## Probabilistic Approximation of Metric Spaces and its Algorithmic Applications (1996)

Venue: | In 37th Annual Symposium on Foundations of Computer Science |

Citations: | 326 - 29 self |

### BibTeX

@INPROCEEDINGS{Bartal96probabilisticapproximation,

author = {Yair Bartal},

title = {Probabilistic Approximation of Metric Spaces and its Algorithmic Applications},

booktitle = {In 37th Annual Symposium on Foundations of Computer Science},

year = {1996},

pages = {184--193}

}

### Years of Citing Articles

### OpenURL

### Abstract

The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to low-distortion embeddings in low-dimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for the analysis of randomized algorithms for optimization problems on metric spaces, by relating the randomized performance ratio for any metric space to the randomized performance ratio for a set of "simple" metric spaces. We define a notion of a set of metric spaces that probabilistically-approximates another metric space. We prove that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion. These metric spaces are "simple" as being: (1) tree metrics. (2) natural for applying a divide-and-conquer algorithmic approach. The technique presented is of particular interest in the context of on-line computation. A large number of on-line al...

### Citations

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Citation Context ...any metric space is at most fffi. This technique is of particular interest in on-line computation. The performance ratio in this case is known as the competitive ratio, proposed by Sleator and Tarjan =-=[ST85]-=-. A large number of on-line algorithmic problems, including metrical task systems [BLS87], server problems [MMS88] , distributed paging [BFR92], and dynamic storage rearrangement [FMRW95], are defined... |

455 | The geometry of graphs and some of its algorithmic applications
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Citation Context ...ir Bartal Abstract The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to low-distortion embeddings in low-dimensional spaces =-=[LLR94]-=-, having many algorithmic applications. This paper provides a novel technique for the analysis of randomized algorithms for optimization problems on metric spaces, by relating the randomized performan... |

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Citation Context ...e in S. Approximations of metric spaces by more simple metric spaces has been intensively studied in various areas of mathematics and computer science. We mention but a few. Johnson and Lindenstrauss =-=[JL84]-=- and Bourgain [Bour85] study embeddings in Hilbert spaces (from a perspective of functional analysis). Graham and Winkler [GW85] study embeddings in ZZ d (from a graph theoretic motivation). Algorithm... |

278 |
On Lipschitz embedding of finite metric spaces in Hilbert space
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Citation Context ...ns of metric spaces by more simple metric spaces has been intensively studied in various areas of mathematics and computer science. We mention but a few. Johnson and Lindenstrauss [JL84] and Bourgain =-=[Bour85]-=- study embeddings in Hilbert spaces (from a perspective of functional analysis). Graham and Winkler [GW85] study embeddings in ZZ d (from a graph theoretic motivation). Algorithmic applications in dis... |

188 | An optimal on-line algorithm for metrical task system
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Citation Context ...-conquer algorithmic approach. The technique presented is of particular interest in the context of on-line computation. A large number of on-line algorithmic problems, including metrical task systems =-=[BLS87]-=-, server problems [MMS88] , distributed paging [BFR92], and dynamic storage rearrangement [FMRW95], are defined in terms of some metric space. Typically for these problems, there are linear lower boun... |

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Citation Context ...ous example is the paging problem with a cache of size K, where the deterministic competitive ratio is K [ST85] while the randomized competitive ratio (against oblivious adversaries) is \Theta(log K) =-=[FKLMSY88]-=-. Many problems that are defined in terms of metric spaces (e.g., [BLS87, MMS88, BFR92, FMRW95]) are still almost entirely open in terms of their randomized competitive ratio (against an oblivious adv... |

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Citation Context ...tion of metric spaces is of particular importance in the case of on-line problems, where the randomization against oblivious adversaries (I.e., adversaries that cannot see the algorithm's coin tosses =-=[BBKTW90]-=-) is very powerful. The most obvious example is the paging problem with a cache of size K, where the deterministic competitive ratio is K [ST85] while the randomized competitive ratio (against oblivio... |

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Citation Context ...oach. The technique presented is of particular interest in the context of on-line computation. A large number of on-line algorithmic problems, including metrical task systems [BLS87], server problems =-=[MMS88]-=- , distributed paging [BFR92], and dynamic storage rearrangement [FMRW95], are defined in terms of some metric space. Typically for these problems, there are linear lower bounds on the competitive rat... |

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Citation Context ...eddings in ZZ d (from a graph theoretic motivation). Algorithmic applications in distributed computation and graph algorithms, have led to the notion of graph spanners, introduced by Peleg and Ullman =-=[PU89]-=- and later studied in many papers including [PS89, ADDJS90, CDNS92], and to low-distortion embeddings in low-dimensional real normed spaces by Linial, London and Rabinovich [LLR94]. We extend this ide... |

