Error Tree: A Tree Structure for Hamming & Edit Distances & Wildcards Matching

TL;DR

The paper introduces Error Tree, a new data structure designed to efficiently solve approximate pattern matching problems involving Hamming distance, edit distance, and wildcards, with optimized query times and space complexity.

Contribution

It presents the Error Tree, a novel tree structure that improves the efficiency of approximate pattern matching for Hamming, edit distances, and wildcards, with detailed complexity analysis.

Findings

Efficient query times for Hamming and wildcards matching.

A space-efficient tree structure for edit distance matching.

Theoretical bounds on performance for various matching problems.

Abstract

Error Tree is a novel tree structure that is mainly oriented to solve the approximate pattern matching problems, Hamming and edit distances, as well as the wildcards matching problem. The input is a text of length $n$ over a fixed alphabet of length $\Sigma$, a pattern of length $m$, and $k$. The output is to find all positions that have $\leq$ $k$ Hamming distance, edit distance, or wildcards matching with $P$. The algorithm proposes for Hamming distance and wildcards matching a tree structure that needs $O(n\frac{log_\Sigma ^{k}n}{k!})$ words and takes $O(\frac {m^k}{k!} + occ$)($O(m + \frac {log_\Sigma ^kn}{k!} + occ$) in the average case) of query time for any online/offline pattern, where $occ$ is the number of outputs. As well, a tree structure of $O(2^{k}n\frac{log_\Sigma ^{k}n}{k!})$ words and $O(\frac {m^k}{k!} + 3^{k}occ$)($O(m + \frac {log_\Sigma ^kn}{k!} + 3^{k}occ$) in the average case) query time for edit distance for any online/offline pattern.