Approximating the Geometric Edit Distance

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.