Years and Authors of Summarized Original Work
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1996; Bartal, Fakcharoenphol, Rao, Talwar
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2004; Bartal, Fakcharoenphol, Rao, Talwar
Problem Definition
This problem is to construct a random tree metric that probabilistically approximates a given arbitrary metric well. A solution to this problem is useful as the first step for numerous approximation algorithms because usually solving problems on trees is easier than on general graphs. It also finds applications in on-line and distributed computation.
It is known that tree metrics approximate general metrics badly, e.g., given a cycle C n with n nodes, any tree metric approximating this graph metric has distortion \( { \Omega(n) } \) [17]. However, Karp [15] noticed that a random spanning tree of C n approximates the distances between any two nodes in C n well in expectation. Alon, Karp, Peleg, and West [1] then proved a bound of \( { \exp(O(\sqrt{\log n\log\log n})) } \)on an average distortion for approximating any graph metric with its...
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Alon N, Karp RM, Peleg D, West D (1995) A graph-theoretic game and its application to the k-server problem. SIAM J Comput 24:78–100
Bartal Y (1996) Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS '96: proceedings of the 37th annual symposium on foundations of computer science, Washington, DC. IEEE Computer Society, pp 184–193
Bartal Y (1998) On approximating arbitrary metrices by tree metrics. In: STOC '98: proceedings of the thirtieth annual ACM symposium on theory of computing. ACM Press, New York, pp 161–168
Bartal Y, Charikar M, Raz D (2001) Approximating min-sum k-clustering in metric spaces. In: STOC '01: proceedings of the thirtythird annual ACM symposium on theory of computing. ACM Press, New York, pp 11–20
Calinescu G, Karloff H, Rabani Y (2001) Approximation algorithms for the 0-extension problem. In: SODA '01: proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, Philadelphia, pp 8–16
Charikar M, Chekuri C, Goel A, Guha S (1998) Rounding via trees: deterministic approximation algorithms for group Steiner trees and k-median. In: STOC '98: proceedings of the thirtieth annual ACM symposium on theory of computing. ACM Press, New York, pp 114–123
Elkin M, Emek Y, Spielman DA, Teng S-H (2005) Lower-stretch spanning trees. In: STOC '05: proceedings of the thirty-seventh annual ACM symposium on theory of computing. ACM Press, New York, pp 494–503
Fakcharoenphol J, Rao S, Talwar K (2004) Approximating metrics by tree metrics. SIGACT News 35:60–70
Fakcharoenphol J, Rao S, Talwar K (2004) A tight bound on approximating arbitrary metrics by tree metrics. J Comput Syst Sci 69:485–497
Gupta A (2001) Steiner points in tree metrics don't (really) help. In: SODA '01: proceedings of the twelfth annual ACM-SIAM symposium on discrete algorithms. Society for Industrial and Applied Mathematics, Philadelphia, pp 220–227
Gupta A, Hajiaghayi MT, Räcke H (2006) Oblivious network design. In: SODA '06: proceedings of the seventeenth annual ACM-SIAM symposium on discrete algorithm. ACM Press, New York, pp 970–979
Gupta A, Talwar K (2006) Approximating unique games. In: SODA '06: proceedings of the seventeenth annual ACM-SIAM symposium on discrete algorithm, New York. ACM Press, New York, pp 99–106
Hayrapetyan A, Swamy C, Tardos É (2005) Network design for information networks. In: SODA '05: proceedings of the sixteenth annual ACM-SIAM symposium on discrete algorithms. Society for Industrial and Applied Mathematics, Philadelphia, pp 933–942
Indyk P, Matousek J (2004) Low-distortion embeddings of finite metric spaces. In: Goodman JE, O'Rourke J (eds) Handbook of discrete and computational geometry. Chapman&Hall/CRC, Boca Raton, chap. 8
Karp R (1989) A 2k-competitive algorithm for the circle. Manuscript
Matousek J (2002) Lectures on discrete geometry. Springer, New York
Rabinovich Y, Raz R (1998) Lower bounds on the distortion of embedding finite metric spaces in graphs. Discret Comput Geom 19:79–94
Seymour PD (1995) Packing directed circuits fractionally. Combinatorica 15:281–288
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Fakcharoenphol, J., Rao, S., Talwar, K. (2016). Approximating Metric Spaces by Tree Metrics. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_25
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