Fully Dynamic Euclidean Bi-Chromatic Matching in Sublinear Update Time

2025-05-13

AI Summary

This paper studies the fully dynamic minimum-cost matching problem for bichromatic point sets in Euclidean space, supporting efficient point insertions and deletions while maintaining a real-time approximation of the Wasserstein distance. We propose the first dynamic algorithm with sublinear update time: under an $O(1/varepsilon)$-approximation guarantee, each update requires only $O(n^varepsilon)$ time—breaking the efficiency bottlenecks of static and semi-dynamic approaches. Our method integrates a geometric hierarchical decomposition, dynamic grid partitioning, local re-optimization, approximate nearest neighbor search, and a matching quality monitoring mechanism. The algorithm provides strong theoretical guarantees on both approximation ratio and update complexity. Experiments on real-world and synthetic datasets demonstrate that our method accelerates Wasserstein distance drift detection by several orders of magnitude over baseline methods, achieving both theoretical rigor and practical efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Dynamic Euclidean bi-chromatic matching with sublinear updates
• Approximating Wasserstein distance via geometric optimization
• Efficient monitoring of distributional drift in real-world data

Innovation

• Fully dynamic algorithm for bi-chromatic matching
• Sub-linear update time with approximation guarantee
• Effective monitoring of Wasserstein distance drift