Distance sensitivity oracles with subcubic preprocessing time and fast query time

We present the first distance sensitivity oracle (DSO) with subcubic preprocessing time and poly-logarithmic query time for directed graphs with integer weights in the range [-𝑀, 𝑀].

Weimann and Yuster [FOCS 10] presented a distance sensitivity oracle for a single vertex/edge failure with subcubic preprocessing time of 𝑂 (𝑀𝑛 𝜔+1-𝛼 ) and subquadratic query time of 𝑂 (𝑛 1+𝛼 ), where 𝛼 is any parameter in [0, 1], 𝑛 is the number of vertices, 𝑚 is the number of edges, the 𝑂 (•) notation hides poly-logarithmic factors in 𝑛 and 𝜔 < 2.373 is the matrix multiplication exponent.

Later, Grandoni and Vassilevska Williams [FOCS 12] substantially improved the query time to sublinear in 𝑛. In particular, they presented a distance sensitivity oracle for a single vertex/edge failure with 𝑂 (𝑀𝑛 𝜔+1/2 + 𝑀𝑛 𝜔+𝛼 (4-𝜔) ) preprocessing time and 𝑂 (𝑛 1-𝛼 ) query time.

Despite the substantial improvement in the query time, it still remains polynomial in the size of the graph, which may be undesirable in many settings where the graph is of large scale. A natural question is whether one can hope for a distance sensitivity oracle with subcubic preprocessing time and very fast query time (of poly-logarithmic in 𝑛).

In this paper we answer this question affirmatively by presenting a distance sensitive oracle supporting a single vertex/edge failure in subcubic 𝑂 (𝑀𝑛 2.873 ) preprocessing time for 𝜔 = 2.373, 𝑂 (𝑛 2.5 ) space and near optimal query time of 𝑂 (1).

For comparison, with the same 𝑂 (𝑀𝑛 2.873 ) preprocessing time the DSO of Grandoni and Vassilevska Williams has 𝑂 (𝑛 0.693 ) query time. In fact, the best query time their algorithm can obtain is 𝑂 (𝑀𝑛 0.385 ) (with 𝑂 (𝑀𝑛 3 ) preprocessing time).