Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time
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We consider the problem of building Distance Sensitivity Oracles (DSOs). Given a directed graph G=(V, E) with edge weights in {1, 2, …, M}, we need to preprocess it into a data structure, and answer the following queries: given vertices u,v∈ V and a failed vertex or edge f∈ (V∪ E), output the length of the shortest path from u to v that does not go through f. Our main result is a simple DSO with Õ(n^2.7233M) preprocessing time and O(1) query time. Moreover, if the input graph is undirected, the preprocessing time can be improved to Õ(n^2.6865M). The preprocessing algorithm is randomized with correct probability ≥ 1-1/n^C, for a constant C that can be made arbitrarily large. Previously, there is a DSO with Õ(n^2.8729M) preprocessing time and polylog(n) query time [Chechik and Cohen, STOC'20]. At the core of our DSO is the following observation from [Bernstein and Karger, STOC'09]: if there is a DSO with preprocessing time P and query time Q, then we can construct a DSO with preprocessing time P+Õ(n^2)· Q and query time O(1). (Here Õ(·) hides polylog(n) factors.)
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