## Approximating a Finite Metric by a Small Number of Tree Metrics (1998)

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Venue: | In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science |

Citations: | 82 - 10 self |

### BibTeX

@INPROCEEDINGS{Charikar98approximatinga,

author = {M. Charikar and C. Chekuri and A. Goel and S. Guha and S. Plotkin},

title = {Approximating a Finite Metric by a Small Number of Tree Metrics},

booktitle = {In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science},

year = {1998},

pages = {379--388}

}

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### Abstract

Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his

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Citation Context ...em. The PST framework requires a certain dual subroutine; when S is the space of tree metrics the dual subroutine turns out to be the minimum communication cost spanning tree problem which is NP-hard =-=[11, 28]-=-. However an approximate dual subroutine can be obtained using graph partitioning techniques from [13, 27, 9], as shown in [5, 8] which give an O(log n log log n) guarantee on the approximation ratio.... |

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Citation Context ...en metric space in a simpler metric space such that the distances are approximately preserved in the embedding. New and improved algorithms have resulted from this idea for several important problems =-=[20, 10, 4, 6]-=-. Based on the work of Karp [16] and Alon et al. [1], Bartal defined probabilistic approximation of metric spaces by a set of simpler metric spaces. Formally, let M be a finite metric space defined on... |

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Citation Context ...mall Number of Tree Metrics M. CHARIKAR C. CHEKURI y A. GOEL z S. GUHA x S. PLOTKIN { Computer Science Department Stanford University Computer Science Department Stanford University Abstract Bartal [=-=4, 5]-=- gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has ... |

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Citation Context ... i.e. x uv 2c uv V d 1 (R)=V d (R) = (2c uv =R)(C d 1 =C d ): The diameter of each cluster is at most 2R. By definition ofs, we now haves 4C d 1 =C d . But C d 1 =C d = (1+ d=2) 2(1=2 + d=2)(3=2) [24]. It is easy to verify that this expression is at most p d=2, which implies thats 2 p d. 2 The same algorithm can also be used to cluster spaces with any l p norm where p 1. Therefore the partition... |

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Citation Context ...he minimum communication cost spanning tree problem which is NP-hard [11, 28]. However an approximate dual subroutine can be obtained using graph partitioning techniques from [13, 27, 9], as shown in =-=[5, 8]-=- which give an O(log n log log n) guarantee on the approximation ratio. Since the best upper bound we have on the optimal value of the fractional packing problem is O(log n log log n), a naive applica... |

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Citation Context ... the best (in the sense of least distortion) probabilistic approximation of M by S as a fractional packing problem (with jSj variables.) We then use the algorithm of Plotkin, Shmoys, and Tardos (PST) =-=[25]-=- for approximately solving this fractional packing problem. The PST framework requires a certain dual subroutine; when S is the space of tree metrics the dual subroutine turns out to be the minimum co... |

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Citation Context ...routine turns out to be the minimum communication cost spanning tree problem which is NP-hard [11, 28]. However an approximate dual subroutine can be obtained using graph partitioning techniques from =-=[13, 27, 9]-=-, as shown in [5, 8] which give an O(log n log log n) guarantee on the approximation ratio. Since the best upper bound we have on the optimal value of the fractional packing problem is O(log n log log... |

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Citation Context ...lustering. The objective is to partition a metric space (induced usually by a weighted undirected graph) into subgraphs such that the diameter of each of the subgraphs is low. Peleg et al. and others =-=[23, 3]-=- studied such partitions motivated by applications in network routing and distributed computing. Leighton and Rao [19] pioneered the use of low diameter clustering for the design of approximation algo... |

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Citation Context ...expectation, within an O(log n log log n) factor of the optimal solution on the graph. This approach has been used to obtain improved approximations for several problems including group Steiner trees =-=[12]-=-, k-median [5, 8], minimum communication cost spanning trees [28], buy-at-bulk network design [2], and vehicle routing [7]. Since the first step in the above approach consists of constructing a tree p... |

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Citation Context ...es are approximately preserved in the embedding. New and improved algorithms have resulted from this idea for several important problems [20, 10, 4, 6]. Based on the work of Karp [16] and Alon et al. =-=[1]-=-, Bartal defined probabilistic approximation of metric spaces by a set of simpler metric spaces. Formally, let M be a finite metric space defined on a vertex set V . For two metric spaces M 1 and M 2 ... |

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Citation Context ...in network routing and distributed computing. Leighton and Rao [19] pioneered the use of low diameter clustering for the design of approximation algorithms based on a divide and conquer approach. See =-=[13, 17, 9]-=- for related results. Bartal [4] used probabilistic partitioning of graphs into low diameter clusters to obtain tree approximations. In this paper we consider low diameter clustering when the graphs a... |

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Citation Context ...sult establishes that finite metrics can be probabilistically approximated by a small number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design =-=[2]-=- and vehicle routing [7]; in addition we subsume results from our earlier work [8] on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a ... |

