1996; Bartal, Fakcharoenphol, Rao, Talwar 2004; Bartal, Fakcharoenphol, Rao, Talwar JITTATFAKCHAROENPHOL1, SATISHRAO2, KUNALTALWAR3
1Department of Computer Engineering, Kasetsart
University, Bangkok, Thailand
2Computer Science Division, University of California at
Berkeley, Berkeley, CA, USA
3Microsoft Research, Silicon Valley Campus, Mountain
View, CA, USA
Keywords and Synonyms
Embedding general metrics into tree metrics 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 com- putation.
It is known that tree metrics approximate general metrics badly, e. g., given a cycle Cn with nnodes, any
tree metric approximating this graph metric has distor- tion˝(n) [17]. However, Karp [15] noticed that a ran- dom spanning tree ofCnapproximates the distances be-
tween any two nodes in Cn well in expectation. Alon,
Karp, Peleg, and West [1] then proved a bound of exp(O(plognlog logn)) on an average distortion for ap- proximating any graph metric with its spanning tree.
Bartal [2] formally defined the notion of probabilistic approximation.
Notations
A graphG= (V;E) with an assignment of non-negative weights to the edges ofGdefines a metric space (V;dG)
where for each pairu;v2V,dG(u;v) is the shortest path
distance betweenuandvinG. A metric (V,d) is atree met- ricif there exists some treeT= (V0;E0) such thatVV0
and for allu;v2V,dT(u;v) =d(u;v). The metric (V,d)
is also called a metric induced byT.
Given a metric (V,d), a distribution D over tree metrics over V ˛-probabilistically approximates d
if every tree metric dT 2D, dT(u;v)d(u;v) and
EdT2D[dT(u;v)] ˛ d(u;v), for everyu;v2V. The quantity˛is referred to as thedistortionof the approxi- mation.
Although the definition of probabilistic approximation uses a distributionDover tree metrics, one is interested in a procedure that constructs a random tree metric dis- tributed according toD, i. e., an algorithm that produces a random tree metric that probabilistically approximates a given metric. The problem can be formally stated as fol- lows.
Problem (APPROX-TREE) INPUT: a metric (V, d)
OUTPUT: a tree metric(V;dT)sampled from a distribution Dover tree metrics that˛-probabilistically approximates (V, d).
Bartal then defined a class of tree metrics, called hierarchi- cally well-separated trees (HST), as follows. Ak-hierarchi- cally well-separated tree(k-HST) is a rooted weighted tree satisfying two properties: the edge weight from any node to each of its children is the same, and the edge weights along any path from the root to a leaf are decreasing by a factor of at leastk. These properties are important to many approximation algorithms.
Bartal showed that any metric on n points can be probabilistically approximated by a set of k-HST’s with O(log2n) distortion, an improvement from exp(O(plognlog logn)) in [1]. Later Bartal [3], follow- ing the same approach as in Seymour’s analysis on the Feedback Arc Set problem [18], improved the distortion down toO(lognlog logn). Using a rounding procedure of Calinescu, Karloff, and Rabani [5], Fakcharoenphol, Rao, and Talwar [9] devised an algorithm that, in expectation, produces a tree withO(logn) distortion. This bound is tight up to a constant factor.
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Approximating Metric Spaces by Tree Metrics Key ResultsA tree metric is closely related to graph decomposi- tion. The randomized rounding procedure of Calinescu, Karloff, and Rabani [5] for the 0-extension problem de- composes a graph into pieces with bounded diameter, cut- ting each edge with probability proportional to its length and a ratio between the numbers of nodes at certain dis- tances. Fakcharoenphol, Rao, and Talwar [9] used the CKR rounding procedure to decompose the graph recur- sively and obtained the following theorem.
Theorem 1 Given an n-point metric (V, d), there exists a randomized algorithm, which runs in time O(n2), that samples a tree metric from the distribution Dover tree metrics that O(logn)-probabilistically approximates (V, d). The tree is also a 2-HST.
The bound in Theorem 1 is tight, as Alon et al. [1] proved the bound of an˝(logn) distortion when (V,d) is induced by a grid graph. Also note that it is known (as folklore) that even embedding a line metric onto a 2-HST requires distortion˝(logn).
If the tree is required to be ak-HST, one can apply the result of Bartal, Charikar, and Raz [4] which states that any 2-HST can be O(k/ logk)-probabilistically ap- proximated byk-HST, to obtain an expected distortion of
O(klogn/ logk).
Finding a distribution of tree metrics that probabilis- tically approximates a given metric has a dual problem that is to find a single treeTwith small average weighted stretch. More specifically, given weightcuvon edges, find
a tree metricdT such that for all u;v 2 VdT(u;v) d(u;v) andPu;v2VcuvdT(u;v)˛Pu;v2Vcuvd(u;v).
Charikar, Chekuri, Goel, Guha, and Plotkin [6] showed how to find a distribution ofO(nlogn) tree met- rics that˛-probabilistically approximates a given metric, provided that one can solve the dual problem. The algo- rithm in Theorem 1 can be derandomized by the method of conditional expectation to find the required tree metric with˛=O(logn). Another algorithm based on modified region growing techniques is presented in [9], and inde- pendently by Bartal.
