research-article

Self-improving algorithms for coordinate-wise maxima

Authors:

Kenneth L. Clarkson,

Wolfgang Mulzer, and

C. SeshadhriAuthors Info & Claims

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

Pages 277 - 286

https://doi.org/10.1145/2261250.2261291

Published: 17 June 2012 Publication History

Abstract

Computing the coordinate-wise maxima of a planar point set is a classic and well-studied problem in computational geometry. We give an algorithm for this problem in the self-improving setting. We have n (unknown) independent distributions cD₁, cD₂, ..., cD_n of planar points. An input pointset (p₁, p₂, ..., p_n) is generated by taking an independent sample p_i from each cD_i, so the input distribution cD is the product prod_i cD_i. A self-improving algorithm repeatedly gets input sets from the distribution cD (which is a priori unknown) and tries to optimize its running time for cD. Our algorithm uses the first few inputs to learn salient features of the distribution, and then becomes an optimal algorithm for distribution cD. Let OPT_cD denote the expected depth of an optimal linear comparison tree computing the maxima for distribution cD. Our algorithm eventually has an expected running time of O(OPT_cD + n), even though it did not know cD to begin with.

Our result requires new tools to understand linear comparison trees for computing maxima. We show how to convert general linear comparison trees to very restricted versions, which can then be related to the running time of our algorithm. An interesting feature of our algorithm is an interleaved search, where the algorithm tries to determine the likeliest point to be maximal with minimal computation. This allows the running time to be truly optimal for the distribution cD.

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Cited By

Gupta RRoughgarden TSudan M(2016)A PAC Approach to Application-Specific Algorithm SelectionProceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science10.1145/2840728.2840766(123-134)Online publication date: 14-Jan-2016
https://dl.acm.org/doi/10.1145/2840728.2840766
Afshani PChekuri C(2014)Fast computation of output-sensitive maxima in a word RAMProceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms10.5555/2634074.2634178(1414-1423)Online publication date: 5-Jan-2014
https://dl.acm.org/doi/10.5555/2634074.2634178
Xing CXiong ZZhang YWu XDan JZhang T(2014)An Efficient Convex Hull Algorithm Using Affine Transformation in Planar Point SetArabian Journal for Science and Engineering10.1007/s13369-014-1365-339:11(7785-7793)Online publication date: 28-Aug-2014
https://doi.org/10.1007/s13369-014-1365-3

Index Terms

Self-improving algorithms for coordinate-wise maxima
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Published In

cover image ACM Conferences

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

436 pages

ISBN:9781450312998

DOI:10.1145/2261250

Program Chairs:
Tamal Dey
The Ohio State University, USA
,
Sue Whitesides
University of Victoria, Canada

Copyright © 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 17 June 2012

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SoCG '12

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SoCG '12: Symposium on Computational Geometry 2012

June 17 - 20, 2012

North Carolina, Chapel Hill, USA

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Cited By

Gupta RRoughgarden TSudan M(2016)A PAC Approach to Application-Specific Algorithm SelectionProceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science10.1145/2840728.2840766(123-134)Online publication date: 14-Jan-2016
https://dl.acm.org/doi/10.1145/2840728.2840766
Afshani PChekuri C(2014)Fast computation of output-sensitive maxima in a word RAMProceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms10.5555/2634074.2634178(1414-1423)Online publication date: 5-Jan-2014
https://dl.acm.org/doi/10.5555/2634074.2634178
Xing CXiong ZZhang YWu XDan JZhang T(2014)An Efficient Convex Hull Algorithm Using Affine Transformation in Planar Point SetArabian Journal for Science and Engineering10.1007/s13369-014-1365-339:11(7785-7793)Online publication date: 28-Aug-2014
https://doi.org/10.1007/s13369-014-1365-3

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