Find the number of occurrences of a subsequence in a string

Question

For example, let the string be the first 10 digits of pi, 3141592653, and the subsequence be 123. Note that the sequence occurs twice:

Answer 1

Great answer, aioobe! To complement your answer, some possible implementations in Python:

1) straightforward, naïve solution; too slow!

def num_subsequences(seq, sub):
    if not sub:
        return 1
    elif not seq:
        return 0
    result = num_subsequences(seq[1:], sub)
    if seq[0] == sub[0]:
        result += num_subsequences(seq[1:], sub[1:])
    return result

2) top-down solution using explicit memoization

def num_subsequences(seq, sub):
    m, n, cache = len(seq), len(sub), {}
    def count(i, j):
        if j == n:
            return 1
        elif i == m:
            return 0
        k = (i, j)
        if k not in cache:
            cache[k] = count(i+1, j) + (count(i+1, j+1) if seq[i] == sub[j] else 0)
        return cache[k]
    return count(0, 0)

3) top-down solution using the lru_cache decorator (available from functools in python >= 3.2)

from functools import lru_cache

def num_subsequences(seq, sub):
    m, n = len(seq), len(sub)
    @lru_cache(maxsize=None)
    def count(i, j):
        if j == n:
            return 1
        elif i == m:
            return 0
        return count(i+1, j) + (count(i+1, j+1) if seq[i] == sub[j] else 0)
    return count(0, 0)

4) bottom-up, dynamic programming solution using a lookup table

def num_subsequences(seq, sub):
    m, n = len(seq)+1, len(sub)+1
    table = [[0]*n for i in xrange(m)]
    def count(iseq, isub):
        if not isub:
            return 1
        elif not iseq:
            return 0
        return (table[iseq-1][isub] +
               (table[iseq-1][isub-1] if seq[m-iseq-1] == sub[n-isub-1] else 0))
    for row in xrange(m):
        for col in xrange(n):
            table[row][col] = count(row, col)
    return table[m-1][n-1]

5) bottom-up, dynamic programming solution using a single array

def num_subsequences(seq, sub):
    m, n = len(seq), len(sub)
    table = [0] * n
    for i in xrange(m):
        previous = 1
        for j in xrange(n):
            current = table[j]
            if seq[i] == sub[j]:
                table[j] += previous
            previous = current
    return table[n-1] if n else 1