We present a two-level hierarchical routing architecture for ad hoc networks. Within each lower level cluster, we describe a self-organizing, 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 up-to-date 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...

Abstract. This paper introduces a near-linear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to near-linear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, approximate shortest paths, and weight- and distance-preserving graph spanners. In particular, an O(log n) approximation to the k-shortest paths problem on an n-vertex, E-edge graph is obtained that runs in Õ (n + E + k) time.

In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter ffl such that 0 ! ffl, 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 + 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 all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph. Research supported by NSF grant CCR-9012357 and NSF PYI award CCR-9157620, together with PYI matching funds from Thinking Machines Corporation and Xerox Corporation. Addit...

Abstract We give a randomized parallel algorithm for computing single-source shortest paths in weighted digraphs. We show that the exact shortest path problem can be efficiently reduced to solving a series of approximate shortest-path subproblems. Our algorithm for the approximate shortest-path problem is based on a technique used by Ullman and Yannakakis in a parallel algorithm for breadth-first search. 1 Introduction One of the most fundamental and ubiquitous problems in combinatorial optimization is finding single-source 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 single-source 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.

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e assigned problem. Clearly the goal is to make "small" the probability that the algorithm solution is incorrect. The other approach consists in giving a suitable probability distribution on the input space and then designing algorithms having an efficient complexity in the average-case according to the previously defined input distribution. There are at least two motivations in studying parallel algorithms, for combinatorial optimization problems, which follow one or both of the probabilistic approaches mentioned above. The first is to speed up sequential, approximation schemes (which work efficiently at least in the average-case) for NP-hard problems of relevant importance. The second motivation is to find efficient parallel (approximated or not) solutions for P-hard problems. A further motivation which will be illustrated in this chapter is the existence of some fundamental searching-problems in combinatorial optimization, like Shortest Path Computations, Breadth and

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

In spite of intensive research, little progress has been made towards fast and work-efficient parallel algorithms for the single source shortest path problem. Our \Delta-stepping algorithm, a generalization of Dial's algorithm and the Bellman-Ford algorithm, improves this situation at least in the following "average-case" 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.

Abstract. In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter ε such that 0 <ε, our algorithm maintains approximate all-pairs 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 all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph. Key Words.