Where’s the Winner? MaxFinding and Sorting with Metric Costs
 Anupam Gupta,
 Amit Kumar
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Abstract
Traditionally, a fundamental assumption in evaluating the performance of algorithms for sorting and selection has been that comparing any two elements costs one unit (of time, work, etc.); the goal of an algorithm is to minimize the total cost incurred. However, a body of recent work has attempted to find ways to weaken this assumption – in particular, new algorithms have been given for these basic problems of searching, sorting and selection, when comparisons between different pairs of elements have different associated costs.
In this paper, we further these investigations, and address the questions of maxfinding and sorting when the comparison costs form a metric; i.e., the comparison costs c _{ uv } respect the triangle inequality c _{ uv } + c _{ vw } ≥ c _{ uw } for all input elements u,v and w. We give the first results for these problems – specifically, we present

An O(log n)competitive algorithm for maxfinding on general metrics, and we improve on this result to obtain an O(1)competitive algorithm for the maxfinding problem in constant dimensional spaces.

An O(log^{2} n)competitive algorithm for sorting in general metric spaces.
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 Title
 Where’s the Winner? MaxFinding and Sorting with Metric Costs
 Book Title
 Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques
 Book Subtitle
 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th International Workshop on Randomization and Computation, RANDOM 2005, Berkeley, CA, USA, August 2224, 2005. Proceedings
 Pages
 pp 7485
 Copyright
 2005
 DOI
 10.1007/11538462_7
 Print ISBN
 9783540282396
 Online ISBN
 9783540318743
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 3624
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Chandra Chekuri ^{(16)}
 Klaus Jansen ^{(17)}
 José D. P. Rolim ^{(18)}
 Luca Trevisan ^{(19)}
 Editor Affiliations

 16. Dept. of Computer Science, University of Illinois
 17. Institute for Computer Science, University of Kiel
 18. Battelle Bâtiment A, Centre Universitaire d’Informatique
 19. UC Berkeley
 Authors

 Anupam Gupta ^{(20)}
 Amit Kumar ^{(21)}
 Author Affiliations

 20. Dept. of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 15213, USA
 21. Dept. of Computer Science & Engineering, Indian Institute of Technology, Hauz Khas, New Delhi, India, 110016
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