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Citation Context ...ese lines begun with Karp's observation [Karp89] that the n-cycle can be 2-probabilisticallyapproximated by an n-line, motivated by its application to the K-server problem. Alon, Karp, Peleg and West =-=[AKPW91]-=- have taken this idea forward in considering probabilistic-approximations of graphs by spanning trees. Their result implies that every metric space can be 2 O( p log n log log n) -probabilistically-ap... |

108 | Optimallycompetitive distributed file allocation - AWERBUCH, BARTAL, et al. - 1993 |

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Citation Context ...ch nearly-optimal randomized algorithms for metrical task systems are known (see more about that in the sequel). A similar approach of using spreading metrics was used by Even, Naor, Rao and Schieber =-=[ENRS95]-=- to obtain approximation algorithms for a large set of optimization problems. However their method is deterministic and therefore does not apply for randomization against an oblivious adversary. The t... |

100 | Competitive algorithms for distributed data management
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Citation Context ... is of particular interest in the context of on-line computation. A large number of on-line algorithmic problems, including metrical task systems [BLS87], server problems [MMS88] , distributed paging =-=[BFR92]-=-, and dynamic storage rearrangement [FMRW95], are defined in terms of some metric space. Typically for these problems, there are linear lower bounds on the competitive ratio of deterministic algorithm... |

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Citation Context ...sts of K-servers in a metric space. Points are requested over time and a server must be moved to the request location. The K-server problem has been extensively studied. Koutsoupias and Papadimitriou =-=[KP94]-=- prove an upper bound of 2K \Gamma 1 on the competitive ratio of WFA (the work function algorithm). However WFA requires computing of optimal costs to end in every configuration and thus requires both... |

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(Show Context)
Citation Context ...ly, Blum, Furst and Tomkins [BFT96] get O(log 2 n) competitive algorithm for the weighted cache by approximating it with a metric space that can also be viewed as a 2-HST. Blum, Raghavan and Schieber =-=[BRS91]-=- give a competitive ratio of 2 O( p log n log log n) for the special case of equally spaced points on the line. We give a randomized algorithm with competitive ratio of 2 O( p log \Delta log log n) (w... |

74 | New results on server problems
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- 1990
(Show Context)
Citation Context ...orithm with a higher competitive ratio. By applying our method on the memoryless K-server algorithm for trees of Chrobak and Larmore [CL91] (an extension of the Double-Coverage algorithm for the line =-=[CKPV90]-=-) we obtain O(K log 2 n)-competitive randomized algorithm for any metric space with time O(K) per request and space ~ O(n +K). Distributed Problems. Our method is also useful for the competitive analy... |

73 | Compact distributed data structures for adaptive routing - Awerbuch, Bar-Noy, et al. - 1989 |

67 | Trees and proximity representations - Barthelemy, Guenoche - 1991 |

65 | Competitive algorithms for replication and migration problems - Black, Sleator - 1989 |

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43 |
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(Show Context)
Citation Context ...s and computer science. We mention but a few. Johnson and Lindenstrauss [JL84] and Bourgain [Bour85] study embeddings in Hilbert spaces (from a perspective of functional analysis). Graham and Winkler =-=[GW85]-=- study embeddings in ZZ d (from a graph theoretic motivation). Algorithmic applications in distributed computation and graph algorithms, have led to the notion of graph spanners, introduced by Peleg a... |

42 | Lower bounds on the distortion of embedding finite metric spaces in graphs, Discrete Comput
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(Show Context)
Citation Context ...) (trivially matched by the upper bound given by MST in a weighted graph defined by the metric space). In particular this is true for the n-cycle as follows from the recent work of Rabinovich and Raz =-=[RR95]-=-. This leads to the notion of probabilistic metric approximations we propose. Seminal work along these lines begun with Karp's observation [Karp89] that the n-cycle can be 2-probabilisticallyapproxima... |

41 | Y.: On-line generalized Steiner problem
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(Show Context)
Citation Context ...ributed randomized K-server algorithm with only O(log 2 n) overhead. We also obtain first randomized distributed on-line algorithms for the generalized Steiner problem and the network-leasing problem =-=[AAB96]-=-, and improved algorithms for the file allocation problem [BFR92, ABF93a], all with competitive ratio O(log n log(minfn; \Deltag)), by exploiting the simplicity of these problems for trees, and in par... |

