## A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics (2003)

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Venue: | In Proceedings of the 35th Annual ACM Symposium on Theory of Computing |

Citations: | 265 - 7 self |

### BibTeX

@INPROCEEDINGS{Fakcharoenphol03atight,

author = {Jittat Fakcharoenphol and Satish Rao and Kunal Talwar},

title = {A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics},

booktitle = {In Proceedings of the 35th Annual ACM Symposium on Theory of Computing},

year = {2003},

pages = {448--455}

}

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### Abstract

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#sto 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.

### Citations

452 | Y.: ”The geometry of graphs and some of its algorithmic applications
- Linial, London, et al.
- 1994
(Show Context)
Citation Context ...eddings. This work also follows the line of research on embeddings, with low distortion, graphs into other "nice" metric spaces, which have good structural properties, such as Euclidean and =-=#1 spaces [37, 26, 18, 43, 23]-=-. The work of Bourgain [14] showed that any finite metric on n nodes can be embedded into #2 with logarithmic distortion with the number of dimensions exponential in n. Linial, London, and Rabinovich ... |

316 | Probabilistic approximations of metric spaces and its algorithmic applications
- Bartal
- 1996
(Show Context)
Citation Context ... smaller than those in the original metric and we would like to bound the distortion or the maximum increase. However, there are simple graphs (e.g. the n-cycle) for which the distortion must be # n) =-=[41, 7, 25]-=-. To circumvent this, Karp [30] considered approximating the cycle by a probability distribution over paths, and showed a simple distribution such that the expected length of each edge is no more than... |

280 |
On Lipschitz embedding of finite metric spaces in Hilbert space
- Bourgain
- 1985
(Show Context)
Citation Context ... research on embeddings, with low distortion, graphs into other "nice" metric spaces, which have good structural properties, such as Euclidean and #1 spaces [37, 26, 18, 43, 23]. The work of=-= Bourgain [14]-=- showed that any finite metric on n nodes can be embedded into #2 with logarithmic distortion with the number of dimensions exponential in n. Linial, London, and Rabinovich [37] modified Bourgain's re... |

279 |
Probabilistic construction of deterministic algorithms: Approximating packing integer programs
- Raghavan
- 1988
(Show Context)
Citation Context ...e. In fact, the computation above can be simplified, replacing the exact value of the expected cost above by the upper bounds used in the analysis (and thus using the method of pessimistic estimators =-=[42]-=-). 3. APPLICATIONS Many problems are easy on trees. The partitioning algorithm we give produces a tree such that the expected stretch of each edge is at most O(log n). By using our result, the approxi... |

253 | On approximating arbitrary metrices by tree metrics
- Bartal
- 1998
(Show Context)
Citation Context ...result improves to O(log n) for planar graphs, and Charikar et.al. [17] showed similar bounds for low dimensional normed spaces. Inspired by ideas from Seymour's work on feedback arc set [45], Bartal =-=[8]-=- improved his earlier result to O(log n log log n). This of course led to improved bounds on the performance ratios of several applications. Bartal also observed that any probabilistic embedding of an... |

239 |
An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms
- Leighton, Rao
- 1988
(Show Context)
Citation Context ... 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(log... |

208 | Concurrent Online Tracking of Mobile Users
- Awerbuch, Peleg
- 1991
(Show Context)
Citation Context ... 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]-=-. We refer the reader to to [8] and [17] for more detailed descriptions of these problems. 4. ACKNOWLEDGEMENTS We would like to thank Yair Bartal for helpful comments on a previous draft of the paper,... |

164 |
Lectures in Discrete Geometry
- Matousek
- 2002
(Show Context)
Citation Context ...-cut gap for multicommodity flow problems. They also gave a lower bound on the distortion of any embeddings of general graphs into #1 . For more details, we point the reader to Chapter 15 in Matousek =-=[38]-=-. Embeddings of special graphs have also been considered by many researchers. Gupta et al. [26] considered embeddings or series-parallel graphs and outerplanar graphs into #1 with constant distortion;... |

159 | Approximation algorithms for classification problems with pairwise relationships: Metric labeling and Markov random fields
- Kleinberg, Tardos
- 1999
(Show Context)
Citation Context ...his distribution. Our trees are also heirarchically well separated, like Bartal's. 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]. We give a more comprehensive list in section 3. 1.2 Related Work Divide and conquer methods have been used to provide polylogarithmic-factor... |

143 | Approximate max-flow min(multi)cut theorems and their applications - Garg, Vazirani, et al. - 1993 |

129 | A polylogarithmic approximation algorithm for the group Steiner tree problem
- Garg, Konjevod, et al.
- 2000
(Show Context)
Citation Context ...o sample a tree from this distribution. Our trees are also heirarchically well separated, like Bartal's. 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]. We give a more comprehensive list in section 3. 1.2 Related Work Divide and conquer methods have been used to prov... |

