Sublinear space algorithms for the longest common substring problem

Abstract

Given m documents of total length n, we consider the problem of finding a longest string common to at least d≥2 of the documents. This problem is known as the longest common substring (LCS) problem and has a classic O(n) space and O(n) time solution (Weiner [FOCS'73], Hui [CPM'92]). However, the use of linear space is impractical in many applications. In this paper we show that for any trade-off parameter 1≤τ≤;n, the LCS problem can be solved in O(τ) space and time O(n2/τ), thus providing the first smooth deterministic time-space trade-off from constant to linear space. The result uses a new and very simple algorithm, which computes a τ-additive approximation to the LCS in O(n2/τ) time and O(1) space. We also show a time-space trade-off lower bound for deterministic branching programs, which implies that any deterministic RAM algorithm solving the LCS problem on documents from a sufficiently large alphabet in O(τ) space must ω (n√ log(n/(τ log n))/ log log(n/(τ log n)) use time. © 2014 Springer-Verlag Berlin Heidelberg.