Approximating General Metric Distances Between a Pattern and a Text

TL;DR

This paper introduces an efficient algorithm that approximates the sum of metric distances between a pattern and all substrings of a text, enabling faster computations in pattern matching tasks.

Contribution

It presents an $ ext{epsilon}$-approximation algorithm for summing metric distances between a pattern and text substrings with improved runtime complexity.

Findings

Algorithm achieves $O(rac{1}{ ext{epsilon}^2} n ext{polylog}(n, | ext{Sigma}|))$ runtime.

Provides a practical method for approximate metric distance calculations in pattern matching.

Enhances efficiency for large-scale text and pattern analysis.

Abstract

Let $T=t_0 ... t_{n-1}$ be a text and $P = p_0 ... p_{m-1}$ a pattern taken from some finite alphabet set $\Sigma$, and let $\dist$ be a metric on $\Sigma$. We consider the problem of calculating the sum of distances between the symbols of $P$ and the symbols of substrings of $T$ of length $m$ for all possible offsets. We present an $\epsilon$-approximation algorithm for this problem which runs in time $O(\frac{1}{\epsilon^2}n\cdot \mathrm{polylog}(n,\abs{\Sigma}))$