## A Nearly Optimal Oracle for Avoiding Failed Vertices and Edges

##### Author(s)

Bernstein, Aaron; Karger, David R.
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We present an improved oracle for the distance sensitivity problem. The goal is to preprocess a directed 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. The previous best algorithm produces an oracle of size ~O(n[superscript 2]) that has an O(1) query time, and an ~O(nn[superscript 2]√m) construction time. It was a randomized Monte Carlo algorithm that worked with high probability. Our oracle also has a constant query time and an ~O(n[superscript 2]) space requirement, but it has an improved construction time of ~O(mn), and it is deterministic. Note that O(1) query, O(n[superscript 2]) space, and O(mn) construction time is also the best known bound (up to logarithmic factors) for the simpler problem of finding all pairs shortest paths in a weighted, directed graph. Thus, barring improved solutions to the all pairs shortest path problem, our oracle is optimal up to logarithmic factors.

##### Date issued

2009##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### Journal

Proceedings of the 41st annual ACM symposium on Theory of computing

##### Publisher

Association for Computing Machinery

##### Citation

Bernstein, Aaron, and David Karger. “A nearly optimal oracle for avoiding failed vertices and edges.” Proceedings of the 41st annual ACM symposium on Theory of computing. Bethesda, MD, USA: ACM, 2009. 101-110.

Version: Author's final manuscript

##### ISBN

978-1-60558-506-2