Introduction to algorithms
High probability parallel transitive-closure algorithms
SIAM Journal on Computing
A parallel randomized approximation scheme for shortest paths
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Efficient parallel shortest-paths in digraphs with a separator decomposition
Journal of Algorithms
Using selective path-doubling for parallel shortest-path computations
Journal of Algorithms
Time-work tradeoffs for parallel algorithms
Journal of the ACM (JACM)
Fast Algorithms for Constructing t-Spanners and Paths with Stretch t
SIAM Journal on Computing
Exact and Approximate Distances in Graphs - A Survey
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
Δ-stepping: a parallelizable shortest path algorithm
Journal of Algorithms
New constructions of (α, β)-spanners and purely additive spanners
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
All-pairs nearly 2-approximate shortest paths in O(n2polylogn) time
Theoretical Computer Science
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Additive spanners and (α, β)-spanners
ACM Transactions on Algorithms (TALG)
Additive spanners in nearly quadratic time
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Lower bounds for local monotonicity reconstruction from transitive-closure spanners
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Transitive-closure spanners: a survey
Property testing
Transitive-closure spanners: a survey
Property testing
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
Improved approximation for the directed spanner problem
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Label cover instances with large girth and the hardness of approximating basic k-spanner
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Approximation algorithms for spanner problems and Directed Steiner Forest
Information and Computation
Runtime guarantees for regression problems
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Parallel graph decompositions using random shifts
Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
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Shortest paths computations constitute one of the most fundamental network problems. Nonetheless, known parallel shortest-paths 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 long-standing open problem. Our main result is an Omne0+s m+n1+e0 work polylog-time 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 &egr;00). This work bound nearly matches the O&d5;sm sequential time. In contrast, previous polylog-time algorithms required nearly minO&d5; n3,O&d5; m2 work (even when s=1), and previous near-linear work algorithms required near-O(n) time. We also present faster sequential algorithms that provide good approximate distances only between “distant” vertices: We obtain an Om+snne 0 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,&egr;)-hop set of a network G=(V,E) is a set E* of new weighted edges such that mimimum-weight d-edge paths in V,E∪E* have weight within (1+&egr;) of the respective distances in G. We construct hop sets of size On1+e0 where &egr;=O(1/polylog n) and d=O(polylog n).