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SWAT '90 Proceedings of the 2nd Scandinavian Workshop on Algorithm Theory
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WDAG '92 Proceedings of the 6th International Workshop on Distributed Algorithms
A Single Source Shortest Path Algorithm for a Planar Distributed Network
STACS '85 Proceedings of the 2nd Symposium of Theoretical Aspects of Computer Science
Compact Oracles for Reachability and Approximate Distances in Planar Digraphs
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ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
f-sensitivity distance Oracles and routing schemes
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Information Processing Letters
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ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We study the s-sources almost shortest paths(abbreviated s-ASP) problem. Given an unweightedgraph G = (V,E),and a subset S ⊆ Vof s nodes, the goal is to compute almostshortest paths between all the pairs of nodes S× V. We devise an algorithm withrunning timeO(∣E∣nρ+ s ·n1 + ζ)for this problem that computes the pathsPu,wfor all pairs (u,w) ∈S × V such that thelength ofPu,wis at most (1 + ε)dG(u,w)+ β(ζ,ρ,ε), andβ(ζ,ρ,ε) is constant whenζ, ρ, and ε are arbitrarily smallconstants.We also devise a distributed protocol for thes-ASP problem that computes the pathsPu,was above, and has time and communication complexities ofO(s ·Diam(G) +n1 +ζ/2) (respectively,O(s ·Diam(G) log3n + n1+ ζ/2 log n)) andO(∣E∣nρ +s · n1+ ζ) (respectively,O(∣E∣nρ +s · n1+ ζ +n1 + ρ +ζ(ρ − ζ/2)/2)) in thesynchronous (respectively asynchronous) setting.Our sequential algorithm, as well as the distributed protocol,is based on a novel algorithm for constructing (1 +ε, β(ζ,ρ, ε))-spannersof size O(n1+ ζ), developed in this article. Thisalgorithm has running time ofO(∣E∣nρ), which issignificantly faster than the previously known algorithm given inElkin and Peleg [2001], whose running time isÕ(n2+ ρ). We also develop the firstdistributed protocol for constructing (1 +ε,β)-spanners. The communication complexity ofthis protocol is near optimal.