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The SmallWorld Phenomenon: An Algorithmic Perspective
 in Proceedings of the 32nd ACM Symposium on Theory of Computing
, 2000
"... Long a matter of folklore, the “smallworld phenomenon ” — the principle that we are all linked by short chains of acquaintances — was inaugurated as an area of experimental study in the social sciences through the pioneering work of Stanley Milgram in the 1960’s. This work was among the first to m ..."
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Cited by 621 (6 self)
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Long a matter of folklore, the “smallworld phenomenon ” — the principle that we are all linked by short chains of acquaintances — was inaugurated as an area of experimental study in the social sciences through the pioneering work of Stanley Milgram in the 1960’s. This work was among the first to make the phenomenon quantitative, allowing people to speak of the “six degrees of separation ” between any two people in the United States. Since then, a number of network models have been proposed as frameworks in which to study the problem analytically. One of the most refined of these models was formulated in recent work of Watts and Strogatz; their framework provided compelling evidence that the smallworld phenomenon is pervasive in a range of networks arising in nature and technology, and a fundamental ingredient in the evolution of the World Wide Web. But existing models are insufficient to explain the striking algorithmic component of Milgram’s original findings: that individuals using local information are collectively very effective at actually constructing short paths between two points in a social network. Although recently proposed network models are rich in short paths, we prove that no decentralized algorithm, operating with local information only, can construct short paths in these networks with nonnegligible probability. We then define an infinite family of network models that naturally generalizes the WattsStrogatz model, and show that for one of these models, there is a decentralized algorithm capable of finding short paths with high probability. More generally, we provide a strong characterization of this family of network models, showing that there is in fact a unique model within the family for which decentralized algorithms are effective.
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...
Distributed Object Location in a Dynamic Network
, 2004
"... Modern networking applications replicate data and services widely, leading to a need for locationindependent routingthe ability to route queries to objects using names independent of the objects' physical locations. Two important properties of such a routing infrastructure are routing locality a ..."
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Cited by 167 (16 self)
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Modern networking applications replicate data and services widely, leading to a need for locationindependent routingthe ability to route queries to objects using names independent of the objects' physical locations. Two important properties of such a routing infrastructure are routing locality and rapid adaptation to arriving and departing nodes. We show how these two properties can be efficiently achieved for certain network topologies. To do this, we present a new distributed algorithm that can solve the nearestneighbor problem for these networks. We describe our solution in the context of Tapestry, an overlay network infrastructure that employs techniques proposed by Plaxton et al. [24].
Compact Routing with Minimum Stretch
 Journal of Algorithms
"... We present the first universal compact routing algorithm with maximum stretch bounded by 3 that uses sublinear space at every vertex. The algorithm uses local routing tables of size O(n 2=3 log 4=3 n) and achieves paths that are most 3 times the length of the shortest path distances for all node ..."
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Cited by 112 (5 self)
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We present the first universal compact routing algorithm with maximum stretch bounded by 3 that uses sublinear space at every vertex. The algorithm uses local routing tables of size O(n 2=3 log 4=3 n) and achieves paths that are most 3 times the length of the shortest path distances for all nodes in an arbitrary weighted undirected network. This answers an open question of Gavoille and Gengler who showed that any universal compact routing algorithm with maximum stretch strictly less than 3 must use\Omega\Gamma n) local space at some vertex. 1 Introduction Let G = (V; E) with jV j = n be a labeled undirected network. Assuming that a positive cost, or distance is assigned with each edge, the stretch of path p(u; v) from node u to node v is defined as jp(u;v)j jd(u;v)j , where jd(u; v)j is the length of the shortest u \Gamma v path. The approximate allpairs shortest path problem involves a tradeoff of stretch against time short paths with stretch bounded by a constant are com...
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...
Routing with Polynomial CommunicationSpace Tradeoff
 SIAM Journal on Discrete Mathematics
, 1993
"... This paper presents a family of memorybalanced routing schemes that use relatively short paths while storing relatively little routing information. The hierarchical schemes H k (for every integer k 1) guarantee a stretch factor of O(k 2 ) on the length of the routes and require storing at most O ..."
