Results 1  10
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20
Probabilistic Approximation of Metric Spaces and its Algorithmic Applications
 In 37th Annual Symposium on Foundations of Computer Science
, 1996
"... The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to lowdistortion embeddings in lowdimensional spaces [LLR94], having many algorithmic applications. This paper provides a novel technique for the analysis of randomized ..."
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Cited by 323 (28 self)
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The goal of approximating metric spaces by more simple metric spaces has led to the notion of graph spanners [PU89, PS89] and to lowdistortion embeddings in lowdimensional 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 probabilisticallyapproximates another metric space. We prove that any metric space can be probabilisticallyapproximated by hierarchically wellseparated trees (HST) with a polylogarithmic distortion. These metric spaces are "simple" as being: (1) tree metrics. (2) natural for applying a divideandconquer algorithmic approach. The technique presented is of particular interest in the context of online computation. A large number of online al...
On Approximating Arbitrary Metrics by Tree Metrics
 In Proceedings of the 30th Annual ACM Symposium on Theory of Computing
, 1998
"... This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hi ..."
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Cited by 260 (13 self)
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This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hierarchically wellseparated tree" metric spaces. This has proved as a useful technique for simplifying the solutions to various problems.
Concurrent Online Tracking of Mobile Users
 J. ACM
, 1991
"... This paper deals with the problem of maintaining a distributed directory server, that enables us to keep track of mobile users in a distributed network in the presence of concurrent requests. The paper uses the graphtheoretic concept of regional matching for implementing efficient tracking mechanis ..."
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Cited by 207 (7 self)
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This paper deals with the problem of maintaining a distributed directory server, that enables us to keep track of mobile users in a distributed network in the presence of concurrent requests. The paper uses the graphtheoretic concept of regional matching for implementing efficient tracking mechanisms. The communication overhead of our tracking mechanism is within a polylogarithmic factor of the lower bound. 1 Introduction Since the primary function of a communication network is to provide communication facilities between users and processes in the system, one of the key problems such a network faces is the need to be able to Department of Mathematics and Lab. for Computer Science, M.I.T., Cambridge, MA 02139, USA. Email: baruch@theory.lcs.mit.edu. Supported by Air Force Contract TNDGAFOSR860078, ARO contract DAAL0386K0171, NSF contract CCR8611442, DARPA contract N0001489J 1988, and a special grant from IBM. y Departmentof Applied Mathematicsand Computer Science, The Weizm...
Excluded Minors, Network Decomposition, and Multicommodity Flow
, 1993
"... In this paper we show that, given a graph and parameters ffi and r, we can find either a Kr;r minor or an edgecut of size O(mr=ffi) whose removal yields components of weak diameter O(r 2 ffi); i.e., every pair of nodes in such a component are at distance O(r 2 ffi) in the original graph. Usi ..."
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Cited by 108 (6 self)
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In this paper we show that, given a graph and parameters ffi and r, we can find either a Kr;r minor or an edgecut of size O(mr=ffi) whose removal yields components of weak diameter O(r 2 ffi); i.e., every pair of nodes in such a component are at distance O(r 2 ffi) in the original graph. Using this lemma, we improve the best known bounds for the mincut maxflow ratio for multicommodity flows in graphs with forbidden small minors. In general graphs, it was known that the ratio is O(log k) for the uniformdemand case (the case where there is a unitdemand commodity between every pair of nodes), and that the ratio is O(log 2 k) for arbitrary demands, where k is the number of commodities. In this paper we show that for graphs excluding any fixed graph as a minor (e.g. planar graphs or boundedgenus graphs), the ratio is O(1) for the uniformdemand case and O(log k) for the arbitrary demand case. For such graphs, our method yields minratio cut approximation algorithms wit...
Nearlinear time construction of sparse neighborhood covers
 SIAM Journal on Computing
, 1998
"... Abstract. This paper introduces a nearlinear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to nearlinear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, ..."
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Cited by 43 (4 self)
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Abstract. This paper introduces a nearlinear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to nearlinear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, approximate shortest paths, and weight and distancepreserving graph spanners. In particular, an O(log n) approximation to the kshortest paths problem on an nvertex, Eedge graph is obtained that runs in Õ (n + E + k) time.
Nearly optimal distributed edge colouring in O(log log n) rounds
 in Proceedings of the Eight Annual ACMSIAM Symposium on Discrete Algorithms (SODA 97
, 1996
"... An extremely simple distributed randomized algorithm is presented which with high probability properly edge colours a given graph using (1+ ")\Delta colours, where \Delta is the maximum degree of the graph and " is any given positive constant. The algorithm is very fast. In particular, for graphs wi ..."
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Cited by 26 (7 self)
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An extremely simple distributed randomized algorithm is presented which with high probability properly edge colours a given graph using (1+ ")\Delta colours, where \Delta is the maximum degree of the graph and " is any given positive constant. The algorithm is very fast. In particular, for graphs with sufficiently large vertex degrees (larger than polylog n, but smaller than any positive power of n), the algorithm requires only O(log log n) communication rounds. The algorithm is inherently distributed, but can be implemented on the PRAM, where it requires O(m\Delta) processors and O(log \Delta log log n) time, or in a sequential setting, where it requires O(m\Delta) time. 1 Introduction The edge colouring problem is a much studied problem in the theory of algorithms, graph theory, and combinatorics, whose relevance to computer science stems from its applications to scheduling and resource allocation problems [6, 11, 14, 17, 19, 12, 24, among others]. Given an input graph, the problem ...
