Fully Dynamic Euclidean Bi-Chromatic Matching in Sublinear Update Time

TL;DR

This work tackles maintaining a high-quality Euclidean bi-chromatic matching under fully dynamic updates, with applications to estimating the 1-Wasserstein distance for evolving distributions. It introduces a bottom-up, nested grid based framework built on a restricted $p$-tree, leveraging implicit representations of intermediate matchings and a Euclidean transportation solver inside cells. The authors prove an expected $O(1/\varepsilon)$-approximation with update time $O(n^{\varepsilon} \cdot \varepsilon^{-1})$, and augment this with an improved dynamic variant using augmenting paths to maintain explicit matchings efficiently. Empirical results on synthetic and real data demonstrate substantial speedups over static recomputation while preserving high accuracy, enabling real-time monitoring of distributional drift via Wasserstein distance estimates.

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.

Figures (5)

• Figure 1: Examples of an instance (left), implicit matching of the aggregated sub-instance (middle), and the resulting matching sub-instance (right). In the left picture, the red points are represented by circles, while the blue points are represented by crosses. The middle picture corresponds to the implicit matching on the aggregated sub-instance, with numbers next to vertices representing their demands and supplies, while numbers next to edges represent the weighting of the edges. The right picture shows the corresponding matching of the input points.
• Figure 2: Convergence and update time on large datasets.
• Figure 3: Drift in the Pickup-Dropoff distributions of the Taxi dataset (left) and update time (right).
• Figure 4: Speedup of dynamic algorithm over static for approximations with $p=8$.
• Figure 5: Estimating the $1$-Wasserstein distance using exact and approximate matchings.

Theorems & Definitions (30)

• theorem 1.1
• theorem 2.1
• lemma 3.1
• lemma 3.2
• theorem 3.3: Static
• theorem 4.1
• theorem C.1: Static
• proof : Proof of Theorem \ref{['thm:static']}
• claim C.1
• proof