126 | An O(log k) approximate min-cut max-flow theorem and approximation algorithm
- Aumann, Rabani
- 1998
(Show Context)
Citation Context ...c distortion with the number of dimensions exponential in n. Linial, London, and Rabinovich [37] modified Bourgain's result to apply for #1 metrics and to use O(log 2 n) dimensions. Aumann and Rabani =-=[3]-=- and Linial, London and Rabinovich [37] gave several applications, including a proof of a logarithmic bound on max-flow min-cut gap for multicommodity flow problems. They also gave a lower bound on th... |

118 | On the placement of internet instrumentation
- Jamin, Jin, et al.
- 2000
(Show Context)
Citation Context ... 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]. We refer the reader to to [8] and [17] for more detailed descriptions of these problems. 4. ACKNOWLEDGEMENTS We would like t... |

117 | A graph-theoretic game and its application to the k-server problem
- Alon, Karp, et al.
- 1995
(Show Context)
Citation Context ...ed length of each edge is no more than twice its original length. This gave a competitive ratio of 2 for the k-server problem (on a cycle) that had motivated this approach. Alon, Karp, Peleg and West =-=[1]-=- looked at approximating arbitrary graph metrics by (a distribution over) spanning trees, and showed an upper bound of 2 O( # log n log log n) on the distortion. Bartal [7] formally defined probabilis... |

104 | Excluded minors, network decomposition, and multicommodity flow
- Klein, Plotkin, et al.
- 1993
(Show Context)
Citation Context ...d the lower bound given by Newman and Rabinovich [39]. Graph decomposition techniques for many interesting classes of graphs have also been extensively studied. For example, Klein, Plotkin, and Rao's =-=[31]-=- result provided a constant factor approximation for graphs that exclude fixed sized minors (which includes planar graphs). Similar results were given by Charikar et al. [17] for geometric graphs. 1.3... |

98 | Buy-at-bulk network design
- Awerbuch, Azar
- 1997
(Show Context)
Citation Context ...heirarchically well separated, like Bartal's. 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]. We give a more comprehensive list in section 3. 1.2 Related Work Divide and conquer methods have been used to provide polylogarithmic-factor approximation algorithms for nu... |

98 | Divide-and-conquer approximation algorithms via spreading metrics
- Even, Naor, et al.
(Show Context)
Citation Context ...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. Even et al.=-=[20]-=- introduced linear programming relaxations for a number of problems and combined them with Seymour 's techniques to give O(log n log log n)-approximation algorithms for many of the problems that previ... |

93 | Approximating the bandwidth via volume respecting embeddings
- Feige
- 1998
(Show Context)
Citation Context ...eddings. This work also follows the line of research on embeddings, with low distortion, graphs into other "nice" metric spaces, which have good structural properties, such as Euclidean and =-=#1 spaces [37, 26, 18, 43, 23]-=-. The work of Bourgain [14] showed that any finite metric on n nodes can be embedded into #2 with logarithmic distortion with the number of dimensions exponential in n. Linial, London, and Rabinovich ... |

83 | Measured descent: A new embedding method for finite metrics - Krauthgamer, Lee, et al. |

82 | Approximating a finite metric by a small number of tree metrics
- Charikar, Chekuri, et al.
- 1998
(Show Context)
Citation Context ...di#ered by a constant factor. This was important for several of his applications. Konjevod, Ravi and Salman [34] showed how Bartal's result improves to O(log n) for planar graphs, and Charikar et.al. =-=[17]-=- showed similar bounds for low dimensional normed spaces. Inspired by ideas from Seymour's work on feedback arc set [45], Bartal [8] improved his earlier result to O(log n log log n). This of course l... |

79 |
Packing Directed Circuits Fractionally
- Seymour
- 1992
(Show Context)
Citation Context ...how Bartal's result improves to O(log n) for planar graphs, and Charikar et.al. [17] showed similar bounds for low dimensional normed spaces. Inspired by ideas from Seymour's work on feedback arc set =-=[45]-=-, Bartal [8] improved his earlier result to O(log n log log n). This of course led to improved bounds on the performance ratios of several applications. Bartal also observed that any probabilistic emb... |

71 | Optimum communication spanning trees
- Hu
- 1974
(Show Context)
Citation Context ...lk network design : Awerbuch and Azar [4] give a O(1)-approximation algorithm on trees. Thus, we can get an O(log n)-approximation algorithm. Minimum cost communication network problem : This problem =-=[28, 40, 46]-=- is essentially the dual problem defined in section 2.5 and hence we get an O(log n) approximation. The group Steiner tree problem : Garg, Konjevod, and Ravi [24] give an O(log k log n)-approximation ... |

