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# A tight bound on approximating arbitrary metrics by tree metrics

Journal of Computer and System Sciences - Special issue: STOC 2003, no. 3 (2004): 485-497

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摘要

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 ...更多

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简介

- 1.1 Metric approximations

The problem of approximating a given graph metric by a “simpler” metric has been a subject of extensive research, motivated from several different perspectives. - Karp, Peleg and West [1] looked at approximating arbitrary graph metrics by spanning trees, and showed an upper bound of 2O on the distortion.
- Bartal [7] formally defined probabilistic embeddings and improved on the previous result by showing how to probabilistically approximate metrics by tree metrics with distortion O.

重点内容

- The problem of approximating a given graph metric by a “simpler” metric has been a subject of extensive research, motivated from several different perspectives
- Point of view, is a tree metric, i.e. a metric arising from shortest path distance on a tree containing the given points
- We show that an arbitrary metic space can be approximated by a distribution over dominating tree metrics with distortion O, closing the gap between the lower and the upper bounds
- The problem of probabilistic approximation by tree metrics asks for a distribution over tree metrics such that the expected stretch of each edge is small

结果

- This gives improved approximation algorithms for various problems including group Steiner tree [24], metric labeling [19, 32], buy-at-bulk network design [4], and vehicle routing [16].
- The authors outline the algorithm for probabilistically embedding an n point metric into a tree, and show that the expected distortion of any distance is O.
- The authors first decompose the graph hierarchically and convert the resulting laminar family to a tree.
- The authors shall describe a random process to define a hierarchical cut decomposition of (V, d), such that the probability that an edge (u, v) is at level i decreases geometrically with i.
- The authors shall bound the probability that w cuts u out of (u, v) at level i.
- This implies that the probability that u gets cut out of edge e is bounded by kiu s=k +1
- The distribution over tree metrics resulting from the algorithm O-probabilistically approximates the metric d.
- A tree T is said to be k-hierarchically well separated if on any root to leaf path the edge lengths decrease by a factor of k in each step.
- The problem of probabilistic approximation by tree metrics asks for a distribution over tree metrics such that the expected stretch of each edge is small.
- The partitioning algorithm the authors give produces a tree such that the expected stretch of each edge is at most O.

结论

- The metric labeling problem : The previous result of Kleinberg and Tardos [32] gives an O-approximation algorithm based on a constant factor approximation for the case that the terminal metric is an HST.
- Metrical Task system : Improving on the result of Bartal, Blum, Burch and Tomkins [10], Fiat and Mendel [22] gave an O-competitive algorithms on HSTs. Bartal and Mendel’s [12] multiembedding result gives an O(λ log n log log n)-competitive ratio, where λ is as defined above.
- The result improves the performance guarantees of several other problems such as vehicle routing [16], min sum clustering [11, 9], covering steiner tree [33], hierarchical placement [35], topology aggregation [6, 44], mirror placement [29], distributed K-server [13], distributed queueing [27] and mobile user [5].

相关工作

- Divide and conquer methods have been used to provide polylogarithmic-factor approximation algorithms for numerous graph problems since the discovery of an O(log n) approximation algorithm for finding a graph separator [36]. The algorithms proceeded by recursively dividing a problem using the above approximation algorithm, and then using the decomposition to find a solution. Typically, the approximation factor was O(log2 n): a logarithmic factor came from the O(log n) separator approximation, another O(log n) factor came from the recursion. Using this framework, polynomial-time approximation algorithms for many problems are given in [36], for example: crossing number, VLSI layout, minimum feedback arc set, and search number.

Independently, Seymour [45] gave an O(log n log log n) bound on the integrality gap for a linear programming relaxation of the feedback arc set problem (for which the above techniques had given an O(log2 n) bound). In doing so, he developed a technique that balanced the approximation factor of his separator based procedure against the cost of the recursion to significantly improve the bounds.

引用论文

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