Approximate Shortest Paths Avoiding a Failed Vertex: Near Optimal Data Structures for Undirected Unweighted Graphs

AbstractLet G=(V,E) be an undirected unweighted graph. A path between any two vertices u,v∈V is said to be t-approximate shortest path if its length is at most t times the length of the shortest path between u and v. We address the problem of building a compact data structure which can efficiently answer the following query for any u,v,x∈V and t>1: Reportt-approximate shortest path betweenuandvwhen vertexxfails. We present data structures for the single source as well as all-pairs versions of this problem. The query time guaranteed by our data structures is optimal up to a constant factor. Moreover, the size of each of them nearly matches the size of the corresponding data structure with no failures.

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