On tail estimates for Randomized Incremental Construction

TL;DR

This paper develops new techniques to derive tail estimates for randomized incremental constructions in computational geometry, improving understanding of their probabilistic running time bounds and limitations.

Contribution

It introduces a novel application of Freedman's inequality for Martingale concentration to obtain tail bounds for RIC algorithms, addressing a longstanding open problem.

Findings

Established tail bounds for RIC algorithms in geometric problems.

Identified cases where inverse polynomial tail bounds do not hold.

Provided probabilistic analysis of trapezoidal map construction time.

Abstract

By combining several interesting applications of random sampling in geometric algorithms like point location, linear programming, segment intersections, binary space partitioning, Clarkson and Shor \cite{CS89} developed a general framework of randomized incremental construction (RIC ). The basic idea is to add objects in a random order and show that this approach yields efficient/optimal bounds on {\bf expected} running time. Even quicksort can be viewed as a special case of this paradigm. However, unlike quicksort, for most of these problems, attempts to obtain sharper tail estimates on the running time had proved inconclusive. Barring some results by \cite{MSW93,CMS92,Seidel91a}, the general question remains unresolved. In this paper we present some general techniques to obtain tail estimates for RIC and and provide applications to some fundamental problems like Delaunay triangulations and construction of Visibility maps of intersecting line segments. The main result of the paper centers around a new and careful application of Freedman's \cite{Fre75} inequality for Martingale concentration that overcomes the bottleneck of the better known Azuma-Hoeffding inequality. Further, we show instances where an RIC based algorithm may not have inverse polynomial tail estimates. In particular, we show that the RIC time bounds for trapezoidal map can encounter a running time of $\Omega (n\log n\log\log n )$ with probability exceeding $\frac{1}{\sqrt{n}}$. This rules out inverse polynomial concentration bounds around the expected running time.