The longest common subsequence problem for small alphabets in the word RAM model

Abstract

Given two strings of lengths m and n, with $m\le n$, the longest common subsequence problem consists of computing a common subsequence of maximum length by deleting symbols from both strings. While the $O\left(mn\right)$ algorithm devised in 1974 is optimal in the most general setting, algorithms that depend on parameters other than m and n have been proposed since then. In the word RAM model, let w be the word size, s be the alphabet size, d be the number of dominant symbol matches between the strings, and p be the length of the longest common subsequence. Fast algorithms for this problem have complexities $O(mn/\log n)$, $O(mn/w)$, $O(ns+\min (p(n-p),pm\left)\right)$, $O(n\log s+d\log \log \min (d,mn/d\left)\right)$, $O(ns+\min (ds,pm\left)\right)$, and $O(ns+s{!}^{2}s+d\log s)$. In this work, we present an $O\left(n\right(s+{\log }^{⁎}n)+\min (d\log s,pm\left)\right)$ algorithm when $s\in O\left(w\right)$, and also an $O\left(n\right(s+{\log }^{⁎}n)+d)$ algorithm when $s\le w$ which uses bitwise instructions that became recently available in modern processors.