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

Let $G=(V,E)$ be any undirected graph on $V$ vertices and $E$ edges. A path $textbf{P}$ between any two vertices $u,vin V$ 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 consider the problem of building a compact data structure for a given graph $G$ which is capable of answering the following query for any $u,v,zin V$ and $t>1$. centerline{em report $t$-approximate shortest path between $u$ and $v$ when vertex $z$ fails} We present data structures for the single source as well all-pairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size {em nearly} match the size of their best static counterparts