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Cited by 75 (13 self)
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This paper presents a family of memorybalanced routing schemes that use relatively short paths while storing relatively little routing information. The hierarchical schemes H k (for every integer k 1) guarantee a stretch factor of O(k 2 ) on the length of the routes and require storing at most O(kn 1 k log n log D) bits of routing information per vertex in an nprocessor network with diameter D. The schemes are nameindependent and applicable to general networks with arbitrary edge weights. This improves on previous designs whose stretch bound was exponential in k. Key words: Communication networks, routing tables, communicationspace tradeoffs, graph covers. Dept. of Mathematics and Lab. for Computer Science, M.I.T., Cambridge, MA 02139; ARPANET: baruch@theory.lcs.mit.edu. Supported by Air Force Contract TNDGAFOSR860078, ARO contract DAAL0386K0171, NSF contract CCR8611442, DARPA contract N0001489J1988, and a special grant from IBM. y Department of Applied Mathemati...
Compact Distributed Data Structures for Adaptive Routing
 In Proc. 21st ACM Symp. on Theory of Computing
, 1989
"... In designing a routing scheme for a communication network it is desirable to use as short as possible paths for routing messages, while keeping the routing information stored in the processors' local memory as succinct as possible. The efficiency of a routing scheme is measured in terms of its stret ..."
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Cited by 71 (7 self)
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In designing a routing scheme for a communication network it is desirable to use as short as possible paths for routing messages, while keeping the routing information stored in the processors' local memory as succinct as possible. The efficiency of a routing scheme is measured in terms of its stretch factor  the maximum ratio between the cost of a route computed by the scheme and that of a cheapest path connecting the same pair of vertices. This paper presents a family of adaptive routing schemes for general networks. The hierarchical schemes HS k (for every fixed k 1) guarantee a stretch factor of O(k 2 \Delta 3 k ) and require storing at most O \Gamma kn 2 k log n \Delta bits of routing information per vertex. The new important features, that make the schemes appropriate for adaptive use, are ffl applicability to networks with arbitrary edge costs; ffl nameindependence, i.e., usage of original names; ffl a balanced distribution of the memory; ffl an efficient onli...
Approximation Algorithms for Steiner and Directed Multicuts
 Journal of Algorithms
, 1996
"... In this paper we consider the steiner multicut problem. This is a generalization of the minimum multicut problem where instead of separating node pairs, the goal is to find a minimum weight set of edges that separates all given sets of nodes. A set is considered separated if it is not contained in ..."
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Cited by 27 (1 self)
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In this paper we consider the steiner multicut problem. This is a generalization of the minimum multicut problem where instead of separating node pairs, the goal is to find a minimum weight set of edges that separates all given sets of nodes. A set is considered separated if it is not contained in a single connected component. We show an O(log 3 (kt)) approximation algorithm for the steiner multicut problem, where k is the number of sets and t is the maximum cardinality of a set. This improves the O(t log k) bound that easily follows from the previously known multicut results. We also consider an extension of multicuts to directed case, namely the problem of finding a minimumweight set of edges whose removal ensures that none of the strongly connected components includes one of the prespecified k node pairs. In this paper we describe an O(log 2 k) approximation algorithm for this directed multicut problem. If k n, this represents and an improvement over the O(logn log ...
Bubbles: Adaptive Routing Scheme for HighSpeed Dynamic Networks
 SIAM Journal on Computing
, 1997
"... This paper presents the first dynamic routing scheme for highspeed networks. The scheme is based on a hierarchical bubbles partition of the underlying communication graph. ..."
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Cited by 24 (9 self)
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This paper presents the first dynamic routing scheme for highspeed networks. The scheme is based on a hierarchical bubbles partition of the underlying communication graph.
Upper and Lower Bounds for Routing Schemes in Dynamic Networks
 Proc. 30th Symp. on Foundations of Computer Science
, 1989
"... We present an algorithm and two lower bounds for the problem of constructing and maintaining routing schemes in dynamic networks. The algorithm distributively assignes addresses to nodes and constructs routing tables in a dynamically growing tree. The resulting routing scheme routes data messages ov ..."
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Cited by 23 (0 self)
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We present an algorithm and two lower bounds for the problem of constructing and maintaining routing schemes in dynamic networks. The algorithm distributively assignes addresses to nodes and constructs routing tables in a dynamically growing tree. The resulting routing scheme routes data messages over the shortest path between any source and destination, assigns addresses of O(log^2 n) bits to each node, and uses in its routing tables O(log^3 n) bits of memory per incident link, where n is the final number of nodes in the tree. The amortized communication cost of the algorithm is O(log n) messages per node. We also give two lower bounds on the tradeoff between the quality of routing schemes (i.e., their stretch factor) and their amortized communication cost in general dynamic networks.