Near-optimal algorithm to count occurrences of subsequences of a given length

Abstract

For k ∈ ℤ+, define Σk as the set of integers {0, 1, …, k − 1}. Given an integer n and a string t of length m ≥ n over Σk, we count the number of times that each one of the kn distinct strings of length n over Σk occurs as a subsequence of t. Our algorithm makes only one scan of t and solves the problem in time complexity mkn−1 and space complexity m + kn. These are very close to best possible.