Oracles for Distances Avoiding a Node or Link Failure∗

We consider the problem of preprocessing an edge-weighted directed graph G to answer queries that ask for the shortest distance from any given node x to any other node y avoiding an arbitrary failed node or link. We describe an oracle (i.e, a simple data structure) for such queries that can be stored in O(n log n) space, and which allows queries to be answered in O(1) time, where n is the number of nodes in G. We also show that if we are willing to use Θ(n) space, we can reduce the preprocessing time by a factor of √ n while maintaining the constant query time. We can also keep track of the shortest path avoiding any failed node or link by maintaining for each node the outgoing edge that should be used to get on such a path. AMS Subject Classifications. [05C85, 68W01] Graph algorithms; [68P05] Data structures; [05C38] Shortest paths; [90B18] Network failures.