An acceleration of FFT-based algorithms for the match-count problem

The match-count problem on strings is a problem of counting the matches of characters for every possible gap of the starting positions between two strings. This problem for strings of lengths m and n (m≤n) over an alphabet of size σ is classically solved in O(σnlog⁡m) time using the algorithm based on the convolution theorem and a fast Fourier transform (FFT). This paper provides a method to reduce the number of computations of the FFT required in the FFT-based algorithm. The algorithm obtained by the proposed method still needs O(σnlog⁡m) time, but the number of required FFT computations is reduced from 3σ to 2σ+1. This practical improvement of the processing time is also applicable to other algorithms based on the convolution theorem, including algorithms for the weighted version of the match-count problem.