A tight bound on approximating arbitrary metrics by tree metrics

In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion Ω(log n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.

[1]  Uriel Feige,et al.  Approximating the bandwidth via volume respecting embeddings (extended abstract) , 1998, STOC '98.

[2]  Yair Bartal,et al.  Probabilistic approximation of metric spaces and its algorithmic applications , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[3]  Noga Alon,et al.  A Graph-Theoretic Game and Its Application to the k-Server Problem , 1995, SIAM J. Comput..

[4]  David Peleg,et al.  Deterministic Polylog Approximation for Minimum Communication Spanning Trees , 1998, ICALP.

[5]  Sudipto Guha,et al.  Approximating a finite metric by a small number of tree metrics , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[6]  Chuan Yi Tang,et al.  A polynomial time approximation scheme for minimum routing cost spanning trees , 1998, SODA '98.

[7]  R. Ravi,et al.  An approximation algorithm for the covering Steiner problem , 2000, SODA '00.

[8]  Amos Fiat,et al.  Better algorithms for unfair metrical task systems and applications , 2000, STOC '00.

[9]  Éva Tardos,et al.  Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[10]  Yuval Rabani,et al.  An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm , 1998, SIAM J. Comput..

[11]  Ilan Newman,et al.  A Lower Bound on the Distortion of Embedding Planar Metrics into Euclidean Space , 2002, SCG '02.

[12]  Sudipto Guha,et al.  Rounding via Trees : Deterministic Approximation Algorithms forGroup , 1998 .

[13]  Yair Bartal,et al.  Multi-embedding and path approximation of metric spaces , 2003, SODA '03.

[14]  Joseph Naor,et al.  Approximation algorithms for the metric labeling problem via a new linear programming formulation , 2001, SODA '01.

[15]  Yossi Azar,et al.  Buy-at-bulk network design , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[16]  Rajmohan Rajaraman,et al.  Placement Algorithms for Hierarchical Cooperative Caching , 2001, J. Algorithms.

[17]  Joseph Naor,et al.  Divide-and-conquer approximation algorithms via spreading metrics , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[18]  Adi Rosén,et al.  The distributed k-server problem-a competitive distributed translator for k-server algorithms , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[19]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[20]  Yair Bartal,et al.  On approximating arbitrary metrices by tree metrics , 1998, STOC '98.

[21]  Joseph Naor,et al.  Divide-and-Conquer Approximation Algorithms via Spreading Metrics (Extended Abstract). , 1995, FOCS 1995.

[22]  Philip N. Klein,et al.  Excluded minors, network decomposition, and multicommodity flow , 1993, STOC.

[23]  Moses Charikar,et al.  Approximating min-sum k-clustering in metric spaces , 2001, STOC '01.

[24]  R. Ravi,et al.  A polylogarithmic approximation algorithm for the group Steiner tree problem , 2000, SODA '98.

[25]  Ran Raz,et al.  Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs , 1998, Discret. Comput. Geom..

[26]  Jirí Matousek,et al.  Low-Distortion Embeddings of Finite Metric Spaces , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[27]  R. Ravi,et al.  On approximating planar metrics by tree metrics , 2001, Inf. Process. Lett..

[28]  Baruch Awerbuch,et al.  Topology aggregation for directed graph , 1998, Proceedings Third IEEE Symposium on Computers and Communications. ISCC'98. (Cat. No.98EX166).

[29]  Maurice Herlihy,et al.  Competitive concurrent distributed queuing , 2001, PODC '01.

[30]  Lixia Zhang,et al.  On the placement of Internet instrumentation , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[31]  Robert Krauthgamer,et al.  Approximate classification via earthmover metrics , 2004, SODA '04.

[32]  Robert Krauthgamer,et al.  Measured Descent: A New Embedding Method for Finite Metrics , 2004, FOCS.

[33]  Yuval Rabani,et al.  Approximation algorithms for the 0-extension problem , 2001, SODA '01.

[34]  Anupam Gupta,et al.  Cuts, Trees and ℓ1-Embeddings of Graphs* , 2004, Comb..

[35]  J. Bourgain.  On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .

[36]  Satish Rao,et al.  Small distortion and volume preserving embeddings for planar and Euclidean metrics , 1999, SCG '99.

[37]  Anupam Gupta,et al.  Embedding k-outerplanar graphs into ℓ1 , 2003, SODA '03.

[38]  Andrew Tomkins,et al.  A polylog(n)-competitive algorithm for metrical task systems , 1997, STOC '97.

[39]  Paul D. Seymour,et al.  Packing directed circuits fractionally , 1995, Comb..

[40]  Anupam Gupta,et al.  Cuts, Trees and ℓ1-Embeddings of Graphs* , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[41]  Baruch Awerbuch,et al.  Concurrent online tracking of mobile users , 1991, SIGCOMM 1991.

[42]  Aravind Srinivasan,et al.  On the Covering Steiner Problem , 2003, FSTTCS.

[43]  Prabhakar Raghavan,et al.  Probabilistic construction of deterministic algorithms: Approximating packing integer programs , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[44]  Satish Rao,et al.  An improved approximation algorithm for the 0-extension problem , 2003, SODA '03.

[45]  Anupam Gupta,et al.  Steiner points in tree metrics don't (really) help , 2001, SODA '01.

[46]  T. C. Hu.  Optimum Communication Spanning Trees , 1974, SIAM J. Comput..

[47]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[48]  Mihalis Yannakakis,et al.  Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications , 1996, SIAM J. Comput..