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Routing in Ad Hoc Networks Using a Spine
, 1997
"... We present a twolevel hierarchical routing architecture for ad hoc networks. Within each lower level cluster, we describe a selforganizing, dynamic spine structure to (a) propagate topology changes, (b) compute updated routes in the background, and (c) provide backup routes in case of transient f ..."
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Cited by 107 (0 self)
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We present a twolevel hierarchical routing architecture for ad hoc networks. Within each lower level cluster, we describe a selforganizing, dynamic spine structure to (a) propagate topology changes, (b) compute updated routes in the background, and (c) provide backup routes in case of transient failures of the primary routes. We analyze and bound the worst case of movements between upper level clusters to show that this hierarchical architecture scales well with network size. 1 Introduction Ad hoc networks are multihop networks in which mobile hosts share a scarce wireless channel. In ad hoc networks, the network topology changes frequently. Hence, routing algorithms must expend overhead either to maintain current routing and topology tables or to discover uptodate routes. Currently, most routing algorithms for ad hoc networks are flat, that is, designed with only one level of hierarchy. These flat routing algorithms can suffer from excessive overhead as network sizes increase. I...
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.
PolylogTime and NearLinear Work Approximation Scheme for Undirected Shortest Paths
 Journal of the ACM
, 2000
"... 1 Shortest paths computations constitute one of the most fundamental network problems. Nonetheless, known parallel shortestpaths algorithms are generally inefficient: they perform significantly more work (product of time and processors) than their sequential counterparts. This gap, known in the lit ..."
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Cited by 31 (1 self)
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1 Shortest paths computations constitute one of the most fundamental network problems. Nonetheless, known parallel shortestpaths algorithms are generally inefficient: they perform significantly more work (product of time and processors) than their sequential counterparts. This gap, known in the literature as the “transitive closure bottleneck, ” poses a longstanding open problem. Our main result is an O(mn ɛ0 + s(m + n 1+ɛ0)) work polylogtime randomized algorithm that computes paths within (1 + O(1 / polylog n)) of shortest from s source nodes to all other nodes in weighted undirected networks with n nodes and m edges (for any fixed ɛ0> 0). This work bound nearly matches the Õ(sm) sequential time. In contrast, previous polylogtime algorithms required min { Õ(n3), Õ(m2)} work (even when s = 1), and previous nearlinear work algorithms required nearO(n) time. We also present faster sequential algorithms that provide good approximate distances only between “distant ” vertices: We obtain an O((m + sn)n ɛ0) time algorithm that computes paths of weight (1 + O(1 / polylog n))dist + O(wmax polylog n), where dist is the corresponding distance and wmax is the maximum edge weight. Our chief instrument, which is of independent interest, are efficient constructions of sparse hop sets. A (d, ɛ)hop set of a network G = (V, E) is a set E ∗ of new weighted edges such that minimumweight dedge paths in (V, E ∪ E ∗ ) have weight within (1 + ɛ) of the respective distances in G. We construct hop sets of size O(n 1+ɛ0) where ɛ = O(1 / polylog n) and d = O(polylog n). 1
A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs
, 1993
"... In this paper we give a fully dynamic approximation scheme for maintaining allpairs shortest paths in planar networks. Given an error parameter ffl such that 0 ! ffl, our algorithm maintains approximate allpairs shortestpaths in an undirected planar graph G with nonnegative edge lengths. The a ..."
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Cited by 19 (1 self)
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In this paper we give a fully dynamic approximation scheme for maintaining allpairs shortest paths in planar networks. Given an error parameter ffl such that 0 ! ffl, our algorithm maintains approximate allpairs shortestpaths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1 + ffl factor. The time bounds for both query and update for our algorithm is O(ffl \Gamma1 n 2=3 log 2 n log D), where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the adds is amortized. Our approximation algorithm is based upon a novel technique for approximately representing allpairs shortest paths among a selected subset of the nodes by a sparse substitute graph. Research supported by NSF grant CCR9012357 and NSF PYI award CCR9157620, together with PYI matching funds from Thinking Machines Corporation and Xerox Corporation. Addit...
