### f-Sensitivity Distance Oracles and Routing Schemes [chapter]

Shiri Chechik, Michael Langberg, David Peleg, Liam Roditty
2010 Lecture Notes in Computer Science
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
more » ... . For an integer parameter k ≥ 1, the size of the data structure is O(f kn 1+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 | · log 2 n · log log n · log log d), where d is the distance between s and t in G(V, E \ F ). Our result differs from previous ones in two major respects: (1) it is the first to consider approximate oracles for general graphs (and thus obtain a succinct data structure); (2) our result holds for an arbitrary number of forbidden edges. In contrast, previous papers concern f -sensitive exact distance oracles, which consequently have size Ω(n 2 ). Moreover, those oracles support forbidden sets F of size |F | ≤ 2. 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(kn 1+1/k log (nW ) log n). To the best of our knowledge, this is the first result obtaining an f -sensitive compact routing scheme for general graphs. *