Constructing the Spanners of Graphs in Parallel
IPPS '96 Proceedings of the 10th International Parallel Processing Symposium
Estimating All Pairs Shortest Paths in Restricted Graph Families: A Unified Approach
WG '01 Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science
As Good as It Gets: Competitive Fault Tolerance in Network Structures
SSS '09 Proceedings of the 11th International Symposium on Stabilization, Safety, and Security of Distributed Systems
Estimating all pairs shortest paths in restricted graph families: a unified approach
Journal of Algorithms
f-sensitivity distance Oracles and routing schemes
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Near linear-work parallel SDD solvers, low-diameter decomposition, and low-stretch subgraphs
Proceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures
Fault-tolerant compact routing schemes for general graphs
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
Fault-tolerant compact routing schemes for general graphs
Information and Computation
Small stretch pairwise spanners
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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The distance between two vertices in a weighted graph is the weight of a minimum-weight path between them. A path has stretch t if its weight is at most t times the distance between its end points. We consider a weighted undirected graph G=(V, E) and present algorithms that compute paths with stretch 2/spl les/t/spl les/log n. We present a O/spl tilde/((m+k)n/sup (2+/spl epsiv///t)) time randomized algorithm that finds paths between k specified pairs of vertices and a O/spl tilde/((m+ns)n/sup 2(1+log(n)/ /sup m+/spl epsiv/)/t/) deterministic algorithm that finds paths from s specified sources to all other vertices (for any fixed /spl epsiv/0), where n=|V| and m=|E|. This improves significantly over the slower O/spl tilde/(min{k, n}m) exact shortest paths algorithms and a previous O/spl tilde/(mn/sup 64/t/+kn/sup 32/t/) time algorithm by Awerbuch et al. A t-spanner of a graph G is a set of weighted edges on the vertices of G such that distances in the spanner are not smaller and within a factor of t from the corresponding distances in G. Previous work was concerned with bounding the size and efficiently constructing t-spanners. We construct t-spanners of size O/spl tilde/(n/sup 1+(2+/spl epsiv///t)) in O/spl tilde/(mn/sup (2+/spl epsiv///t)) expected time (for any fixed /spl epsiv/0), what constitutes a faster construction (by a factor of n/sup (3+2//t)) of sparser spanners than was previously attainable. We also provide efficient parallel constructions. Our algorithms are based on new structures called pairwise-covers and a novel approach to construct them efficiently.