BARTAL, Y. 1998. On approximating arbitrary metrics by tree metrics. In Proceedings of the ACM Symposium on Theory of Computing. ACM, New York.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... bounded in n, our upper bound on the local memory requirement simplifies to O(S log n log n) ffl The amortized adaptability of our tracking scheme is O(log n) The logarithmic factors in the stretch are partially derived from the properties of a network decomposition of Bartal [9], as we will see in Sections 3 and 4. Utilizing improved clustering techniques may result in better stretch factors. For example, for planar graphs one can divide the bounds for the stretch by a log log n factor by employing a clustering algorithm of [15] Furthermore, increasing the bound on the ....

....then one can drop another log n factor in all bounds for stretch and memory requirement (a simplified such scheme, which only provides guarantees on the expected cost of a read operation, is presented in Section 3) 1. 2 Related Work The clustering and decomposition techniques of Bartal [8, 9, 15] build on the seminal work of Awerbuch and Peleg [6] see also [4] who provide the first low diameter hierarchical decomposition for arbitrary networks. These clustering techniques have found several applications in distributed networks, and network algorithms such as maintaining routing tables ....

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pages 161--168, 1998.

....algorithm for virtual private network design. Previously, constant factor approximation algorithms were known only for special cases of this problem [9, 13] the best known algorithm for the general case was a O(log n log log n) algorithm obtained by applying the tree embeddings of [6]. 3. We give a simple constant factor approximation algorithm for the single sink buy at bulk network design problem. Our performance guarantee improves over what was previously known [26] by roughly a factor of 3, and gives an evengreater improvement over previous combinatorial approximation ....

....studied from the viewpoint of approximation algorithms over the past few years. After the problem was introduced by Salman et al. 24] a long line of papers that we will not review in detail here have presented successively superior algorithms for increasingly general versions of the problem [1, 2, 6, 10, 11, 12, 18, 20, 21, 26]. For the SSBB problem considered here, the first nontrivial approximation was found by Awerbuch and Azar [2] using the tree embeddings of Bartal [5] and the first constant factor approximation was given by Guha et al. 12] The performance guarantee of the combinatorial algorithm of [12] was ....

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Yair Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161--168, 1998.

....nontrivial to obtain an o(log n) approximation bound, even on some seemingly special graphs, such as trees. Some of our results are speci c to star metrics and ultrametrics , which arise as an important tool in obtaining approximation algorithms in more general metric spaces, as shown, e.g. in [2, 3, 7]. We also study a geometric variant of the problem where the robots are located at points of a geometric space and travel times are given by geometric distances. It turns out that geometry helps substantially in being able to compute good approximate solutions. For example, we give a nearly ....

Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proc. 30th Annual ACM Symposium on Theory of Computing, pages 161-168, 1998.

....itself is a tree, then we can nd a tree with stretch of 1. Finding good trees to execute the Arrow protocol is studied by Peleg and Reshef in [8] They note that if the adversary (who decides where requests occur) is oblivious, then one can use approximation of metric spaces by tree metrics [1, 2, 3] to choose a tree with an expected overhead of O(log n log log n) for general graphs and O(log n) expected overhead for constant dimensional Euclidean graphs. When combined with results from the previous section, this gives us an O(log n log log n log r) competitive ratio for the Arrow protocol ....

Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161-168, 1998.

....14. An important characteristic of the amortizing algorithm is that it can be implemented eciently, even in a distributed setting. We discuss such an implementation towards the end of the paper. It is worth noting that the recent results on the approximation of general metrics by tree metrics [5, 6, 12] imply that any hierarchical placement algorithm can be used to obtain a placement algorithm for general metrics giving up an extra O(log n log log n) factor in the approximation. 1.3 Related work Dowdy and Foster [15] initiated the study of cooperative caching in the context of allocating les ....

Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pages 184-193, October 1996.

