License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2019.58
URN: urn:nbn:de:0030-drops-102977
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10297/
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Sen, Sandeep

A Unified Approach to Tail Estimates for Randomized Incremental Construction

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LIPIcs-STACS-2019-58.pdf (0.7 MB)

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 [Kenneth L. Clarkson and Peter W. Shor, 1989] 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 expected running time. Even quicksort can be viewed as a special case of this paradigm. However, unlike quicksort, for most of these problems, sharper tail estimates on their running times are not known. Barring some promising attempts in [Kurt Mehlhorn et al., 1993; Kenneth L. Clarkson et al., 1992; Raimund Seidel, 1991], the general question remains unresolved.
In this paper we present a general technique 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 is derived from a new and careful application of Freedman's [David Freedman, 1975] inequality for Martingale concentration that overcomes the bottleneck of the better known Azuma-Hoeffding inequality. Further, we explore 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 1/(sqrt{n)}. This rules out inverse polynomial concentration bounds within a constant factor of the O(n log n) expected running time.

BibTeX - Entry

@InProceedings{sen:LIPIcs:2019:10297,
  author =	{Sandeep Sen},
  title =	{{A Unified Approach to Tail Estimates for Randomized Incremental Construction}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{58:1--58:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Rolf Niedermeier and Christophe Paul},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10297},
  doi =		{10.4230/LIPIcs.STACS.2019.58},
  annote =	{Keywords: ric, tail estimates, martingale, lower bound}
}

Keywords: ric, tail estimates, martingale, lower bound
Collection: 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)
Issue Date: 2019
Date of publication: 12.03.2019

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