Approximating the Geometric Edit Distance

TL;DR

This paper introduces the first sublinear approximate algorithm for computing the geometric edit distance between point sequences in Euclidean space, achieving near-linear time with high probability.

Contribution

It presents the first strictly sublinear, near-linear time randomized algorithms for approximating the geometric edit distance in constant-dimensional Euclidean space.

Findings

Achieves a randomized O(n log^2 n) time algorithm with a √n approximation ratio.

Generalizes to an α-approximation with adjustable trade-offs, running in O(n^2 / α^2 log n) time.

Algorithms are Monte Carlo, providing high-probability approximate solutions.

Abstract

Edit distance is a measurement of similarity between two sequences such as strings, point sequences, or polygonal curves. Many matching problems from a variety of areas, such as signal analysis, bioinformatics, etc., need to be solved in a geometric space. Therefore, the geometric edit distance (GED) has been studied. In this paper, we describe the first strictly sublinear approximate near-linear time algorithm for computing the GED of two point sequences in constant dimensional Euclidean space. Specifically, we present a randomized (O(n\log^2n)) time (O(\sqrt n))-approximation algorithm. Then, we generalize our result to give a randomized $\alpha$-approximation algorithm for any $\alpha\in [\sqrt{\log n}, \sqrt{n / \log n}]$, running in time $O(n^2/\alpha^2 \log n)$. Both algorithms are Monte Carlo and return approximately optimal solutions with high probability.