Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings

TL;DR

This paper introduces linear-time algorithms for geometric graphs with sublinearly many crossings, enabling efficient solutions for problems like Voronoi diagrams and shortest paths without complex assumptions.

Contribution

The authors develop a planarization-based approach that achieves linear-time algorithms for geometric graphs with sublinear crossings, solving an open problem in triangulating self-intersecting polygons.

Findings

Algorithms run in O(n) time for graphs with k sublinear crossings.

The approach extends planar separator methods to complex geometric graphs.

Linear-time triangulation of self-intersecting polygons with sublinear crossings.

Abstract

We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated logarithmic factor. Specific problems we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in the standard comparison-based computational model; hence, we make no assumptions about the distribution or bit complexities of edge weights, nor do we utilize unusual bit-level operations on memory words. Instead, our algorithms are based on a planarization method that "zeroes in" on edge crossings, together with methods for extending planar separator decompositions to geometric graphs with sublinearly many crossings. Incidentally, our planarization algorithm also solves an open computational geometry problem of Chazelle for triangulating a self-intersecting polygonal chain having n segments and k crossings in linear time, for the case when k is sublinear in n by an iterated logarithmic factor.