Aaron Bernstein

A nearly optimal oracle for avoiding failed vertices and edges

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(n2) that has an O(1) query time, and an ~O(n2√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(n2) space requirement, but it has an improved construction time of ~O(mn), and it is deterministic. Note that O(1) query, O(n2) 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.

Fully Dynamic (2 + epsilon) Approximate All-Pairs Shortest Paths with Fast Query and Close to Linear Update Time

For any fixed

Fully Dynamic Matching in Bipartite Graphs

1 ≫ \eps ≫ 0

Maintaining shortest paths under deletions in weighted directed graphs: [extended abstract]

we present a fully dynamic algorithm for maintaining

Deterministic decremental single source shortest paths: beyond the o(mn) bound

(2 + \eps)

Deterministic partially dynamic single source shortest paths for sparse graphs

-approximate all-pairs shortest paths in undirected graphs with positive edge weights. We use a randomized (Las Vegas) update algorithm (but a deterministic query procedure), so the time given is the \emph{expected} amortized update time. \\\indent Our query time

Online bipartite matching with amortized O (log 2 n ) replacements

O(\log \log \log n)

Deterministic Partially Dynamic Single Source Shortest Paths in Weighted Graphs.

. The update time is

Distance-Preserving Graph Contractions

\widetilde{O}(mn^{O(1/\sqrt{\log n})}\log(nR))

General Bounds for Incremental Maximization

, where