Worst case efficient single and multiple string matching in the Word-RAM model

TL;DR

This paper develops worst-case efficient data structures for single and multiple string matching in the word RAM model, achieving near-optimal query times with linear space, advancing theoretical bounds in pattern matching algorithms.

Contribution

It introduces new linear space data structures that approach the optimal query time for string matching problems in the word RAM model, including techniques to make query times independent of pattern length.

Findings

Single pattern matching query time: O(n(1/m+log sigma/w)+occ)

Multiple pattern matching query time: O(n((log d+log y+log log d)/y+log sigma/w)+occ)

Techniques to achieve pattern-length-independent query times using the four Russian method

Abstract

In this paper, we explore worst-case solutions for the problems of single and multiple matching on strings in the word RAM model with word length w. In the first problem, we have to build a data structure based on a pattern p of length m over an alphabet of size sigma such that we can answer to the following query: given a text T of length n, where each character is encoded using log(sigma) bits return the positions of all the occurrences of p in T (in the following we refer by occ to the number of reported occurrences). For the multi-pattern matching problem we have a set S of d patterns of total length m and a query on a text T consists in finding all positions of all occurrences in T of the patterns in S. As each character of the text is encoded using log sigma bits and we can read w bits in constant time in the RAM model, we assume that we can read up to (w/log sigma) consecutive characters of the text in one time step. This implies that the fastest possible query time for both problems is O((n(log sigma/w)+occ). In this paper we present several different results for both problems which come close to that best possible query time. We first present two different linear space data structures for the first and second problem: the first one answers to single pattern matching queries in time O(n(1/m+log sigma/w)+occ) while the second one answers to multiple pattern matching queries to O(n((log d+log y+log log d)/y+log sigma/w)+occ) where y is the length of the shortest pattern in the case of multiple pattern-matching. We then show how a simple application of the four russian technique permits to get data structures with query times independent of the length of the shortest pattern (the length of the only pattern in case of single string matching) at the expense of using more space.