When Crossings Count - Approximating the Minimum Spanning Tree

TL;DR

This paper introduces an approximation algorithm for the minimum spanning tree based on crossing metrics in planar line arrangements and demonstrates how to embed these metrics into high-dimensional spaces for various geometric problems.

Contribution

It presents a novel (1+μ)-approximation algorithm for crossing-based MSTs and a method to embed crossing metrics into high-dimensional spaces for broader algorithmic applications.

Findings

Provides an efficient approximation algorithm with expected runtime O((n/μ^5) alpha^3(n) log^5 n)

Shows how to embed crossing metrics into high-dimensional spaces preserving distances

Enables use of existing subquadratic algorithms for problems like matching and clustering with crossing metrics

Abstract

In the first part of the paper, we present an (1+\mu)-approximation algorithm to the minimum-spanning tree of points in a planar arrangement of lines, where the metric is the number of crossings between the spanning tree and the lines. The expected running time is O((n/\mu^5) alpha^3(n) log^5 n), where \mu > 0 is a prescribed constant. In the second part of our paper, we show how to embed such a crossing metric, into high-dimensions, so that the distances are preserved. As a result, we can deploy a large collection of subquadratic approximations algorithms \cite im-anntr-98,giv-rahdg-99 for problems involving points with the crossing metric as a distance function. Applications include matching, clustering, nearest-neighbor, and furthest-neighbor.