Efficient Oracles and Routing Schemes for Replacement Paths

Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in a graph G, the replacement path πG−e(s, t) is a shortest s−t path that avoids e. In this paper we focus on the general case in which (s, t) ∈ S × T , where S, T ⊆ V (G), and we present several efficient constructions maintaining the collection of all πG−e(s, t) for every e ∈ E(G), either implicitely (i.e., through compact data structures a.k.a. distance sensitivity oracles (DSO)), or explicitely (i.e., through sparse subgraphs a.k.a. fault-tolerant preservers (FTP)). More precisely, we provide the following results: (1) DSO: For every S, T ⊆ V , we construct a DSO for maintaining S × T distances under single edge (or vertex) faults. This DSO has size Õ(n √ |S||T |) and query time of O( √ |S||T |). At the expense of having quasi-polynomial query time, the size of the oracle can can be improved to Õ(|T | √ |S|n), which is optimal for |T | = Ω( √ n|S|). When |T | = Ω(n 4 |S| 1 4 ), the construction can be further refined in order to get a polynomial query time. We also consider the approximate additive setting, and show a family of DSOs, that exhibits a tradeoff between the additive stretch and the size of the oracle. Finally, for the meaningful single-source case, the above result is complemented by a lower bound conditioned on the Set-Intersection conjecture. These lower bound results establish a separation between the oracle and the subgraph settings. (2) FTP: We show the construction of a path-reporting DSO of size Õ(n4/3(|S||T |)1/3) reporting πG−e(s, t) in O(|πG−e(s, t)|+ (n|S||T |)1/3) time. Such a DSO can be transformed into a FTP having the same size, and moreover it can be elaborated in order to make it optimal both in space and query time for the special case in which T = V (G). Our FPT improves over previous constructions when |T | = O( √ |S|n). (3) Routing and Labeling Schemes: For the well-studied single source setting, we present a novel routing scheme, that allows to route messages on πG−e(s, t) by using edge labels and routing tables of size Õ( √ n), and a header message of poly-logarithmic size. We also provide optimal labeling scheme for the setting. 1998 ACM Subject Classification G.2.2 Graph Algorithms