Given a weighted planar bipartite graph G ( A ∪ B , E ) where each edge has an integer edge cost, we give an Õ( n 4/3 log nC ) time algorithm to compute minimumcost perfect matching; here C is the maximum edge cost in the graph. The previous bestknown planarity exploiting algorithm has a running time of O ( n 3/2 log n ) and is achieved by using planar separators (Lipton and Tarjan ’80). Our algorithm is based on the bitscaling paradigm (Gabow and Tarjan ’89). For each scale, our algorithm first executes O ( n 1/3 ) iterations of Gabow and Tarjan’s algorithm in O ( n 4/3 ) time leaving only O ( n 2/3 ) vertices unmatched. Next, it constructs a compressed residual graph H with O ( n 2/3 ) vertices and O ( n ) edges. This is achieved by using an r division of the planar graph G with r = n 2/3 . For each partition of the r division, there is an edge between two vertices of H if and only if they are connected by a directed path inside the partition. Using existing efficient shortestpath data structures, themore »
An O(n5/4) Time ∊Approximation Algorithm for RMS Matching in a Plane
The 2Wasserstein distance (or RMS distance) is a useful measure of similarity between probability distributions with exciting applications in machine learning. For discrete distributions, the problem of computing this distance can be expressed in terms of finding a minimumcost perfect matching on a complete bipartite graph given by two multisets of points A, B ⊂ ℝ2, with A = B = n, where the ground distance between any two points is the squared Euclidean distance between them. Although there is a nearlinear time relative ∊approximation algorithm for the case where the ground distance is Euclidean (Sharathkumar and Agarwal, JACM 2020), all existing relative ∊approximation algorithms for the RMS distance take Ω(n3/2) 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 ∊approximation algorithm that runs in O(n^5/4 poly{log n, 1/∊}) time. Our algorithm is inspired by a recent approach for finding a minimumcost perfect matching in bipartite planar graphs (Asathulla et al, TALG 2020). Their algorithm depends heavily on the existence of sublinear sized vertex separators as well as shortest path data structures that require planarity. Surprisingly, we are able to design a similar more »
 Award ID(s):
 1909171
 Publication Date:
 NSFPAR ID:
 10232551
 Journal Name:
 Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms
 Page Range or eLocationID:
 869  888
 ISSN:
 15579468
 Sponsoring Org:
 National Science Foundation
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