# 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(m √(n)). Here we provide the first deterministic algorithm for this problem, with the same O(m √(n)) time. 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 a combinatorial randomized f-DSO with O(mn^4-α) preprocessing time and subquadratic O(n^2-2(1-α)/f) 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.

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