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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The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected n-vertex graph G = (V, E) and an integer k ≥ 1, the subgraph u, v) denotes the distance between u and v in G . Graph spanners were extensively studied since their introduction over two decades ago. It is known how to efficiently construct a (2k −1)-spanner of size O(n 1+1/k ), and this sizestretch tradeoff is conjectured to be tight.

A spanner H of a weighted undirected graph G is a "sparse" subgraph that approximately preserves

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. (Our algorithm can also be modified to work in the incremental setting, where the graph is initially empty and subject to a sequence of online adversarial edge insertions.)In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Later papers presented conditional lower bounds showing that O(mn) is optimal up to log factors.In SODA 2011, Bernstein and Roditty showed how to bypass these lower bounds and improve upon the Even and Shiloach O(mn) total update time bound by allowing a (1 + ) approximation. This triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a (1 + )-approximate algorithm whose total update time is near linear: O(m 1+O(1/ √ log n) ). However, every single one of these improvements over the Even-Shiloach algorithm was randomized and assumed a non-adaptive adversary. This additional assumption meant that the algorithms were not suitable for certain settings and could not be used as a black box data structure. Very recently Bernstein and Chechik presented in STOC 2016 the first deterministic improvement over Even and Shiloach, that did not rely on randomization or assumptions about the adversary: in an undirected unweighted graph the algorithm maintains (1+ )-approximate distances and has total update timeÕ(n 2 ). In this paper, we present a new deterministic algorithm for the problem with total update timeÕ(n 1.25 √ m) = O(mn 3/4 ): it returns a (1 + ) approximation, and is limited to undirected unweighted graphs. Although this result is still far from matching the randomized near-linear total update time, it presents important progress towards that direction, because unlike the STOC 2016Õ(n 2 ) algorithm it beats the Even and Shiloach O(mn) bound for all graphs, not just sufficiently dense ones. In particular, theÕ(n 2 ) algorithm relied entirely on a new sparsification technique, and so could not hope to yield an improvement for sparse graphs. We present the first deterministic improvement for sparse graphs

International audienceThis paper considers fully dynamic (1 + ε) distance oracles and (1 + ε) forbidden-set labeling schemes for pla- nar graphs. For a given n-vertex planar graph G with edge weights drawn from [1,M] and parameter ε > 0, our forbidden-set labeling scheme uses labels of length λ = O(ε−1 log2 n log (nM ) * (ε−1 + log n)). Given the labels of two vertices s and t and of a set F of faulty vertices/edges, our scheme approximates the distance between s and t in G \ F with stretch (1 + ε), in O(|F|2λ) time. We then present a general method to transform (1 + ε) forbidden-set labeling schemas into a fully dynamic (1 + ε) distance oracle. Our fully dynamic (1 + ε) distance oracle is of size O(n log n * (ε−1 + log n)) and has O ̃(n1/2) query and update time, both the query and the update time are worst case. This improves on the best previously known (1 + ε) dynamic distance oracle for planar graphs, which has worst case query time O ̃(n2/3) and amortized update time of O ̃(n2/3). Our (1 + ε) forbidden-set labeling scheme can also be extended into a forbidden-set labeled routing scheme with stretch (1 + ε)