....simultaneous optimization and is the main subject of study in this paper. Our algorithm for problem R2 can be derandomized using the method of conditional expectations to obtain a deterministic guarantee for problem Det. Related Work: When f is known in advance, a sequence of interesting papers [3, 7, 2, 10, 15] led to a constant factor approximation for this problem by Guha, Meyerson, and Munagala [11] the problem is known to be NP Hard and MAX SNP Hard since it contains the Steiner tree problem as a special case) When f is known in advance, but can be different for different links, Meyerson, ....

....and Plotkin [15] gave a randomized O(log k) approximation algorithm which was derandomized by Chekuri, Khanna, and Naor [8] When f is not known, Awerbuch and Azar [3] demonstrated how the tree embeddings of Bartal [6] can be used to solve problem R1 . Using the more recent result of Bartal [7], this results in an O(log n log log n) guarantee for problem R1. However, it is not clear how these techniques can be extended to give similar results for the more interesting problem R2. In fact, it is conceivable a priori that none of the trees produced by Bartal s algorithm would be ....

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Y. Bartal. On approximating arbitrary metrics by tree metrics. 30th ACM Symposium on Theory of Computing, 1998.

....in turn is at most N) and ra denotes the number of groups. For the group Steiner problem where K 1, this approximation ra tio matches the best known ratio. We can transform the problem in any metric to one on a tree, with a worsening in the performance ratio, by using the technique of Bar tal [5] and Charikar, Chekuri, Goel and Guha [7] This then leads to an approximation algorithm for the general covering Steiner problem with approximation guarantee of O(lognloglognlogNlogmlogK) Using im proved metric approximations for graphs that exclude Ks,s as a minor (such as planar graphs, ....

....factor of optimal) 2 Linear programming formulation First, we make several assumptions that do not reduce the generality of the problem, but make it easier to formulate. 2.1 Assumptions. We assume the following. 1) The graph G is a weighted tree (we can use the results of Bartal [5] on probabilistic approximation of general metrics by tree metrics to reduce the original problem to the problem in a weighted tree with a slight loss in the performance ratio the details are in Section 4.1) 2) The groups are disjoint (a vertex belonging to several groups may be expanded ....

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161-168, 1998.

....distortion for any edge small. Alon et al. 2] extended this result to show that every metric on n points can be probabilistically approximated by tree metrics with distortion 20(1g n log log n) Bartal [4] achieved a better distortion of O (log 2 n) and improved this bound to O (log n log log n) [5]. The technique underlying his results is an efficient computation of probabilistic (strong) partitions of the graph, where the induced graph on each cluster of the partition has a small diameter. The idea is to define a distribution over the set of strong partitions such that the probability of ....

Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30thAnnual ACM Symposium on Theory of Computing, pages 161-168, 1998.

....this ratio is at most 4. Our Results: Let n be the number of nodes in the input graph (number of points in which there is an object initially or is a destination point) Let k be the capacity of the vehicle. We use the recent results on probabilistic approximation of metric spaces by tree metrics [5, 6, 10] to reduce the problem on a general metric space to an instance of a special form on a class of trees called height balanced trees. We give an algorithm for the Capacitated Dial a Ride problem with approximation ratio O( p k) for special instances on height balanced trees. This gives us a ....

....De nition 3 A k hierarchically well separated tree (k HST) is de ned as a rooted weighted tree such that (1) The edge weight from any node to each of its children is the same. 2) The edge weights along any path from the root to a leaf are decreasing by a factor of at least k. Theorem 1 (Bartal [6]) Every metric space M over V can be probabilistically approximated by the set of 2 HSTs, where = O(logn log log n) Here the points in V occur as the leaves of the 2 HST, the internal nodes (and the root) are dummy nodes. Actually, we will use a slightly di erent kind of a tree, which we ....

Y. Bartal. \On approximating arbitrary metrics by tree metrics", Proc. 30th STOC, (1998), pages 161-168. 11

....Awerbuch and Azar [AA] gave a randomized O(log 2 n) approximation algorithm for the buy at bulk problem with many cable types and many sources and sinks, where n is the number of nodes in the input graph. This improves to O(log n log log n) using the improved tree metric construction of Bartal [Bar]. Salman et al. also gave a constant approximation in [SCR ] for the single cable type case using a LAST construction [KRY] in place of the spanner construction used in [MP] The approximation ratio was later improved by Hassin, Ravi and Salman [HRS] Andrews and Zhang [AZ] studied a special case ....

