A Fast Heuristic for Exact String Matching

TL;DR

This paper introduces a randomized heuristic for exact string matching that preprocesses a pattern to identify a sparse substring, enabling faster search times especially for patterns with certain character distributions.

Contribution

The paper proposes a novel randomized heuristic that preprocesses patterns to efficiently find all occurrences in the text, improving search times based on sparse substring identification.

Findings

Preprocessing time is $O(n ext{delta})$.

Expected search time is $O( m / ext{min}(|sparse(P)|, ext{Delta}) )$.

Expected sparse substring length is $ ext{Omega}( ext{Delta} imes ext{log}(rac{2 ext{Delta}}{2 ext{Delta}- ext{delta}}))$ for random patterns.

Abstract

Given a pattern string $P$ of length $n$ consisting of $\delta$ distinct characters and a query string $T$ of length $m$, where the characters of $P$ and $T$ are drawn from an alphabet $\Sigma$ of size $\Delta$, the {\em exact string matching} problem consists of finding all occurrences of $P$ in $T$. For this problem, we present a randomized heuristic that in $O(n\delta)$ time preprocesses $P$ to identify $sparse(P)$, a rarely occurring substring of $P$, and then use it to find all occurrences of $P$ in $T$ efficiently. This heuristic has an expected search time of $O( \frac{m}{min(|sparse(P)|, \Delta)})$, where $|sparse(P)|$ is at least $\delta$. We also show that for a pattern string $P$ whose characters are chosen uniformly at random from an alphabet of size $\Delta$, $E[|sparse(P)|]$ is $\Omega(\Delta log (\frac{2\Delta}{2\Delta-\delta}))$.