Pattern Matching in Multiple Streams

TL;DR

This paper explores efficient deterministic pattern matching algorithms in multiple streaming texts, providing optimal space bounds and solutions for exact, k-mismatch, and k-difference problems in a streaming context.

Contribution

It introduces a new streaming model for pattern matching, offering near-optimal algorithms and space bounds for multiple pattern matching problems.

Findings

Exact matching solved in constant time per symbol with O(m+s) space.

k-mismatch and k-difference problems solved in O(k) time with O(m+ks) space.

Proved space bounds are nearly optimal, up to a logarithmic factor.

Abstract

We investigate the problem of deterministic pattern matching in multiple streams. In this model, one symbol arrives at a time and is associated with one of s streaming texts. The task at each time step is to report if there is a new match between a fixed pattern of length m and a newly updated stream. As is usual in the streaming context, the goal is to use as little space as possible while still reporting matches quickly. We give almost matching upper and lower space bounds for three distinct pattern matching problems. For exact matching we show that the problem can be solved in constant time per arriving symbol and O(m+s) words of space. For the k-mismatch and k-difference problems we give O(k) time solutions that require O(m+ks) words of space. In all three cases we also give space lower bounds which show our methods are optimal up to a single logarithmic factor. Finally we set out a number of open problems related to this new model for pattern matching.