Computing Exact Bottleneck Distance on Random Point Sets

• Main Author: Ye, Jiacheng
• Other Authors: Computer Science
• Format: Others
• Published: Virginia Tech 2020
• Subjects: bipartite graph; bottleneck matching
• Online Access: http://hdl.handle.net/10919/98669

Given a complete bipartite graph on two sets of points containing n points each, in a bottleneck matching problem, we want to find an one-to-one correspondence, also called a matching, that minimizes the length of its largest edge; the length of an edge is simply the Euclidean distance between its end-points. As an application, consider matching taxis to requests while minimizing the largest distance between any request to its matched taxi. The length of the largest edge (also called the bottleneck distance) has numerous applications in machine learning as well as topological data analysis. One can use the classical Hopcroft-Karp (HK-) Algorithm to find the bottleneck matching. In this thesis, we consider the case where A and B are points that are generated uniformly at random from a unit square. Instead of the classical HK-Algorithm, we implement and empirically analyze a new algorithm by Lahn and Raghvendra (Symposium on Computational Geometry, 2019). Our experiments show that our approach outperforms the HK-Algorithm based approach for computing bottleneck matching. === Master of Science === Consider the problem of matching taxis to an equal number of requests. While matching them, one objective is to minimize the largest distance between a request and its match. Finding such a matching is called the bottleneck matching problem. In addition, this optimization problem arises in topological data analysis as well as machine learning. In this thesis, I conduct an empirical analysis of a new algorithm, which is called the FAST-MATCH algorithm, to find the bottleneck matching. I find that, when a large input data is randomly generated from a unit square, the FAST-MATCH algorithm performs substantially faster than the classical methods