Theorem 2 Given an n-point metric (V, d), there exists a polynomial-time deterministic algorithm that finds a dis- tribution D over O(nlogn) tree metrics that O(logn)- probabilistically approximates (V, d).
Note that the tree output by the algorithm contains Steiner nodes, however Gupta [10] showed how to find another tree metric without Steiner nodes while preserving all dis- tances within a constant factor.
Applications
Metric approximation by random trees has applications in on-line and distributed computation, since randomiza- tion works well against oblivious adversaries, and trees are easy to work with and maintain. Alon et al. [1] first used tree embedding to give a competitive algorithm for thek- server problem. Bartal [3] noted a few problems in his pa- per: metrical task system, distributed paging, distributed
k-server problem, distributed queuing, and mobile user. After the paper by Bartal in 1996, numerous applica- tions in approximation algorithms have been found. Many approximation algorithms work for problems on tree met- rics or HST metrics. By approximating general metrics with these metrics, one can turn them into algorithms for general metrics, while, usually, losing only a factor of O(logn) in the approximation factors. Sample prob- lems are metric labeling, buy-at-bulk network design, and group Steiner trees. Recent applications include an ap- proximation algorithm to the Unique Games [12], infor- mation network design [13], and oblivious network de- sign [11].
The SIGACT News article [8] is a review of the metric approximation by tree metrics with more detailed discus- sion on developments and techniques. See also [3,9], for other applications.
Open Problems
Given a metric induced by a graph, some application, e. g., solving a certain class of linear systems, does not only re- quire a tree metric, but a tree metric induced by a spanning tree of the graph. Elkin, Emek, Spielman, and Teng [7] gave an algorithm for finding a spanning tree with aver- age distortion ofO(log2nlog logn). It remains open if this bound is tight.
Cross References Metrical Task Systems
Sparse Graph Spanners
Recommended Reading
1. Alon, N., Karp, R.M., Peleg, D., West, D.: A graph-theoretic game and its application to thek-server problem. SIAM J. Comput. 24, 78–100 (1995)
2. Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: FOCS ’96: Proceedings of the 37th Annual Symposium on Foundations of Computer Sci- ence, Washington, DC, USA, IEEE Computer Society, pp. 184– 193 (1996)
3. Bartal, Y.: On approximating arbitrary metrices by tree metrics. In: STOC ’98: Proceedings of the thirtieth annual ACM sympo-
Approximations of Bimatrix Nash Equilibria
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53 sium on Theory of computing, pp. 161–168. ACM Press, NewYork (1998)
4. Bartal, Y., Charikar, M., Raz, D.: Approximating min-sum k-clus- tering in metric spaces. In: STOC ’01: Proceedings of the thirty- third annual ACM symposium on Theory of computing, pp. 11– 20. ACM Press, New York (2001)
5. Calinescu, G., Karloff, H., Rabani, Y.: Approximation algorithms for the 0-extension problem. In: SODA ’01: Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, Philadelphia, PA, USA, Society for Industrial and Applied Math- ematics, pp. 8–16. (2001)
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7. Elkin, M., Emek, Y., Spielman, D.A., Teng, S.-H.: Lower-stretch spanning trees. In: STOC ’05: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pp. 494– 503. ACM Press, New York (2005)
8. Fakcharoenphol, J., Rao, S., Talwar, K.: Approximating metrics by tree metrics. SIGACT News35, 60–70 (2004)
9. Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approx- imating arbitrary metrics by tree metrics. J. Comput. Syst. Sci. 69, 485–497 (2004)
10. Gupta, A.: Steiner points in tree metrics don’t (really) help. In: SODA ’01: Proceedings of the twelfth annual ACM-SIAM sym- posium on Discrete algorithms, Philadelphia, PA, USA, Society for Industrial and Applied Mathematics, pp. 220–227. (2001) 11. Gupta, A., Hajiaghayi, M.T., Räcke, H.: Oblivious network de-
sign. In: SODA ’06: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pp. 970–979. ACM Press, New York (2006)
12. Gupta, A., Talwar, K.: Approximating unique games. In: SODA ’06: Proceedings of the seventeenth annual ACM-SIAM sym- posium on Discrete algorithm, New York, NY, USA, pp. 99–106. ACM Press, New York (2006)
13. Hayrapetyan, A., Swamy, C., Tardos, É.: Network design for information networks. In: SODA ’05: Proceedings of the six- teenth annual ACM-SIAM symposium on Discrete algorithms, Philadelphia, PA, USA, Society for Industrial and Applied Math- ematics, pp. 933–942. (2005)
14. Indyk, P., Matousek, J.: Low-distortion embeddings of finite metric spaces. In: Goodman, J.E., O’Rourke, J. (eds.) Hand- book of Discrete and Computational Geometry. CRC Press, Inc., Chap. 8 (2004), To appear
15. Karp, R.: A 2k-competitive algorithm for the circle. Manuscript (1989)
16. Matousek, J.: Lectures on Discrete Geometry. Springer, New York (2002)
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18. Seymour, P.D.: Packing directed circuits fractionally. Combina- torica15, 281–288 (1995)