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...

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We show how to decompose efficiently in parallel any graph into a number, ~ fl, of outerplanar subgraphs (called hammocks) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G. Frederickson and the parallel ear decomposition technique, thus we call it the hammock-on-ears decomposition. We mention that hammock-on-ears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We achieve this decomposition in O(logn log log n) time using O(n + m) CREW PRAM processors, for an n-vertex, m-edge graph or digraph. The hammock-on-ears decomposition implies a general framework for solving graph problems efficiently. Its value is demonstrated by a variety of applications on a significant class of (di)graphs, namely that of sparse (di)graphs. This class consists of all (di)graphs which have a ~ fl between 1 and \Theta(n...

Abstract. A (1 + ɛ)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The relevant measures for a distance-oracle construction are: space, query time, and preprocessing time. There are strong distance-oracle constructions known for planar graphs (Thorup) and, subsequently, minor-excluded graphs (Abraham and Gavoille). However, these require Ω(ɛ −1 n lg n) space for n-node graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, bounded-genus graphs, and minorexcluded graphs we give distance-oracle constructions that require only

Abstract. In this paper we present space-query trade-offs for external memory data structures that answer shortest path queries on planar directed graphs. For any S = Ω(N 1+ɛ)andS = O(N 2 /B), our main result is a family of structures that use S space and answer queries in O ( N2 SB) I/Os, thus obtaining optimal space-query product O(N2 /B). An S space structure can be constructed in O ( √ S · sort(N)) I/Os, where sort(N) is the number of I/Os needed to sort N elements, B is the disk block size, and N is the size of the graph. 1

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