66 | Approximation Algorithms for the 0– Extension Problem
- Calinescu, Karloff, et al.
- 2001
(Show Context)
Citation Context ...Similar results were given by Charikar et al. [17] for geometric graphs. 1.3 Our techniques The algorithm relies on techniques from the algorithm for 0-extension given by Calinescu, Karlo# and Rabani =-=[15]-=-, and improved by Fakcharoenphol, Harrelson, Rao and Talwar [21]. The CKR procedure implies a randomized algorithm that outputs clusters of diameter about # such that the probability of an edge e bein... |

65 | Approximation algorithms for the metric labeling problem via a new linear programming formulation
- Chekuri, Khanna, et al.
- 2001
(Show Context)
Citation Context ...his distribution. Our trees are also heirarchically well separated, like Bartal's. 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]. We give a more comprehensive list in section 3. 1.2 Related Work Divide and conquer methods have been used to provide polylogarithmic-factor... |

62 | Small Distortion and Volume Preserving Embeddings for Planar and Euclidean Metrics - Rao - 1999 |

54 |
Approximating min-sum k-clustering in metric spaces
- Bartal, Charikar, et al.
- 2001
(Show Context)
Citation Context ...tem[10] and metric labeling [32]. We note that the trees we construct are 2-HSTs. Bartal [8] also observed that a 2-HST can be converted to a k-HST with distortion O(k), later improved to O(k/ log k) =-=[11]-=-. This combined with our result implies a probabilistic embedding into k-HSTs with distortion O(k log n/ log k). In fact, a slight modification of our technique (details omitted) can be used to direct... |

54 | Rounding via trees: deterministic approximation algorithms for group Steiner trees and k-median
- Charikar, Chekuri, et al.
- 1999
(Show Context)
Citation Context ...ated, like Bartal's. 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]-=-. We give a more comprehensive list in section 3. 1.2 Related Work Divide and conquer methods have been used to provide polylogarithmic-factor approximation algorithms for numerous graph problems sinc... |

54 | Placement algorithms for hierarchical cooperative caching
- Korupolu, Rajaraman
- 1999
(Show Context)
Citation Context ...adversaries. The result also 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]. We refer the reader to to [8] and [17] for more detailed descriptions o... |

49 | Low-distortion embeddings of finite metric spaces - Indyk, Matouˇsek |

48 | A polylog(n)-competitive algorithm for metrical task systems
- Bartal, Blum, et al.
- 1997
(Show Context)
Citation Context ...ected stretch of each edge, not just the average. This led to polylogarithmic competitive ratio algorithms for a number of online problems (against oblivious adversaries) such as metrical task system =-=[10]-=-. Charikar et.al. [16, 17] showed how to derandomize the approximation algorithms that follow from Bartal's embeddings. This work also follows the line of research on embeddings, with low distortion, ... |

46 |
distortion and volume preserving embeddings for planar and Euclidean metrics
- Small
- 1999
(Show Context)
Citation Context ...eddings. This work also follows the line of research on embeddings, with low distortion, graphs into other "nice" metric spaces, which have good structural properties, such as Euclidean and =-=#1 spaces [37, 26, 18, 43, 23]-=-. The work of Bourgain [14] showed that any finite metric on n nodes can be embedded into #2 with logarithmic distortion with the number of dimensions exponential in n. Linial, London, and Rabinovich ... |

41 | Lower bounds on the distortion of embedding finite metric spaces in graphs
- RABINOVICH, RAZ
- 1998
(Show Context)
Citation Context ... smaller than those in the original metric and we would like to bound the distortion or the maximum increase. However, there are simple graphs (e.g. the n-cycle) for which the distortion must be # n) =-=[41, 7, 25]-=-. To circumvent this, Karp [30] considered approximating the cycle by a probability distribution over paths, and showed a simple distribution such that the expected length of each edge is no more than... |

36 | A Lower Bound on the Distortion of Embedding Planar Metrics into Euclidean Space
- Newman, Rabinovich
- 2002
(Show Context)
Citation Context ...how a constantdistortion embedding for k-outerplanar graphs. For planar graph, Rao [43] gave an O( # log n)-distortion embedding into #2 , which matched the lower bound given by Newman and Rabinovich =-=[39]-=-. Graph decomposition techniques for many interesting classes of graphs have also been extensively studied. For example, Klein, Plotkin, and Rao's [31] result provided a constant factor approximation ... |

29 | Topology aggregation for directed graphs
- Awerbuch, Shavitt
- 2001
(Show Context)
Citation Context ...o 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]. We refer the reader to to [8] and [17] for more detailed descriptions of these problems. 4. ACKNOWLED... |

