String Matching with Variable Length Gaps

TL;DR

This paper introduces an efficient, simple algorithm for string matching with variable length gaps, optimizing time and space complexity for applications like computational biology.

Contribution

It presents a new algorithm that achieves near-optimal time and space bounds for pattern matching with variable length gaps, improving upon previous methods.

Findings

Achieves time complexity O(n log k + m + α)

Uses space O(m + A) where A is gap length sum

Provides algorithms for position finding and encoding

Abstract

We consider string matching with variable length gaps. Given a string $T$ and a pattern $P$ consisting of strings separated by variable length gaps (arbitrary strings of length in a specified range), the problem is to find all ending positions of substrings in $T$ that match $P$. This problem is a basic primitive in computational biology applications. Let $m$ and $n$ be the lengths of $P$ and $T$, respectively, and let $k$ be the number of strings in $P$. We present a new algorithm achieving time $O(n\log k + m +\alpha)$ and space $O(m + A)$, where $A$ is the sum of the lower bounds of the lengths of the gaps in $P$ and $\alpha$ is the total number of occurrences of the strings in $P$ within $T$. Compared to the previous results this bound essentially achieves the best known time and space complexities simultaneously. Consequently, our algorithm obtains the best known bounds for almost all combinations of $m$, $n$, $k$, $A$, and $\alpha$. Our algorithm is surprisingly simple and straightforward to implement. We also present algorithms for finding and encoding the positions of all strings in $P$ for every match of the pattern.