f-Sensitivity Distance Oracles and Routing Schemes

An f-sensitivity distance oracle for a weighted undirected graph G(V, E) is a data structure capable of answering restricted distance queries between vertex pairs, i.e., calculating distances on a subgraph avoiding some forbidden edges. This paper presents an efficiently constructible f-sensitivity distance oracle that given a triplet (s, t, F),where s and t are vertices and F is a set of forbidden edges such that |F| ≤ f, returns an estimate of the distance between s and t in G(V, E\ F). For an integer parameter k ≥ 1, the size of the data structure is O(fkn1+1/k log (nW)), where W is the heaviest edge in G, the stretch (approximation ratio) of the returned distance is (8k-2)(f +1), and the query time is O(|F|ċlog2nċlog log nċlog log d), where d is the distance between s and t in G(V, E\F). The paper also considers f-sensitive compact routing schemes, namely,routing schemes that avoid a given set of forbidden (or failed) edges. It presents a scheme capable of withstanding up to two edge failures. Given a message M destined to t at a source vertex s, in the presence of a forbidden edge set F of size |F| ≤ 2 (unknown to s), our scheme routes M from s to t in a distributed manner, over a path of length at most O(k) times the length of the optimal path (avoiding F). The total amount of information stored in vertices of G is O(kn1+1/k log (nW) log n).