position of character i in the given string.

A null string has one subsequence, so dp[0] = 1.

read a
n = strlen(a)
for i = 1 to n
  dp[i] = sum[i - 1] - sum[last[a[i]] - 1]
  sum[i] = sum[i - 1] + dp[i]
  last[a[i]] = i

return sum[n]

Explanation

dp[i] = sum[i - 1] - sum[last[a[i]] - 1]

Initially, we assume we can append a[i] to all subsequences ending on previous characters, but this might violate the condition that the counted subsequences need to be distinct. Remember that last[a[i]] gives us the last position a[i] appeared on until now. The only subsequences we overcount are those that the previous a[i] was appended to, so we subtract those.

sum[i] = sum[i - 1] + dp[i]
last[a[i]] = i

Update these values as per their definition.

If your indexing starts from 0, use a[i - 1] wherever I used a[i]. Also remember to wrap your computations in a mod function if you're going to submit code. This should be implemented like this:

mod(x) = (x % m + m) % m

In order to correctly handle negative values in some languages (such as C/C++).