Y. Bartal, \On approximating arbitrary metrics by tree metrics", Proc. 30th Ann. ACM Symposium on Theory of Computing, 1998.

....Awerbuch and Azar [4] gave a randomized O(log 2 n) approximation algorithm for the buy at bulk problem with many cable types and many sources and sinks, where n is the number of nodes in the input graph. This improves to O(log n log log n) using the improved tree metric construction of Bartal [5]. For the single sink case with many cable types, an O(log n) approximation was obtained by Meyerson, Munagala and Plotkin [12] based on their work on the Cost Distance two metric network design problem. Salman et al. also gave a constant approximation in [14] for the single cable type case using a ....

Bartal, Y.: On approximating arbitrary metrics by tree metrics. Proc. 30th Ann. ACM Symposium on Theory of Computing, 1998.

....connecting pieces of the network into larger ones, considering each type of cable in turn and constructing alternate shortest path and Steiner forests. This approach is very di erent from the techniques used earlier by [13, 2] which relied on approximating metrics using trees via the results of [3, 11, 4, 5]. Our algorithm utilizes more structure speci c to the problem, and we are in the process of comparing it to previous approaches. Our algorithm is randomized and combinatorial with a running time of O(kn 2 log n) for n demand points and k cable types. This is signi cantly better than the ....

....by Salman et al. in [13] Related work includes the result of Awerbuch and Azar [2] which gives an O(log 2 n) approximation even in the case where di erent demand points must communicate with di erent sinks. This result may be improved to O(log n log log n) using subsequent results of Bartal [5] on probabilistic approximation of metrics using trees, and may be derandomized using the results of [6, 7] Independent of this work, Garg et al. [8] obtained a O(K) approximation (where K is the number of di erent cable types) to the single sink buy at bulk problem by rounding the natural LP ....

Y. Bartal. On approximating arbitrary metrics by tree metrics. 30th ACM Symposium on Theory of Computing, 1998.

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Yair Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 183--193, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th STOC, pages 183{ 193, 1998.

....in T . The notion of 1 HST coincides with that of an ultrametric. Any k HST is also a 1 HST, i.e. an ultrametric. However, for every k 1 the class of k HST is a proper subclass of ultrametrics. Ultrametrics and k HSTs have played a key role in recent work on embeddings of finite metric spaces [2 5]. Let f : X Y be an embedding of the metric space (X, dX ) into the metric space (Y, d Y ) We define the distortion of f by dist(f) sup dX (x, y) sup d Y (f(x) f(y) We denote by c Y (X) the least distortion with which X may be embedded in Y . When c Y (X) # we say that X ....

....which X may be embedded in Y . When c Y (X) # we say that X # embeds into Y . When there is a bijection f between two metric spaces X and Y with dist(f) # we say that X and Y are # equivalent. The following proposition provides a comparison between ultrametrics and k HSTs. Proposition 1 ([2]) For any k 1, any ultrametric is k equivalent to a k HST. A basic property of ultrametrics is: Proposition 2 ( 6] Any ultrametric is isometrically embeddable in # 2 . Since # 2 isometrically embeds into L p for every 1 a similar result follows for embeddings in # p . Moreover, a ....

Y. Bartal, On approximating arbitrary metrics by tree metrics, in: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 183--193.

....Ramsey type problems. In spite of the similarity of these problems, the results in the metric setting di er markedly from those for the linear setting. Finite metric spaces and their embeddings in other metric spaces have been intensively investigated in recent years. See for example the papers [3, 4, 13, 14, 23, 40, 50], the surveys [29, 35] and the book [41] for an exposition of most of the known results. We would like to particularly point out the fundamental article of M. Gromov [28] in which many of the modern concepts and problems in this area were systematically studied for the rst time (in fact, the ....

