Optimal pattern matching algorithms

TL;DR

This paper introduces a class of finite state machines called $w$-matching machines to analyze and optimize pattern matching algorithms' asymptotic speed on iid texts, providing a method to find the best algorithm within a certain class.

Contribution

It defines $w$-matching machines and proves the existence of an optimal machine that maximizes asymptotic speed for a given pattern and iid model.

Findings

Existence of an optimal $w$-matching machine for given parameters.

Method to compute the maximum asymptotic speed among a class of algorithms.

Construction of an algorithm achieving the optimal speed.

Abstract

We study a class of finite state machines, called \defi{$w$-matching machines}, which yield to simulate the behavior of pattern matching algorithms while searching for a pattern $w$. They can be used to compute the asymptotic speed, i.e. the limit of the expected ratio of the number of text accesses to the length of the text, of algorithms while parsing an iid text to find the pattern $w$. Defining the order of a matching machine or of an algorithm as the maximum difference between the current and accessed positions during a search (standard algorithms are generally of order $|w|$), we show that being given a pattern $w$, an order $k$ and an iid model, there exists an optimal $w$-matching machine, i.e. with the greatest asymptotic speed under the model among all the machines of order $k$, of which the set of states belongs to a finite and enumerable set. It shows that it is possible to determine: 1) the greatest asymptotic speed among a large class of algorithms, with regard to a pattern and an iid model, and 2) a $w$-matching machine, thus an algorithm, achieving this speed.