# 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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research
06/29/2021

### Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles

Given a graph with a source vertex s, the Single Source Replacement Path...
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### Deterministic Replacement Path Covering

In this article, we provide a unified and simplified approach to derando...
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### Fast Distance Sensitivity Oracle for Multiple Failures

When a network is prone to failures, it is very expensive to compute the...
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### Deep Distance Sensitivity Oracles

One of the most fundamental graph problems is finding a shortest path fr...
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05/19/2020

### Multiple Source Replacement Path Problem

One of the classical line of work in graph algorithms has been the Repla...
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09/15/2022

### Algorithms and Lower Bounds for Replacement Paths under Multiple Edge Failures

This paper considers a natural fault-tolerant shortest paths problem: fo...
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12/06/2021

### Fixed-Parameter Sensitivity Oracles

We combine ideas from distance sensitivity oracles (DSOs) and fixed-para...