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Abstract

Let $G=(V,E)$ be an unweighted undirected graph on $n$ vertices. Let $\delta(u,v)$ denote the distance between vertices $u,v\inV$. An algorithm is said to compute all-pairs $t$-approximate shortest -paths/distances, for some $t\ge 1$, if for each pair of vertices $u,v\in V$, the path/distance reported by the algorithm is not longer/greater than $t\delta(u,v)$.\\ This paper presents two randomized algorithms for computing all-pairs nearly 2-approximate shortest distances. The first algorithm takes expected $O(m^{2/3}n\log n + n^2)$ time, and for any $u,v\in V$ reports distance no greater than $2\delta(u,v)+1$. Our second algorithm requires expected $O(n^2\log^{3/2} n)$ time, and for any $u,v\in V$, reports distance bounded by $2\delta(u,v) + 3$.\\ This paper also presents the first expected $O(n^2)$ time algorithm to compute all-pairs 3-approximate distances.