An Õ(n^5/4) Time ε-Approximation Algorithm for RMS Matching in a Plane
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The 2-Wasserstein distance (or RMS distance) is a useful measure of similarity between probability distributions that has exciting applications in machine learning. For discrete distributions, the problem of computing this distance can be expressed in terms of finding a minimum-cost perfect matching on a complete bipartite graph given by two multisets of points A,B ⊂ℝ^2, with |A|=|B|=n, where the ground distance between any two points is the squared Euclidean distance between them. Although there is a near-linear time relative ε-approximation algorithm for the case where the ground distance is Euclidean (Sharathkumar and Agarwal, JACM 2020), all existing relative ε-approximation algorithms for the RMS distance take Ω(n^3/2) time. This is primarily because, unlike Euclidean distance, squared Euclidean distance is not a metric. In this paper, for the RMS distance, we present a new ε-approximation algorithm that runs in O(n^5/4poly{log n,1/ε}) time. Our algorithm is inspired by a recent approach for finding a minimum-cost perfect matching in bipartite planar graphs (Asathulla et al., TALG 2020). Their algorithm depends heavily on the existence of sub-linear sized vertex separators as well as shortest path data structures that require planarity. Surprisingly, we are able to design a similar algorithm for a complete geometric graph that is far from planar and does not have any vertex separators. Central components of our algorithm include a quadtree-based distance that approximates the squared Euclidean distance and a data structure that supports both Hungarian search and augmentation in sub-linear time.
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