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Citation Context ...icated analysis of the partitioning procedure first used by Seymour [27], and subsequently developed into a more broadly applicable divide and conquer paradigm called spreading metrics by Even et al. =-=[9]-=-. It is worth noting that the above mentioned partitioning methods have been applied to problems where the underlying metric is obtained by solving a linear program, while the weights are capacities g... |

93 | Approximating the bandwidth via volume respecting embeddings
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Citation Context ...en metric space in a simpler metric space such that the distances are approximately preserved in the embedding. New and improved algorithms have resulted from this idea for several important problems =-=[20, 10, 4, 6]-=-. Based on the work of Karp [16] and Alon et al. [1], Bartal defined probabilistic approximation of metric spaces by a set of simpler metric spaces. Formally, let M be a finite metric space defined on... |

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Citation Context ...routine turns out to be the minimum communication cost spanning tree problem which is NP-hard [11, 28]. However an approximate dual subroutine can be obtained using graph partitioning techniques from =-=[13, 27, 9]-=-, as shown in [5, 8] which give an O(log n log log n) guarantee on the approximation ratio. Since the best upper bound we have on the optimal value of the fractional packing problem is O(log n log log... |

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Citation Context ...lustering. The objective is to partition a metric space (induced usually by a weighted undirected graph) into subgraphs such that the diameter of each of the subgraphs is low. Peleg et al. and others =-=[23, 3]-=- studied such partitions motivated by applications in network routing and distributed computing. Leighton and Rao [19] pioneered the use of low diameter clustering for the design of approximation algo... |

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Citation Context ...in applications (see [4] for details). We note that the trees guaranteed by Theorem 2.2 are also HSTs. It is possible to use the parallel algorithm for positive linear programming of Luby and Nisan [=-=21]-=- in place of [25] at the cost increasing the number of trees generated by a polylogarithmic factor. However we do not have a parallel tree construction procedure to take advantage of the algorithm of ... |

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Citation Context ...all number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design [2] and vehicle routing [7]; in addition we subsume results from our earlier work =-=[8]-=- on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a deterministic optimization problem via linear programming. This view also provides... |

48 | A polylog(n)-competitive algorithm for metrical task systems - Bartal, Blum, et al. - 1997 |

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Citation Context ...olution on the graph. This approach has been used to obtain improved approximations for several problems including group Steiner trees [12], k-median [5, 8], minimum communication cost spanning trees =-=[28]-=-, buy-at-bulk network design [2], and vehicle routing [7]. Since the first step in the above approach consists of constructing a tree probabilistically, all the approximation algorithms which result a... |

41 | Lower bounds on the distortion of embedding finite metric spaces in graphs - RABINOVICH, RAZ - 1998 |

21 | On approximating planar metrics by tree metrics
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Citation Context ...ification to PST (Theorem 2.3) is of independent interest. Using improved graph partitioning results, we show that planar graphs can be O(log n)-probabilistically approximated (matching the result of =-=[18]-=- but using a small distribution) , and points in d-dimensional Euclidean spaces can be O( p d log n)-probabilistically approximated using a distribution on O(n log n) and O(dn log n) trees, respective... |

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A 2k-competitive algorithm for the circle
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Citation Context ...such that the distances are approximately preserved in the embedding. New and improved algorithms have resulted from this idea for several important problems [20, 10, 4, 6]. Based on the work of Karp =-=[16]-=- and Alon et al. [1], Bartal defined probabilistic approximation of metric spaces by a set of simpler metric spaces. Formally, let M be a finite metric space defined on a vertex set V . For two metric... |

15 |
Deterministic polylog approximation for minimum communication spanning trees (extended abstract
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Citation Context ...O( p d log n)-probabilistically approximated using a distribution on O(n log n) and O(dn log n) trees, respectively. For Euclidean spaces our result improves on the earlier known result of O(d log n) =-=[22]-=-. The improvement is obtained by a new graph partitioning procedure for Euclidean spaces that we describe below. One of our contributions, apart from obtaining a polynomial size distribution, are our ... |

8 |
personal communication
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Citation Context ...uced by points insd p ( d equipped with the l p norm). We prove a lower bound ofsd) onsfor graphs induced bysd 1 ; an identical lower bound has recently been proved by Indyk for graphs induced bysd 1 =-=-=-[14]. These two lower bounds and our upper bound forsd 2 can be combined to obtain matching upper and lower bounds of (f(d; p)) ons, where f(d; p) = d 1=p for 1 p 2 and f(d; p) = d 1 1=p for p 2. W... |

7 |
Divideand -conquer approximation algorithms via spreading metrics
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- 1995
(Show Context)
Citation Context ...icated analysis of the partitioning procedure first used by Seymour [27], and subsequently developed into a more broadly applicable divide and conquer paradigm called spreading metrics by Even et al. =-=[9]-=-. It is worth noting that the above mentioned partitioning methods have been applied to problems where the underlying metric is obtained by solving a linear program, while the weights are capacities g... |

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The Finite Capacity Dial-a-Ride Problem", these proceedings
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Citation Context ...nite metrics can be probabilistically approximated by a small number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design [2] and vehicle routing =-=[7]-=-; in addition we subsume results from our earlier work [8] on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a deterministic optimizati... |