Fully Dynamic Euclidean Bi-Chromatic Matching in Sublinear Update Time

TL;DR

This paper introduces the first fully dynamic Euclidean bi-chromatic matching algorithm with sublinear update time, enabling efficient distributional analysis and outperforming existing methods in real-world and synthetic datasets.

Contribution

It presents a novel fully dynamic algorithm for Euclidean bi-chromatic matching with sublinear update time and provable approximation guarantees.

Findings

Achieves $O(1/\varepsilon)$-approximation with $O(n^{\varepsilon})$ update time.

Enables effective monitoring of distributional drift in Wasserstein distance.

Outperforms baseline methods in runtime on real and synthetic data.

Abstract

We consider the Euclidean bi-chromatic matching problem in the dynamic setting, where the goal is to efficiently process point insertions and deletions while maintaining a high-quality solution. Computing the minimum cost bi-chromatic matching is one of the core problems in geometric optimization that has found many applications, most notably in estimating Wasserstein distance between two distributions. In this work, we present the first fully dynamic algorithm for Euclidean bi-chromatic matching with sub-linear update time. For any fixed $\varepsilon > 0$, our algorithm achieves $O(1/\varepsilon)$-approximation and handles updates in $O(n^{\varepsilon})$ time. Our experiments show that our algorithm enables effective monitoring of the distributional drift in the Wasserstein distance on real and synthetic data sets, while outperforming the runtime of baseline approximations by orders of magnitudes.