28 | Steiner points in tree metrics don’t (really) help
- Gupta
- 2001
(Show Context)
Citation Context ... smaller than those in the original metric and we would like to bound the distortion or the maximum increase. However, there are simple graphs (e.g. the n-cycle) for which the distortion must be # n) =-=[41, 7, 25]-=-. To circumvent this, Karp [30] considered approximating the cycle by a probability distribution over paths, and showed a simple distribution such that the expected length of each edge is no more than... |

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- Fiat, Mendel
- 2003
(Show Context)
Citation Context ...og k) result by Bartal and Mendel [12], where # = O(min{log n log log log n, log # log log #}). Metrical Task system : Improving on the result of Bartal, Blum, Burch and Tomkins [10], Fiat and Mendel =-=[22]-=- gave an O(log n log log n)-competitive algorithms on HSTs. Bartal and Mendel's [12] multiembedding result thus gives an O(# log n log log n)-competitive ratio, where # is as defined above. Our result... |

21 | On approximating planar metrics by tree metrics
- Konjevod, Ravi, et al.
(Show Context)
Citation Context ...ed hierarchically well separated. This meant that weights on successive levels of the tree differed by a constant factor. This was important for several of his applications. Konjevod, Ravi and Salman =-=[34]-=- showed how Bartal’s result improves to O(log n) for planar graphs, and Charikar et.al. [17] showed similar bounds for low dimensional normed spaces. Inspired by ideas from Seymour’s work on feedback ... |

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Cuts, trees and l1-embeddings of graphs
- Gupta, Newman, et al.
(Show Context)
Citation Context ...eddings. This work also follows the line of research on embeddings, with low distortion, graphs into other “nice” metric spaces, which have good structural properties, such as Euclidean and ℓ1 spaces =-=[37, 26, 18, 43, 23]-=-. The work of Bourgain [14] showed that any finite metric on n nodes can be embedded into ℓ2 with logarithmic distortion with the number of dimensions exponential in n. Linial, London, and Rabinovich ... |

19 |
A 2k-competitive algorithm for the circle
- Karp
- 1989
(Show Context)
Citation Context ...tric and we would like to bound the distortion or the maximum increase. However, there are simple graphs (e.g. the n-cycle) for which the distortion must be # n) [41, 7, 25]. To circumvent this, Karp =-=[30]-=- considered approximating the cycle by a probability distribution over paths, and showed a simple distribution such that the expected length of each edge is no more than twice its original length. Thi... |

18 | Embedding k-outerplanar graphs into ℓ1 - Chekuri, Gupta, et al. - 2006 |

17 | Cuts, trees and `1-embeddings of graphs
- Gupta, Newman, et al.
- 1999
(Show Context)
Citation Context |

17 | Approximate classification via earthmover metrics - Archer, Fakcharoenphol, et al. - 2004 |

16 | An approximation algorithm for the covering steiner problem
- Konjevod, Ravi
- 2000
(Show Context)
Citation Context ...tive ratio against oblivious adversaries. The result also 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]. We refer the reader to to [8] and [17] for... |

15 |
Deterministic polylog approximation for minimum communication spanning trees (extended abstract
- Peleg, Reshef
- 1998
(Show Context)
Citation Context ...lk network design : Awerbuch and Azar [4] give a O(1)-approximation algorithm on trees. Thus, we can get an O(log n)-approximation algorithm. Minimum cost communication network problem : This problem =-=[28, 40, 46]-=- is essentially the dual problem defined in section 2.5 and hence we get an O(log n) approximation. The group Steiner tree problem : Garg, Konjevod, and Ravi [24] give an O(log k log n)-approximation ... |

12 | Multi-embeddings and path-approximation of metric spaces
- Bartal, Mendel
(Show Context)
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9 | The χt coloring problem - Kaller, Gupta, et al. - 1995 |

7 |
The distributed k-server problem—a competitive distributed translator for k-server algorithms
- Bartal, Rosén
- 1997
(Show Context)
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6 | Embedding k-outerplanar graphs into `1 - Chekuri, Gupta, et al. - 2003 |

4 |
On approximating planar metrics by tree metrics
- Konjevod, Ravi, et al.
- 1997
(Show Context)
Citation Context ...med hierarchically well separated. This meant that weights on successive levels of the tree di#ered by a constant factor. This was important for several of his applications. Konjevod, Ravi and Salman =-=[34]-=- showed how Bartal's result improves to O(log n) for planar graphs, and Charikar et.al. [17] showed similar bounds for low dimensional normed spaces. Inspired by ideas from Seymour's work on feedback ... |

3 |
Competitive Concurrent Distributed Queueing
- Herlihy, Tirthapura, et al.
- 2001
(Show Context)
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