Randomization yields simple O(n log star n) algorithms for difficult Omega(n) problems

TL;DR

This paper demonstrates that by leveraging influence graph results and additional information, randomized algorithms can be optimized from O(n log n) to O(n log* n) for certain computational geometry problems.

Contribution

It introduces a technique that improves the expected complexity of randomized algorithms for specific geometric problems using influence graph insights.

Findings

Expected complexity reduced to O(n log* n)

Applicable to triangulation and Delaunay triangulation with extra info

Enhances efficiency of geometric algorithms

Abstract

We use here the results on the influence graph by Boissonnat et al. to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from O(n log n) to O(n log star n). This technique applies in the following applications: triangulation of a simple polygon, skeleton of a simple polygon, Delaunay triangulation of points knowing the EMST (euclidean minimum spanning tree).