Fast algorithms for finding nearest common ancestors
SIAM Journal on Computing
Fibonacci heaps and their uses in improved network optimization algorithms
Journal of the ACM (JACM)
Recursive star-tree parallel data structure
SIAM Journal on Computing
Oracles for distances avoiding a link-failure
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Mathematics for the Analysis of Algorithms
Mathematics for the Analysis of Algorithms
Vickrey Prices and Shortest Paths: What is an Edge Worth?
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
A new approach to dynamic all pairs shortest paths
Journal of the ACM (JACM)
Dual-failure distance and connectivity oracles
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
A nearly optimal oracle for avoiding failed vertices and edges
Proceedings of the forty-first annual ACM symposium on Theory of computing
Fault-tolerant spanners for general graphs
Proceedings of the forty-first annual ACM symposium on Theory of computing
A near-linear-time algorithm for computing replacement paths in planar directed graphs
ACM Transactions on Algorithms (TALG)
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
f-sensitivity distance Oracles and routing schemes
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Fault Tolerant Spanners for General Graphs
SIAM Journal on Computing
Finding Alternative Shortest Paths in Spatial Networks
ACM Transactions on Database Systems (TODS)
Replacement Paths and Distance Sensitivity Oracles via Fast Matrix Multiplication
ACM Transactions on Algorithms (TALG)
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We present improved oracles for the distance sensitivity problem. The goal is to preprocess a graph G = (V,E) with non-negative edge weights to answer queries of the form: what is the length of the shortest path from x to y that does not go through some failed vertex or edge f. There are two state of the art algorithms for this problem. The first produces an oracle of size Õ(n2) that has an O(1) query time, and an Õ(mn2) construction time. The second oracle has size O(n2.5), but the construction time is only Õ(mn1.5). We present two new oracles that substantially improve upon both of these results. Both oracles are constructed with randomized, Monte Carlo algorithms. For directed graphs with non-negative edge weights, we present an oracle of size Õ(n2), which has an O(1) query time, and an Õ(n2√m) construction time. For unweighted graphs, we achieve a more general construction time of Õ(√n3 · APSP + mn), where APSP is the time it takes to compute all pairs shortest paths in an aribtrary subgraph of G.