A randomized parallel algorithm for singlesource shortest paths
 Journal of Algorithms
, 1997
"... Abstract We give a randomized parallel algorithm for computing singlesource shortest paths in weighted digraphs. We show that the exact shortest path problem can be efficiently reduced to solving a series of approximate shortestpath subproblems. Our algorithm for the approximate shortestpath prob ..."
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Cited by 16 (1 self)
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Abstract We give a randomized parallel algorithm for computing singlesource shortest paths in weighted digraphs. We show that the exact shortest path problem can be efficiently reduced to solving a series of approximate shortestpath subproblems. Our algorithm for the approximate shortestpath problem is based on a technique used by Ullman and Yannakakis in a parallel algorithm for breadthfirst search. 1 Introduction One of the most fundamental and ubiquitous problems in combinatorial optimization is finding singlesource shortest paths in a weighted graph. Aside from being important in its own right, the problem arises in algorithms for many other problems, especially those related to flow. In view of the importance of the singlesource shortest paths problem, it is unfortunate that all known parallel algorithms for this problem are very inefficient on sparse graphs. This inability to make efficient use of parallelism in computing shortest paths is of both theoretical and practical significance. A fast and efficient parallel algorithm for this problem remains a major goal in the design of parallel graph algorithms.
Randomized Parallel Algorithms
 IN SOLVING COMBINATORIAL PROBLEMS IN PARALLEL, VOLUME 1054 OF LNCS
, 1996
"... ..."
Parallel and Dynamic ShortestPath Algorithms for Sparse Graphs
, 1995
"... ere capable of anything and instilling in us a desire to be the best in whatever we did. I would also like to thank my high school teachers Mr. Jaypal Chandra and Ms. Bhuvaneshvari for showing me that education could be fun, and Professors. M.V. Tamhankar, and H. Subramanian for some truly inspiring ..."
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ere capable of anything and instilling in us a desire to be the best in whatever we did. I would also like to thank my high school teachers Mr. Jaypal Chandra and Ms. Bhuvaneshvari for showing me that education could be fun, and Professors. M.V. Tamhankar, and H. Subramanian for some truly inspiring courses in mathematics. At Brown, I would like to thank Professors Philip Klein, Roberto Tamassia, and Jeff Vitter for advising this thesis and for teaching me much of what I know. I would like to thank Prof. Vitter for introducing me to research and for his confidence in my abilities. His constant encouragement kept me motivated during times when the going was tough. I would like to thank Prof. Tamassia for encouraging my interest in dynamic graph algorithms and for suggesting the problem solved in Chapter 5. A large portion of the results in this thesis were obtained in joint work with Prof. Phil Klein. I would like to thank him for his boundless enthusiasm for research and for the innume
ΔStepping: A Parallel Single Sourche Shortest . . .
 IN ESA ’98: PROCEEDINGS OF THE 6TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS
, 1998
"... In spite of intensive research, little progress has been made towards fast and workefficient parallel algorithms for the single source shortest path problem. Our \Deltastepping algorithm, a generalization of Dial's algorithm and the BellmanFord algorithm, improves this situation at least in t ..."
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In spite of intensive research, little progress has been made towards fast and workefficient parallel algorithms for the single source shortest path problem. Our \Deltastepping algorithm, a generalization of Dial's algorithm and the BellmanFord algorithm, improves this situation at least in the following "averagecase" sense: For random directed graphs with edge probability n and uniformly distributed edge weights a PRAM version works in expected time O using linear work. The algorithm also allows for efficient adaptation to distributed memory machines. Implementations show that our approach works on real machines. As a side effect, we get a simple linear time sequential algorithm for a large class of not necessarily random directed graphs with random edge weights.
© 1998 SpringerVerlag New York Inc. A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs 1
"... Abstract. In this paper we give a fully dynamic approximation scheme for maintaining allpairs shortest paths in planar networks. Given an error parameter ε such that 0 <ε, our algorithm maintains approximate allpairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The ..."
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Abstract. In this paper we give a fully dynamic approximation scheme for maintaining allpairs shortest paths in planar networks. Given an error parameter ε such that 0 <ε, our algorithm maintains approximate allpairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1 + ε factor. The time bounds for both query and update for our algorithm is O(ε−1n2/3 log2 n log D), where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the additions is amortized. Our approximation algorithm is based upon a novel technique for approximately representing allpairs shortest paths among a selected subset of the nodes by a sparse substitute graph. Key Words.