Advances in metric embedding theory
 IN STOC ’06: PROCEEDINGS OF THE THIRTYEIGHTH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2006
"... Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians ..."
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Cited by 26 (8 self)
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Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians as well as computer scientists and has been applied in many algorithmic applications. A cornerstone of the field is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(log n) distortion. Bourgain’s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is possible that an embedding can do much better in terms of the average distortion. Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs. In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain’s theorem. In fact, our embedding possesses a much stronger property. We define the ℓqdistortion of a uniformly distributed pair of points. Our embedding achieves the best possible ℓqdistortion for all 1 ≤ q ≤ ∞ simultaneously. These results have several algorithmic implications, e.g. an O(1) approximation for the unweighted uncapacitated quadratic assignment problem. The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result im
Sparser: A paradigm for running distributed algorithms
, 1990
"... This paper introduces a transformer for improving the communication complexity of several classes of distributed algorithms. The transformer takes a distributed algorithm whose message complexity is O(f \Delta m) and produces a new distributed algorithm to solve the same problem with O(f \Delta n lo ..."
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Cited by 25 (0 self)
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This paper introduces a transformer for improving the communication complexity of several classes of distributed algorithms. The transformer takes a distributed algorithm whose message complexity is O(f \Delta m) and produces a new distributed algorithm to solve the same problem with O(f \Delta n log n+m log n) message complexity, where n and m are the total number of nodes and links in the network, and f is an arbitrary function of n and m. Applying our paradigm to the standard all shortest paths algorithm [Gal76, Gal82, Seg83] yields a new algorithm which solves the problem in O(n 2 log n) messages (The previous best that we know of is O(m \Delta n) messages). When applied to the O(m \Delta log 3 n) breadthfirst search algorithm of Awerbuch and Peleg [AP90a] our paradigm yields an O(m+ n \Delta log 4 n) messages algorithm. 1 introduction One way to run a distributed algorithm is to collect all its inputs to one node, run a sequential algorithm on all the inputs at this ...
Fast Distributed Network Decompositions and Covers
 Journal of Parallel and Distributed Computing
, 1996
"... This paper presents deterministic sublineartime distributed algorithms for network decomposition and for constructing a sparse neighborhood cover of a network. The latter construction leads to improved distributed preprocessing time for a number of distributed algorithms, including allpairs shorte ..."
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Cited by 25 (4 self)
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This paper presents deterministic sublineartime distributed algorithms for network decomposition and for constructing a sparse neighborhood cover of a network. The latter construction leads to improved distributed preprocessing time for a number of distributed algorithms, including allpairs shortest paths computation, load balancing, broadcast, and bandwidth management. A preliminary version of this paper appeared in the Proceedings of the Eleventh Annual ACM Symposium on the Principles of Distributed Computing. y Lab. for Computer Science, MIT, Cambridge, MA 02139. Supported by Air Force Contract AFOSR F4962092J0125, NSF contract 9114440CCR, DARPA contracts N0001491J1698 and N00014J921799, and a special grant from IBM. z Dept. of Mathematics and Lab. for Computer Science, MIT. Supported in part by an NSF Postdoctoral Research Fellowship and an ONR grant provided to the Radcliffe Bunting Institute. x Dept. of Math Sciences, Johns Hopkins University, Baltimore, MD 21...
NearLinear Cost Sequential and Distributed Constructions of Sparse Neighborhood Covers
 in Proceedings of the 34th Annual Symposium on Foundations of Computer Science (FOCS
, 1993
"... This paper introduces the first nearlinear (specifically, O(E log n + n log 2 n)) time algorithm for constructing a sparse neighborhood cover in sequential and distributed environments. This automatically implies analogous improvements (from quadratic to nearlinear) to all the results in the li ..."
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Cited by 20 (0 self)
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This paper introduces the first nearlinear (specifically, O(E log n + n log 2 n)) time algorithm for constructing a sparse neighborhood cover in sequential and distributed environments. This automatically implies analogous improvements (from quadratic to nearlinear) to all the results in the literature that rely on network decompositions, both in sequential and distributed domains, including adaptive routing schemes with ~ O (1) 1 stretch and memory, small edge cuts in planar graphs, sequential algorithms for dynamic approximate shortest paths with ~ O (E) cost for edge insertion/deletion and ~ O (1) time to answer shortestpath queries, weight and distancepreserving graph spanners with ~ O (E) running time and space, and distributed asynchronous "fromscratch" BreadthFirstSearch and network synchronizer constructions with ~ O (1) message and space overhead (down from O(n)). Lab. for Computer Science, MIT, Cambridge, MA 02139. Supported by Air Force Contract AFOSR F4962092 ...