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 max-finding 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
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An O(log n)-competitive algorithm for max-finding on general metrics, and we improve on this result to obtain an O(1)-competitive algorithm for the max-finding problem in constant dimensional spaces.
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An O(log2 n)-competitive algorithm for sorting in general metric spaces.
Our main technique for max-finding is to run two copies of a simple natural online algorithm (that costs too much when run by itself) in parallel. By judiciously exchanging information between the two copies, we can bound the cost incurred by the algorithm; we believe that this technique may have other applications to online algorithms.
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References
Knuth, D.E.: The art of computer programming. Sorting and searching, vol. 3. Addison- Wesley Publishing Co., Reading (1973)
Charikar, M., Fagin, R., Guruswami, V., Kleinberg, J., Raghavan, P., Sahai, A.: Query strategies for priced information. In: Proc. 32nd ACM STOC, pp. 582–591 (2000)
Gupta, A., Kumar, A.: Sorting and selection with structured costs. In: Proc. 42nd IEEE FOCS, pp. 416–425 (2001)
Kannan, S., Khanna, S.: Selection with monotone comparison costs. In: Proc. 14th ACM SIAM SODA, pp. 10–17 (2003)
Bartal, Y.: Probabilistic approximations of metric spaces and its algorithmic applications. In: Proc. 37th IEEE FOCS, pp. 184–193 (1996)
Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: Proc. 35th ACM STOC, pp. 448–455 (2003)
Hartline, J., Hong, E., Mohr, A., Rocke, E., Yasuhara, K.: As reported in [3]
Kleinberg, J.: Detecting a network failure. Internet Math 1, 37–55 (2003)
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© 2005 Springer-Verlag Berlin Heidelberg
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Gupta, A., Kumar, A. (2005). Where’s the Winner? Max-Finding and Sorting with Metric Costs. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_7
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DOI: https://doi.org/10.1007/11538462_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28239-6
Online ISBN: 978-3-540-31874-3
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