Deterministic combinatorial replacement paths and distance sensitivity oracles

In this work we derandomize two central results in graph algorithms, replacement paths and distance sensitivity oracles (DSOs) matching in both cases the running time of the randomized algorithms. For the replacement paths problem, let G = (V, E) be a directed unweighted graph with n vertices and m edges and let P be a shortest path from s to t in G. The replacement paths problem is to find for every edge e ∈ P the shortest path from s to t avoiding e. Roditty and Zwick [ICALP 2005] obtained a randomized algorithm with running time of O^e(m√n). Here we provide the first deterministic algorithm for this problem, with the same O^e(m√n) time. Due to matching conditional lower bounds of Williams et al. [FOCS 2010], our deterministic combinatorial algorithm for the replacement paths problem is optimal up to polylogarithmic factors (unless the long standing bound of O^e(mn) for the combinatorial boolean matrix multiplication can be improved). This also implies a deterministic algorithm for the second simple shortest path problem in O^e(m√n) time, and a deterministic algorithm for the k-simple shortest paths problem in O^e(km√n) time (for any integer constant k > 0). For the problem of distance sensitivity oracles, let G = (V, E) be a directed graph with real-edge weights. An f-Sensitivity Distance Oracle (f-DSO) gets as input the graph G = (V, E) and a parameter f, preprocesses it into a data-structure, such that given a query (s, t, F) with s, t ∈ V and F ⊆ E ∪ V, |F| ≤ f being a set of at most f edges or vertices (failures), the query algorithm efficiently computes the distance from s to t in the graph G \ F (i.e., the distance from s to t in the graph G after removing from it the failing edges and vertices F). For weighted graphs with real edge weights, Weimann and Yuster [FOCS 2010] presented several randomized f-DSOs. In particular, they presented a combinatorial f-DSO with O^e(mn⁴−^α) preprocessing time and subquadratic O^e(n²−2(1−α)^/f) query time, giving a tradeoff between preprocessing and query time for every value of 0 < α < 1. We derandomize this result and present a combinatorial deterministic f-DSO with the same asymptotic preprocessing and query time.