Optimal diameter computation within bounded clique-width graphs
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Coudert et al. (SODA'18) proved that under the Strong Exponential-Time Hypothesis, for any ϵ >0, there is no O(2^o(k)n^2-ϵ)-time algorithm for computing the diameter within the n-vertex cubic graphs of clique-width at most k. We present an algorithm which given an n-vertex m-edge graph G and a k-expression, computes all the eccentricities in O(2^ O(k)(n+m)^1+o(1)) time, thus matching their conditional lower bound. It can be modified in order to compute the Wiener index and the median set of G within the same amount of time. On our way, we get a distance-labeling scheme for n-vertex m-edge graphs of clique-width at most k, using O(klog^2n) bits per vertex and constructible in O(k(n+m)logn) time from a given k-expression. Doing so, we match the label size obtained by Courcelle and Vanicat (DAM 2016), while we considerably improve the dependency on k in their scheme. As a corollary, we get an O(kn^2logn)-time algorithm for computing All-Pairs Shortest-Paths on n-vertex graphs of clique-width at most k. This partially answers an open question of Kratsch and Nelles (STACS'20).
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