....in the rooted tree are labelled by real numbers. The labels decrease by a factor k as you go down the levels away from the root. The distance between two leaves is the label of their lowest common ancestor. These decomposable metrics were introduced by Bartal [3] Subsequently, he showed [4] that any n point metric can be O(log n log log n) probabilistically embeddable in ultrametrics. It is also shown in [4] that any ultrametric is k equivalent to a k HST. These results have found many unexpected algorithmic applications in recent years, mostly in providing computationally ecient ....

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 183-193, 1998.

....Previous lower bounds were e 1 log n) KRR94] and log n= log log n) BKRS00] for MTS and similar lower bounds for K server in n point space, when n K. Our paper improve these lower bounds, following major advancements in the analysis of the upper bounds for MTS, appearing in [Bar96, Bar98, BBBT97, FM00] The best current upper bound for MTS is O(log n log log n) Currently there is no general randomized upper bound for the K server problem better than 2K 1 [KP95] Seiden [Sei01] has a promising result in this direction, showing sub linear bounds for certain spaces with ....

....algorithm for N then there is c competitive algorithm for M . This notion can be generalized to a probabilistic metric approximation [Bar96] by considering a set of approximating metric spaces that dominate the original metric space and bounding the expectation of the distances. In [Bar96, Bar98] it is proved that any metric space on n points can be O(log n log log n) probabilistically approximated by an HST, thus reducing the problem of devising algorithm for MTS on any metric space to devising an algorithm for HSTs only [BBBT97, FM00] HSTs and their probabilistic approximation of ....

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Yair Bartal, On approximating arbitrary metrics by tree metrics, Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 183-193.

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BARTAL, Y. 1998. On approximating arbitrary metrics by tree metrics. In Proceedings of the ACM Symposium on Theory of Computing. ACM, New York.

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Yair Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th ACM Symposium on Theory of Computing (STOC), pages 161--168, 1998.

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Bartal, Y.: On approximating arbitrary metrics by tree metrics. In: Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science (IEEE FOCS). (1996)

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Y. Bartal, "On approximating arbitrary metrics by tree metrics," in STOC, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161 -- 168, Dallas, TX, May 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161 -- 168, Dallas, TX, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161-168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In 30th Annual ACM Symposium on Theory of Computing, pages 161--168. ACM, 1998.

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Y. Bartal, On approximating arbitrary metrics by tree metrics. In STOC '98 (Dallas, TX), pages 161-168. ACM, New York, 1998.

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Y. Bartal, "On approximating arbitrary metrics by tree metrics", STOC, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. 30th STOC, 161--168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161-168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings, ACM Symposium on Theory of Computing, pages 161--168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. 30th ACM Symposium on Theory of Computing, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. Proceedings of 30th ACM Symposium on Theory of Computing, 1998.

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Y. Bartal, On approximating arbitrary metrics by tree metrics, STOC, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings, ACM Symposium on Theory of Computing, pages 161--168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of 30th STOC, pages 161--168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proc. 30th Annu. ACM Sympos. Theory Comput., pages 161-168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In 30th STOC, pages 161--168, 1998.

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Yair Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161--168, 1998.

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Yair Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161--168, 1998.

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Y. Bartal, "On approximating arbitrary metrics by tree metrics", In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 161-168.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proc. 30th Annu. ACM Sympos. Theory Comput., pages 161-168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pages 184--193, October 1996.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th ACM Symposium on the Theory of Computation (STOC'98), pages 161--168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In STOC, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In 30th Annual ACM Symposium on Theory of Computing, pages 161-168. ACM, 1998.

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Y. Bartal, \On approximating arbitrary metrics by tree metrics", Proceedings of STOC 1998, 161-168.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In 30th Annual ACM Symposium on Theory of Computing, pages 161--168. ACM, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pages 184--193, October 1996.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the Thirtieth Annual pages 161--168, 1998.

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Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of 30th STOC, pages 161--168, 1998.

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