34 | Competitive Distributed Job Scheduling - Awerbuch, Kutten, et al. - 1992 |

33 | On-line distributed data management - Lund, Reingold, et al. - 1994 |

30 | On page migration and other relaxed task systems - Bartal, Charikar, et al. - 1997 |

30 |
Dynamic file migration in distributed computer systems
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(Show Context)
Citation Context ...istributed paging problem [BFR92, ABF93b, ABF96] where files may also be replicated, but is of special interest since many systems allow only file migrations, to avoid issues of copy consistency (see =-=[GS90]-=- for a survey). Let a network be represented by a weighted graph G = (V; E; w). A set of files F resides in different nodes in the network. Processor v can accommodate in its local memory up to K v fi... |

30 | A theory of competitive analysis of distributed algorithms - Ajtal, Aspnes, et al. - 1994 |

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20 | Randomized algorithms for metrical task systems
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(Show Context)
Citation Context ...otal move and task costs. The problem has deterministic competitive ratio of 2n \Gamma 1 and the best randomized competitive ratio for general networks is en\Gamma1 e\Gamma1 , due to Irani and Seiden =-=[IS95]-=-. The metrical task system problem is closely related to the the (n \Gamma 1)-server problem [MMS88] on n points also called the pursuit-evasion game. Both upper and lower bounds for one problem trans... |

19 |
A 2k-competitive algorithm for the circle
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(Show Context)
Citation Context ... as follows from the recent work of Rabinovich and Raz [RR95]. This leads to the notion of probabilistic metric approximations we propose. Seminal work along these lines begun with Karp's observation =-=[Karp89]-=- that the n-cycle can be 2-probabilisticallyapproximated by an n-line, motivated by its application to the K-server problem. Alon, Karp, Peleg and West [AKPW91] have taken this idea forward in conside... |

16 |
A Decomposition Theorem and Bounds for Randomized Server Problems
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(Show Context)
Citation Context ... lower bounds for one problem translate to the other. \Theta(log n) bounds for the uniform metric space were given in [FKLMSY88, BLS87]. A lower bound of \Omega\Gamma q log n log log n ) was given in =-=[BKRS93]-=-. They also give O(log n) upper bounds in graphs that can be viewed as binary k-HST's for very large k. Recently, Blum, Furst and Tomkins [BFT96] have given an O(logn) upper bound for a special case o... |

16 | Tree spanners: Spanning trees that approximate the distances - Cai - 1992 |

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- Seiden
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(Show Context)
Citation Context ... very large k. Recently, Blum, Furst and Tomkins [BFT96] have given an O(logn) upper bound for a special case of a 2-HST which is a line graph. Similar results have been independently given by Seiden =-=[Seiden96]-=-. Finally, Blum, Furst and Tomkins [BFT96] get O(log 2 n) competitive algorithm for the weighted cache by approximating it with a metric space that can also be viewed as a 2-HST. Blum, Raghavan and Sc... |

10 |
migration with limited local memory capacity
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- 1995
(Show Context)
Citation Context ...a m) lower bound on the deterministic competitive ratio and an\Omega\Gamma/30 m) lower bound on the randomized competitive ratio, where m is the total aggregate memory in the network. Albers and Koga =-=[AK95]-=- solve the problem in the uniform metric case giving a randomized upper bound of O(logK). We give a randomized algorithm for k-HST's with competitive ratio O(log m), and thus get a randomized algorith... |

7 | Heat & Dump: Randomized competitive distributed paging - Awerbuch, Bartal, et al. - 1993 |

7 |
An Optimal On-Line k-Server Algorithm for Trees
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- 1991
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Citation Context ...elta . Therefore, it may be preferable to have a more efficient algorithm with a higher competitive ratio. By applying our method on the memoryless K-server algorithm for trees of Chrobak and Larmore =-=[CL91]-=- (an extension of the Double-Coverage algorithm for the line [CKPV90]) we obtain O(K log 2 n)-competitive randomized algorithm for any metric space with time O(K) per request and space ~ O(n +K). Dist... |

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4 | Competitive Access Time via Dynamic Storage Rearrangement
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- 1995
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Citation Context ...f on-line computation. A large number of on-line algorithmic problems, including metrical task systems [BLS87], server problems [MMS88] , distributed paging [BFR92], and dynamic storage rearrangement =-=[FMRW95]-=-, are defined in terms of some metric space. Typically for these problems, there are linear lower bounds on the competitive ratio of deterministic algorithms. Although randomization against an oblivio... |

4 | Randomized Algorithms for Multiprocess or Page Migration - Westbrook - 1991 |

1 | What To Do With You Free Time: On-line Algorithms for Infrequent Requests - Blum, Furst, et al. - 1996 |