Sublinear Space Algorithms for the Longest Common Substring Problem

TL;DR

This paper introduces a novel deterministic algorithm that offers a smooth trade-off between space and time for the longest common substring problem, reducing space usage while maintaining efficiency.

Contribution

It presents the first deterministic time-space trade-off algorithm for LCS, achieving sublinear space with a simple approach and establishing lower bounds for deterministic algorithms.

Findings

Achieves $O( au)$ space and $O(n^2/ au)$ time for LCS

Provides a simple $ au$-additive approximation algorithm

Establishes lower bounds for deterministic RAM algorithms

Abstract

Given $m$ documents of total length $n$, we consider the problem of finding a longest string common to at least $d \geq 2$ of the documents. This problem is known as the \emph{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 $1 \leq \tau \leq n$, the LCS problem can be solved in $O(\tau)$ space and $O(n^2/\tau)$ time, 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 $\tau$-additive approximation to the LCS in $O(n^2/\tau)$ 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 $O(\tau)$ space must use $\Omega(n\sqrt{\log(n/(\tau\log n))/\log\log(n/(\tau\log n)})$ time.