Vassilevska W. [STOC 13] show that inÕ (m √ n) time, one can compute for each v ∈ V in an undirected graph, an estimate e (v) for the eccentricity (v) such that max {R, 2 /3 • (v)} ≤ e (v) ≤ min {D, 3 /2 • (v)} where R = minv (v) is the radius of the graph. Here we improve the approximation guarantee by showing that a variant of the same algorithm can achieve estimates (v) with 3 /5 • (v) ≤ (v) ≤ (v).

We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worstcase guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates).Our main result is a simple randomized algorithm that for any parameter c > 1 has a worst-case update time of O(cn 2+ 2 /3 log 4 /3 n) and answers distance queries correctly with probability 1 − 1/n c , against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time ofÕ(n 2+ 3 /4 ) and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds.

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